Topics in algebraic geometry and geometric modelingfolk.uio.no/hermunn/PhD.pdf · The vision of the...

Topics in algebraic geometry

and geometric modeling

Pal Hermunn Johansen

Contents

1 Introduction 1

2 The tangent developable 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Tangent developables . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Local properties of a real tangent developable . . . . . . . . . . . 72.4 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 The tangent developable of a complex algebraic curve . . . . . . 12

3 Closest points, moving surfaces and algebraic geometry 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 The underlying idea . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Degrees of the moving surfaces . . . . . . . . . . . . . . . . . . . 243.4 Implementation of a test algorithm . . . . . . . . . . . . . . . . . 273.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Solving a closest point problem by subdivision 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 The basic method . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Quality of the output and special cases . . . . . . . . . . 35

4.3 Improving the basic method . . . . . . . . . . . . . . . . . . . . . 364.3.1 Changing the subdivision . . . . . . . . . . . . . . . . . . 374.3.2 Changing the multiplication algorithm to allow an early

exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3.3 Introducing a box test and a plane test . . . . . . . . . . 384.3.4 Using the second order derivatives . . . . . . . . . . . . . 38

iii

iv CONTENTS

4.3.5 The recursive algorithm explained . . . . . . . . . . . . . 394.3.6 The basic method with the box and plane tests . . . . . . 404.3.7 Doing a preconditioned constant sign test . . . . . . . . . 404.3.8 Speed measurements . . . . . . . . . . . . . . . . . . . . . 41

4.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Monoid hypersurfaces 455.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 Monoid surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4 Quartic monoid surfaces . . . . . . . . . . . . . . . . . . . . . . . 56

6 The strata of quartic monoids 716.1 Definition of the strata . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Types 1 to 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.3 Type 9 - the tangent cone is smooth . . . . . . . . . . . . . . . . 95

Chapter 1

Introduction

This PhD thesis was started as a part of the European Community fundedproject “Intersection algorithms for geometry based IT-applications using ap-proximate algebraic methods”. The project was coordinated by Tor Dokken ofSINTEF in Oslo and included partners from many countries: The University ofCantabria from Spain, INRIA and the University of Nice from France, think3from Italy and France, the University of Linz from Austria, and SINTEF andthe University of Oslo from Norway.

The vision of the project was to bring algebraic geometry and approximationtheory together and apply it to problems in Computer Aided Geometric Design(CAGD). The imperfect quality of intersection algorithms in CAGD systemsimposes high costs on the product creation process in industry, and it wasdeemed necessary to find new and better methods for solving these problems.A better understanding of the geometrical objects in CAGD is needed to fulfillthis goal.

The vision and goal of the project has certainly influenced my work andfocus as a PhD student. As a direct result of that, the topics in this thesis coverdifferent parts of the project plan. Chapters 2, 5 and 6 provide insight intoobjects that are interesting in CAGD and geometric modelling.

Chapters 3 and 4, on the other hand, investigate an important problem inCAGD and other industrial applications, the closest point problem. The inves-tigation covers both theoretical aspects and pure optimalization. The closestpoint problem is so common that having fast algorithms is of great importance.The topics of the chapters may seem unrelated, but they all address centralproblems in applied geometry.

1

2 CHAPTER 1. INTRODUCTION

Chapter 2 is a study of tangent developables and was published [15] in theproceedings of the first COMPASS workshop. Developable surfaces are commonin CAGD. A good understanding of these objects is helpful for programmers thatwish to use them in CAGD application programs.

The tangent developable of a curve C ⊂ P3 is a singular surface with cuspidaledges along C and the flex tangents of C. It also contains a multiple curve,typically double. We express the degree of this curve in terms of the invariantsof C. In many cases we can describe the intersections of C with the multiplecurve, and pictures of these cases are provided.

Chapter 3 describes a method of computing closest points to a parametricsurface patch. The chapter is the result of collaboration with Jan B. Thomassenand Tor Dokken and was published [41] in the Proceedings after the conference“Spline curves and surfaces” in Tromsø. The article was a collaboration betweenthree people and my responsibility was mainly the degree formulas and text inSection 3.3.

The method for computing closest points to a given parametric surface patchis based on “moving surfaces”. For each parametric surface there are two naturalmoving surfaces, one for each parameter direction. These two objects let usreduce the closest point problem for a given point to solving two univariatepolynomial equations. We also describe an implementation of our algorithmwhich – although not being fast – is very reliable.

Chapter 4 is a written and improved version of a talk given at the MEGA2005 conference. This chapter deals with the same problem as the previouschapter, but solves the problem by subdivision techniques. Different ways ofsolving this problem through subdivision are explored, and different optimiza-tions are timed. An error analysis for subdivision methods is carried out, andthis gives the user full control over the guaranteed accuracy of the subdivisionmethods.

Chapter 5 is an article written with my advisor Ragni Piene and one of herother students, Magnus Løberg. This article has been accepted for the proceed-ings of the COMPASS 2 workshop, and is a study of monoid hypersurfaces.

A monoid hypersurface is an irreducible hypersurface of degree d which has asingular point of multiplicity d− 1. Any monoid hypersurface admits a rationalparameterization, and is hence of potential interest in computer aided geometricdesign. We study properties of monoids in general and of monoid surfaces inparticular. The main results include a description of the possible real forms ofthe singularities on a monoid surface other than the point of multiplicity d− 1.These results are applied to the classification of singularities on quartic monoidsurfaces, complementing earlier work on the subject.

3

My contribution has been formulating and proving the lemmas and propo-sitions in this chapter, building on the work started by the other authors. Inparticular, Proposition 5.8 and its constructive proof has been my contribution,and also extending work in [38] into a complete classification of singularitiesaway from the triple point.

In Chapter 6 the classification of monoids is continued by considering thespace of quartic monoids in P3 with only isolated singularities. This space has anatural stratification based on geometric invariants related to the singularites ofthe monoids. The strata of monoids are defined by first defining the invariants,and then defining when two different monoids are considered to have the sameset of invariants.

The result is a very high number of strata of monoids. By using the classi-fication in the previous chapter, we are able to calculate the dimension of eachstratum. Also, if a stratum is associated to a singular tangent cone, then thestratum can be expressed as an image of a certain map, and this constructionlet us recover the number of components of the stratum.

During the work on the thesis over the last four years I have met manyinteresting people and made several new friends. Many of these have inspiredand helped me complete my work, and I am happy to mention some of themhere.

First of all I would like to thank my advisor, Ragni Piene, for always beingpositive and supportive in my efforts. I will also thank her for answering lots ofquestions, for asking me the right questions, and for providing small hints whenmy research has been incomplete or temporarily stuck.

I would also like to thank my cand.scient. advisor Jan Christophersen for hiseffort in turning me into a worthy PhD candidate. Many thanks to Tor Dokkenfor leading the successful GAIA II project and providing insight into the worldof CAGD.

I would like to thank Mohammed Elkadi, Bernard Mourrain and AndreGalligo for help and advice during my stay in Nice.

Finally, I would like to thank the many fellow students with whom I sharedan office, Torquil Macdonald Sørensen, Tore Halsne Flatten, Le Thi Ha, AnTa Thi Kieu, Guillaume Cheze, Ola Nilsson, my good friends George HarryHitching and Oliver Labs, and, most of all, my girlfriend Maria Samuelsen. Youhave all made the work on this thesis a better experience.

4 CHAPTER 1. INTRODUCTION

Chapter 2

The tangent developable

2.1 Introduction

If we have a curve on which tangents can be defined, then the associated tangentdevelopable is the surface swiped out by the tangents. Tangent developableshave a cuspidal edge, and are easy to generate. Since most developable surfacesare tangent developables, the Computer Aided Geometric Design communityshould be interested in their properties. This article describes the local andglobal geometry of tangent developables.

For the local study of tangent developables we consider analytic real curves.Cleave showed in [5] that for most curves the tangent developable has a cuspidaledge along most of the curve. This was extended by Mond in [23] and [24]where he analyzed the tangent developable of more special curves. This workwas further extended by Ishikawa in [13], and results from that article are usedin section 2.3.

The following section contains figures illustrating the local behavior of tant-gent developables, and one may want to have a brief look at these before readingthe rest of the text.

In section 2.5 the tangent developables of complex projective algebraic curvesare described. Algebraic geometrical invariants are introduced and relationsbetween these invariants are taken from [31]. We also show that tangent devel-opables of rational curves of degree ≥ 4 have a double curve.

Many thanks goes to Ragni Piene for lots of good advice and considerablehelp with this article.

5

6 CHAPTER 2. THE TANGENT DEVELOPABLE

2.2 Tangent developables

Given a curve in some space, its tangent developable is the union of the tangentlines to the curve. The tangent line at a singular point is defined as the limit oftangent lines at non-singular points. If the curve is algebraic, then its tangentdevelopable will be an algebraic surface.

Assume we have a parameterization of a curve with a non-vanishing deriva-tive. Then we can make a map that parameterizes the corresponding tangentdevelopable. Let U ⊂ R, γ : U → R3 be a map with a non-vanishing derivative.Define the map Γ : U × R → R3 by

Γ(t, u) = γ(t) + uγ′(t) (2.1)

In this case the tangent developable of γ(U) is the image of Γ. The follow-ing example uses this technique to calculate the implicit equation of a tangentdevelopable.

Example 2.1 (The tangent developable of the twisted cubic). Consider thetwisted cubic curve parameterized by γ : R → R3 where γ(t) = (t, t2, t3).The tangent developable is then the image of Γ : R2 → R3 where Γ(t, u) =(t+ u, t2 + 2ut, t3 + 3ut2). The algebra program Singular [10] can calculate theimplicit equation of the surface:

z2 − 6xyz + 4x3z + 4y3 − 3x2y2 = 0.

In this case the implicit equation describe the same set of points as thethe image of Γ. However, when dealing with real parameterizations this is notalways true.

Calculating the Jacobian ideal shows us that the tangent developable issingular exactly at γ(R). Moreover, if the surface is intersected with a generalplane, the resulting curve will have a cusp singularity at each intersection pointwith γ(R).

Definition 2.2 (The type of a germ). Let γ be a smooth (C∞) curve germ,γ : (R, p) → (R3, q). We say that the germ is of finite type if the vectors

γ′(p), γ′′(p), γ′′′(p), γ(4)(p), . . .

span R3. In this case, let ai = min{k | dim〈γ′(p), γ′′(p), . . . , γ(k)(p)〉 = i} anddefine the type of the germ to be the triple (a1, a2, a3).

2.3. LOCAL PROPERTIES OF A REAL TANGENT DEVELOPABLE 7

In this article we will only look at parameterizations where all the germs areof finite type.

What does a tangent developable look like? Along most of the curve, thetangent developable has a cuspidal edge singularity, so it is never smooth.

2.3 Local properties of a real tangent developable

We now want to study the local properties of the tangent developable close tothe curve. Now we are no longer forced to use complex numbers, so we chooseto study only real tangent developables. Since this is a local study, we now lookat germs of curves γ : (R, 0) → (R3, 0), as in definition 2.2.

Cleave shows in [5] that the tangent developable of most smooth curves γhave a cuspidal edge along most of the curve. That is, the cuspidal edge existsat intervals of points of type (1, 2, 3). We have already decided only to look atcurves where all the points are of finite type, and for all of these curves we willhave a cuspidal edge along most of the curve.

In the language of Cleave: Given a curve with nonzero curvature and torsionat a point γ(t0). If the tangent developable is intersected with a general planethrough γ(t0), the resulting curve will have a cusp at that point. In [24] Mondprovides drawings of the tangent developable at points of type (1, 2, k) for 3 ≤k ≤ 7. This is (in the language of differential geometry) when the torsionvanishes to order ≤ 4.

This was extended by Goo Ishikawa in [13], where he proves the following:The local diffeomorphism class of the tangent developable is determined by thetype of the point if and only if the type is one of the following: (1, 2, 2+r) wherer is a positive integer, (1, 3, 4), (1, 3, 5), (2, 3, 4) or (3, 4, 5).

In other words, for these types we can restrict our study to curves on theform

x = tl1+1 =: ta

y = tl2+2 =: tb

z = tl3+3 =: tc

at the origin. For other types we have to include more terms (of the powerseries) in the local parameterizations to study the point. In these cases we canget several different real pictures, but since points of other types are quite exotic,they will not be analyzed here.


Knowing this we can calculate local self intersection curves at points of type(1, 2, k) quite easily:

Example 2.3 ((a, b, c) = (1, 2, k) for k ≥ 3). To find local self intersectioncurves we need to solve the equation Γ(t, u) = Γ(s, v) where Γ is defined as inequation (2.1), Γ(t, u) = (t + u, t2 + 2tu, tk + ktk−1u). Some straightforwardcalculations leads us to solving

−(t2 − s2) + 2w(t− s) = 0(1− k)(tk − sk) + kw(tk−1 − sk−1) = 0,

where w = t+ u = s+ v.Assuming s 6= t we (eventually) get

0 = 2(1− k)(tk − sk) + k(t+ s)(tk−1 − sk−1)= (2− k)(tk−1 + sk−1) + 2(tk−2s+ tk−3s2 + . . .+ tsk−2).

It is not hard to prove that s = −t is the only possible real self intersection byanalyzing the polynomial f(t) = (2− k)(tk−1 + 1) + 2(tk−2 + tk−3 + . . .+ t) andits derivative. The real self intersection occurs exactly when k is even. This iscompatible with what Mond found in [23], but since Mond looked at C∞ curveshe could only draw the conclusion for k ≤ 7. Note that we have complex selfintersections for all k ≥ 5.

Example 2.4 (Types (1, 3, 4), (1, 3, 5), (2, 3, 4) and (3, 4, 5)). Points of types(1, 3, 4), (1, 3, 5) and (2, 3, 4) each have one local real self intersection curve,while points of type (3, 4, 5) have no real self intersection curves. This wascalculated using Singular [10].

The following section contains pictures of all of these types.

2.4 Illustrations

This section contains figures of tangent developables of different curves, eachparameterized by a map t→ (ta, tb, tc) for some triple (a, b, c). For each of thecurves, the origin is of type (a, b, c) and all other points (close to the origin)are of type (1, 2, 3). For all the figures, we have drawn the points that are at adistance of ≤ 2 from the origin, so the figures illustrate the local properties ofthe tangent developable.

The first five figures show points of type (1, 2, k). We can see that we haveself intersection curves exactly when k is even, as calculated in example 2.3.

In the first figure, all points are of type (1, 2, 3):

2.4. ILLUSTRATIONS 9

For most curves, the only types are (1, 2, 3) and (1, 2, 4). The following figureshows a point of type (1, 2, 4):

The following figures show points of type (1, 2, k).


The tangent developable of the curve (t, t2, t5)

The tangent developable of the curves (t, t2, t6) and (t, t2, t7)

The rest of the figures come from example 2.4. Note that for the pointswhere k1(0) = 1 (types (1, 3, 4) and (1, 3, 5)) the line which is a cuspidal edge,but not part of the curve, is an inflectional tangent line. This corresponds tothe Plucker formula mentioned in section 2.5, c = r0 + k1, where c is the degreeof the cuspidal edge.


2.4. ILLUSTRATIONS 11


The tangent developable of the curve (t2, t3, t4)

The tangent developable of the curve (t3, t4, t5)


2.5 The tangent developable of a complex alge-braic curve

To any projective algebraic curve, there are associated several invariants, mostimportantly the degree and genus of the curve. Classical algebraic geometrygives many relations between these values and the geometry of the curve. In[31] Piene obtained results for the tangent developable, and the formulas havebeen taken from that article.

In this section a curve will be a reduced algebraic curve C0 in the projectivecomplex 3-space, P3

C. We also assume that the curve spans the space. LetX ⊂ P3

C denote the tangent developable of C0.Let h : C → C0 be the normalization map, so that C is the desingularization

of C0. Let g denote the (geometric) genus of the curve and r0 the degree.The rank r1 is defined as the number of tangents that intersect a general line.Clearly this is the same as the degree of the tangent developable. The classr2 is defined as the number of osculating planes to C0 that contain a generalpoint. The osculating plane at a point on the curve is the plane with the highestorder of contact with the curve at that point. Another point of view is that theosculating plane at a point x0 is the limit of the planes containing x0, x1 andx2 as x1, x2 → x0.

For each point p ∈ C, we can choose affine coordinates around h(p) suchthat the branch of C0 determined by p has a (formal) parameterization at h(p)equal to

x = tl1+1 + . . .

y = tl2+2 + . . .

z = tl3+3 + . . .

with l0 := 0 ≤ l1 ≤ l2 ≤ l3. This (formal) parameterization is also a curve germγ : (C, 0) → (C3, 0). Because of this we extend the notion of the type to thecomplex domain, and say that the type of the germ determined by p is equal to(l1 + 1, l2 + 2, l3 + 3).

The coordinates are chosen such that p is the origin, the tangent is the liney = z = 0, and the osculating plane is z = 0. We call ki(p) = li+1 − li the ithstationary index of p. Since ki(p) 6= 0 only for a finite number of points p wecan define ki =

∑p∈C ki(p).

If l1 = 0, then the germ is nonsingular. If l1 ≥ 1 we say that the germ hasa cusp, and if l1 = 1 the cusp is said to be ordinary. If l1 = 0 and l2 ≥ 1 we

2.5. THE TANGENT DEVELOPABLE OF A COMPLEX CURVE 13

call the point h(p) an inflection point or flex, and if l2 = 1 the flex is ordinary.If l1 = l2 = 0 and l3 ≥ 1 we say that the curve has a stall or a point ofhyperosculation. For most curves we will have no cusps and no flexes.

Now it is time to state the relations between these values, all taken from[31]:

r1 = 2r0 + 2g − 2− k0 (2.2)r2 = 3(r0 + 2g − 2)− 2k0 − k1 (2.3)k2 = 4(r0 + 3g − 3)− 3k0 − 2k1 (2.4)

Note that r1 ≥ 3 since since r1 is the degree of the tangent developable, and noquadric surface with a cuspidal edge exists. Furthermore, r2 ≥ 3 since r2 is thedegree of the dual curve, and the dual curve must span the space. From thedefinition we get k2 ≥ 0.

The tangent developable X of C0 has degree µ0 = r1, rank µ1 = r2 (definedas the class of the intersection of the tangent developable with a general plane,a plane curve) and class µ2 = 0 (defined as the number of tangent planescontaining a general line). Its cuspidal edge consists of C0 and the flex tangentsof C0. The cuspidal edge has degree c = r0 + k1.

Formulas involving algebraic invariants, as those above, are often calledPlucker formulas, and such formulas is central in enumerative algebraic ge-ometry. There are lots of Plucker formulas, relating many different algebraicinvariants.

In addition to the cuspidal edge, X has a double (or higher order multiple)curve, some times called the nodal curve of C0. It consists of points that are onmore than one tangent of C0. Eventual bitangents are part of the nodal curve.

Let b denote the degree of the nodal curve. If the nodal curve is double andthe flexes of C0 are ordinary, then [31] gives the following expressions for b:

2b = µ0(µ0 − 1)− µ1 − 3c = r1(r1 − 1)− r2 − 3(r0 + k1)= r1(r1 − 4)− k0 − 2k1

= (2r0 + 2g − 2− k0)(2r0 + 2g − 6− k0)− k0 − 2k1.

For rational curves, g = 0, so then

2b = (2r0 − 2− k0)(2r0 − 6− k0)− k0 − 2k1.

In this case we see that

k2 = 4(r0 − 3)− 3k0 − 2k1 ≥ 0


impliesk0 = 4

3r0 − 4− 23k1 − 1

3k2 ≤ 43r0 − 4

We can find a lower bound for b for rational curves of degree r0 ≥ 4 by firsteliminating k1 (using equation (2.4)) in the expression for b:

2b = (2r0 − 2− k0)(2r0 − 6− k0)− k0 − 2k1

= (2r0 − 2− k0)(2r0 − 6− k0) + 2k0 + k2 − 4r0 + 12≥ (2r0 − 2− k0)(2r0 − 6− k0) + 2k0 − 4r0 + 12 (using k2 ≥ 0).

As a function in k0 the expression above is strictly decreasing (for k0 ≤ 43r0−4).

In other words, we can set k0 = 43r0 − 4 and not break the inequality:

2b ≥ (2r0 − 2− ( 43r0 − 4))(2r0 − 6− ( 4

3r0 − 4)) + 2( 43r0 − 4)− 4r0 + 12

= 49r0(r0 − 3).

We conclude that rational curves with b = 0 must have degree ≤ 3, and thetwisted cubic is the only one of these that is not planar. It follows that everyrational curve C0 of degree greater than 3 gives a tangent developable with anodal curve of positive degree.

We want to check if b = 1 is possible. If g = 0 and b = 1, then 2b ≥ 49r0(r0−3)

implies r0 = 4. Also, k0 ≤ 43r0 − 4 = 4

3 . This leads us to consider two cases,k0 = 0 and k0 = 1. If k0 = 0 the formula for b implies k1 = 5 and equation (2.4)gives k2 = −6. If k0 = 1 the formula for b implies k1 = 1 and equation (2.4)give k2 = −1. The second stationary index k2 cannot be negative, so b = 1 isimpossible.

The following example shows that b = 2 actually can occur for g = 0 andr0 ≥ 4:

Example 2.5 (A singular curve of degree 4). Let the curve γ0 : C → C3 begiven by γ0(t) = (t, t2, t3+t4). This is an imbedding that is one to one on points,so the degree is 4. Note that γ0 is nonsingular, but if we take the projectivecompletion γ : P1 → P3 given by

γ(s; t) = (s4; s3t; s2t2; st3 + t4)

we get a singular curve. In fact, setting t = 1 yields the local parameterizationat (0; 1), s→ (s4; s3; s2; s+ 1). Let (w;x; y; z) be the projective coordinates forP3

C. Since 1/(1+ s) = 1− s+ s2− s3 + . . . in a neighborhood of 0, setting z = 1


gives the local parameterization

w = s2 − s3 + s4 − s5 + . . .

x = s3 − s4 + s5 − s6 + . . .

y = s4 − s5 + s6 − s7 + . . . .

We see that the type of the local parameterization is (2, 3, 4), and thus k0(γ(0; 1)) =1 and k1(γ(0; 1)) = k2(γ(0; 1)) = 0. At any other point we see that the first andsecond derivative are linearly independent, so each of them are of type (1, 2, n)for some value of n. This means that we have k0 = 1 and k1 = 0. The degreeof the curve is r0 = 4, and the genus of the curve is g = 0 since the curve isrational. Now we can calculate the rest of the invariants mentioned above.

From the formulas we get the rank of the curve, r1 = 5, the class of thecurve, r2 = 4, the second stationary index, k2 = 1, the degree of the surfaceµ0 = r1 = 5, the rank of the surface µ1 = r2 = 4, and finally the degree of thenodal curve, b = 2.

Using Singular [10], we can verify some of the results. A Grobner basescomputation gives us the implicit equation of the surface:

F = 3wx2y2 + 12x3y2 − 4w2y3 − 14wxy3 + 8x2y3 − 9wy4 − 4wx3z

−16x4z + 6w2xyz + 24wx2yz − 6w2y2z − w3z2

This equation is, predictably, of degree µ0 = r1 = 5. We can find the singularlocus by setting the four partial derivatives equal to zero. The last one,

12· ∂F∂z

= −2wx3 − 8x4 + 3w2xy + 12wx2y − 3w2y2 − w3z,

leads us to consider two cases, w = 0 and w 6= 0.The first case implies x = 0 from ∂F/∂z = 0, and then ∂F/∂w = 0 gives

y = 0. This leaves us with one point, namely (0; 0; 0; 1) = γ(0; 1), the singularpoint of the curve.

If w 6= 0 we can choose w = 1 and solve the system of equation quite easily.This is because ∂F/∂z = 0 becomes

0 = −2x3 − 8x4 + 3xy + 12x2y − 3y2 − z, (2.5)

so we can substitute z into the other equations. In other words, assuming∂F/∂z = 0, the equation ∂F/∂y = 0 gives

0 = 16x6 + 8x5 − 32x4y + x4 − 16x3y + 16x2y2 − 2x2y + 8xy2 + y2

= (4x+ 1)2(x2 − y)2.


If x2 − y = 0, then equation (2.5) gives z = x3 + x4, as expected.Setting x = −1/4 in the rest of the equations gives us a solution for every

y, so z is a polynomial of degree 2 in y given by (2.5). This is the degree of thenodal curve that we calculated earlier.

Note that most curves will have k0 = k1 = 0, with a nodal curve of degreeb = 2(r0 + g− 1)(r0 + g− 3). Unless r0 = 3 and g = 0, the nodal curve will notbe empty.

The cuspidal edge and the nodal curve may both be singular, and they willusually intersect. If the nodal curve is double and the flexes are ordinary, Xwill have a finite number of points with multiplicity ≥ 3. These points can beof different types.

If the nodal curve has a node at q outside the cuspidal edge, then q mustlie on at least 3 tangents, and therefore the nodal curve must have at leastmultiplicity 3 at q since any selection of two out of three tangents will give abranch in the nodal curve.

The total number T of triple points of the tangent developable X of C0 isgiven in [31] and is

T = 16 (r1 − 4)((r1 − 3)(r1 − 2)− 6g). (2.6)

The formula (2.6) is valid when the nodal curve is double. When the nodalcurve is more than double we have to use a generalized formula for the degreeof the multiple curves (also found in [31]). If the nodal curve consists of curvesDj , where Dj is ordinary j-multiple, then the degrees bj of Dj satisfy∑

j

j(j − 1)bj = r1(r1 − 1)− r2 − 3(r0 + k1), (2.7)

still assuming the flexes to be ordinary. Note that this is a very special case,and that producing interesting examples with high j may be hard.

An example where the nodal curve is triple can be found in [40, p. 65], andwe have calculated, using Singular [10], the details1.

Example 2.6 (The equianharmonic rational quartic). Let α = 13

√−3, let C0

be the rational curve defined by the map γ : P1C → P3

C where

γ(s; t) = (αs4 − s2t2;αs3t;αst3;αt4 − s2t2),

1There is an error in [40], m is not supposed to be√−3, but the same as α in the example,

m = 13

√−3.


and let X be its tangent developable. A Grobner bases computation gives usthe implicit equation F = 0 of the surface X. Here F is a polynomial of degree6 in the projective coordinates (w;x; y; z):

F = 12w2x3y + 3w4y2 − 72αw2x2y2 + 12w2xy3 − 256αx3y3

+18αw3xyz + 24wx3yz + 6w3y2z + 48αwx2y2z + 24wxy3z

+3w2x2z2 − 12αw2xyz2 + 12x3yz2 + 3w2y2z2 − 72αx2y2z2

+12xy3z2 + 4αw3z3 + 6wx2z3 + 18αwxyz3 + 3x2z4

Taking a primary decomposition of the Jacobian ideal of F , we find that thesingular locus of X consists of two components, the curve C0 and the conic Ddefined by z2 + 4xy = 0 in the plane w + z = 0.

We want to show that D is a triple curve of X. The conic D can be param-eterized by θ : P1

C → P3C where

θ(u; v) = (−2uv;−v2;u2; 2uv).

Using this parameterization we find the following: The point θ(u; v) lies on thetangent to C0 at γ(s; t) if and only if

G(s, t, u, v) := s3u− 3αst2u+ 3αs2tv − t3v = 0.

For a fixed (u; v) ∈ P1C, zeros of G(s, t, u, v) corresponds to points on C0 whose

tangent contain θ(u; v). For most (u; v) ∈ P1C we will get three distinct tangents.

In fact, let ∆(u, v) denote the discriminant of G with respect to (s; t). In thiscase

∆(u, v) = (u2 + (3α+ 1)uv − v2)(u2 − (3α+ 1)uv − v2).

If ∆(u, v) 6= 0, then the point θ(u; v) lies on three distinct tangents to C0.Let A denote the four points on D corresponding to ∆(u, v) = 0. We con-

clude that each point on D not in A lies on exactly three tangents of C0. Thismeans that D is a triple curve of X.

Moreover, A is exactly the intersection of D and C0, and these four pointsare the only points on C0 whose local parameterization is not of type (1, 2, 3).In fact, the local parameterizations in each point of A is of type (1, 2, 4). Thismeans that k0 = k1 = 0 and k2 = 4. Furthermore, the degree of C0 is r0 = 4 andthe formulas give the rank r1 = 6 and the class r2 = 6 of the curve. The multiplecurve only have one component, the triple curve D, and this corresponds tob3 = 2 in equation (2.7).


The set A form an equianharmonic set on D, and that is why C0 is calledthe equianharmonic rational quartic. Note that this example is very specialand arise from the thorough study [40] of the rational normal curve in P4

C. Thecurve C0 is constructed by projecting the rational normal curve in P4

C from ageneral point on a quadric called the nucleus of the polarity. The equation ofthe nucleus of the polarity is x0x4 − 4x1x3 + 3x2

2 and the projection centre ofthis example is (1, 0, α, 0, 1).

All the formulas in this section holds for curves in P3C. We can not make

similar equalities for real curves, but the projective invariants of the complexcurve give results for the real part in the form of inequalities. However, theseinequalities will not be made explicit in this article.

Chapter 3

Closest points, MovingSurfaces and AlgebraicGeometry1

Jan B. Thomassen, Pal H. Johansen, and Tor Dokken

3.1 Introduction

In this paper, we present a new method for calculating closest points to a para-metric surface. The method is based on algebraic techniques, in particular onmoving surfaces. Moving surfaces are objects that have previously been usedfor implicitization [34], but the closest point problem now provides another ap-plication of these.

Recently, there has been renewed interest in exploring links between geomet-ric modeling and algebraic geometry [9]. The work presented in this paper isa part of this trend, and extends work from the European Commission projectGAIA II. Algebraic geometry has many uses in geometric modeling, includingsuch applications as point classification, implicitization, intersection and self-intersection problems, ray-tracing, etc. It was therefore natural to ask whether

1This chapter is the article [15]. My contribution is mainly the theory in Section 3.3.

19

20 CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES

algebraic geometry also can be used in algorithms for computing closest points.The closest point problem is a generic problem in CAGD. Applications in-

clude surface smoothing, surface fitting, and curve or surface selection. Theclosest point problem can be described in the following way. We are given aparametric surface p(u, v) and a point x0 in space. We want to find the pointpcl on the surface that is closest to x0, or more precisely, we want to find theparameters (ucl, vcl) of pcl.

The conventional way to compute closest points involves iterative methods,like Newton’s method, to minimize the distance function from x0 to a point onthe surface. This leads to solving a set of two polynomial equations in u and v,

(x0 − p(u, v)) · pu(u, v) = 0, (3.1)(x0 − p(u, v)) · pv(u, v) = 0,

for the footpoints to x0. We recall that a footpoint p to x0 is a point on thesurface such that the vector (x0 − p) is orthogonal to the tangent plane at p.Eqs. (3.1) express an orthogonality condition: The vector (x0−pcl) is orthogonalto the tangent vectors pu(ucl, vcl) and pv(ucl, vcl) at the closest point. This isillustrated in Fig. 3.1.

pclup (u,v)

p (u,v )v

x0

Figure 3.1: Orthogonality conditions for the closest point.

One disadvantage of iterative methods is that we need an initial guess. Itis a problem to come up with a good initial guess [20]. A bad guess may givea sequence of iterations that does not converge, or that converges to the wrongsolution. Furthermore, if a large number of closest points needs to be computed,the method may be slow.

3.2. THE UNDERLYING IDEA 21

Another way to solve Eqs. (3.1) is to use subdivision techniques. An exampleof this is Bezier clipping [27]. These methods are often robust and effective, butmay be unstable and use a long time to converge for some difficult surfaces, likesurfaces with singularities. Such methods are probably the methods of choicein real applications, but we will not discuss them further here.

The method we propose in this paper for solving the closest point problemuses moving surfaces, as already mentioned. A moving surface in our setting is aone-parameter family of surfaces. We construct two such surfaces: one movingin the u-direction and one moving in the v-direction. The two moving surfacesgive us two polynomial equations that are univariate. Univariate polynomialequations can be solved fast with a recursive solver and all roots may be found onthe interval of interest within a predefined accuracy. This will give an algorithmthat does not need any initial guess, has no convergence problems, and is fastwhen many closest points are to be calculated.

We may use elimination theory and Sylvester resultants to construct themoving surfaces in our method. From this construction, we obtain formulasfor the algebraic degrees of the geometric objects involved when the surfacesaddressed are Bezier surfaces.

We also describe an implementation of an algorithm for computing closestpoints based on the moving surface method. In this implementation, we con-struct the moving surfaces by solving a system of linear equations, rather thanby using resultants. The implementation produced accurate results when runon test cases of biquadratic Bezier surfaces. Unfortunately, it couldn’t be ap-plied to bicubic surfaces due to memory shortage when building certain matricesnecessary for the construction of the moving surfaces.

The organization of the paper is the following. In the following section, wedescribe the way we use moving surfaces and the idea behind our method. InSection 3.3, we analyze the method by using elimination theory and Sylvester’sresultant, which gives us formulas for the algebraic degrees of the moving sur-faces in the scheme. In Section 3.4, we present an algorithm for our method anddescribe some results we have obtained from implementing it. Finally, Section3.5 is a discussion of these results.

3.2 The underlying idea

Our method involves moving surfaces, which have been introduced by Sederbergfor implicitization [34]. In that context, a moving surface is an implicit surfacedepending on two parameters, but in our setting a moving surface is a one-


parameter family of implicit surfaces. Let us make the assumption that weare dealing with parametric surfaces that are single rational patches. Thus, amoving surface q(x;u), depending on the parameter u, is given by

q(x;u) =N∑i=0

qi(x)Bi,N (u). (3.2)

Here, Bi,N (u) are Bernstein basis polynomials of degree N , and qi(x) is a setof N + 1 algebraic functions. In other words, q is given in terms of a Bernsteinpolynomial in u of degree N , where the coefficients qi are implicit surfaces.Furthermore, the moving surface q follows a surface p(u, v) (in the parameteru) if

q(p(u, v);u) = 0. (3.3)

Moving surfaces may in this way follow a parametric surface in either the u orthe v direction.

How can we make use of such moving surfaces? Suppose we find a movingsurface q1(x;u) with the following properties:

• q1 follows the given surface p(u, v) in u. This means that the surfacedefined by q1(x;u) = 0 intersects p in u-isoparameter curves.

• q1 is orthogonal to p for each u.

• q1 is ruled for each u. More precisely, it is swept out by lines spanned bythe normal n(u, v) along the u-isoparameter curves.

Then, for a given point x0 in space, the equation

q1(x0;u) = 0 (3.4)

is a univariate equation for the u-parameter of all footpoints to x0. An exampleof a moving surface with these properties is shown in Figure 3.2. Clearly, wemay have a similar moving surface q2 in the v direction. In the following, thesubscript 1 or 2 on q refers to either u or v.

A possible exception to this situation is that we are dealing with certain non-generic surfaces, like surfaces of revolution. For these surfaces some points (likethose lying on the axis of revolution) may give, not footpoints, but footcurves.I.e. the set of points with the same distance to x0 is a curve on the surface.This is presumably a problem for most methods of computing closest points,

3.2. THE UNDERLYING IDEA 23

p(u,v)

n(u,v)q1(x0;u)=0

u

v

Figure 3.2: A moving surface q1 that intersects p at u-isocurves, is orthogonalto it, and is ruled.

and requires a separate discussion. For simplicity we assume that all the surfaceswe consider are sufficiently generic for this to happen.

Based on the considerations above, we propose a method for computingclosest points in two steps:

1. Preprocessing. Construct two moving surfaces: q1 for the u-direction, andq2 for the v-direction. This is done once for each surface.

2. For each given point x0, use the two moving surfaces to get two univariateequations in u and v:

q1(x0;u) = 0, (3.5)q2(x0; v) = 0.

Check each pair of solutions (u, v) to these equations, along with theclosest point on the border, to find which one corresponds to the closestpoint.

Finding the closest point on the border amounts to running a similar algorithmfor the four border curves.


A sketch of a situation where we get a solution u0 and v0 from Step 2 isshown in Figure 3.3.

pcl

x0

( ;uq1 x0 0)=0

pv(u0;v0)

pu(u0;v0)

( ;vq2 x0 0)=0

Figure 3.3: When the solutions u0 and v0 are found in Step 2, we can draw themoving surfaces for these two parameter values. The point x0, the closest pointpcl, and the straight line between them, lie on both of these surfaces.

Let us also make a remark about curves. A similar construction works forcurves, both in 2D and 3D. In 2D we have moving lines, while in 3D we havemoving planes. Since lines and planes are described implicitly by algebraicfunctions that are linear, the algorithms become simpler. Furthermore, forcurves there is only one equation in Step 2. This equation is in fact equivalentto the orthogonality condition (x0 − p(t)) · p′(t) = 0.

3.3 Degrees of the moving surfaces

The two moving surfaces described in the previous section can be analyzed moreformally. In this section we will use elimination theory, in particular Sylvester’sresultant, to perform this analysis [6]. We will assume that the surface p is asingle polynomial patch, i.e. a Bezier patch. In this case we obtain formulasfor the algebraic degrees involved in q1 and q2 given a parametric surface ofbidegree (nu, nv). Referring back to the form (3.2) of a moving surface, the

3.3. DEGREES OF THE MOVING SURFACES 25

required degrees are:

d1 ≡ degx(q1) = the degree of q1 (or q1,i) in x

d2 ≡ degx(q2) = the degree of q2 (or q2,i) in x

N ≡ degu(q1) = the degree of q1 (or Bi,N ) in u

It turns out that degu(q1) is equal to degv(q2) so we need only one N . This isconnected with the fact that N counts the number of possible footpoints, andthis is given by the number of roots of q1 and q2, respectively.

Thus we have a parameterized surface p : R2 → R3, where p is given bythree polynomials p1, p2, p3 ∈ R[u, v] of degree (nu, nv). We assume that thisdescription of the surface is sufficiently general, so that the degrees cannot bereduced. Now let V be the set of points (u, v,x) ∈ R × R × R3 such that x ison the normal of p given by the parameter values (u, v).

The set V is described by the two equations

F1(u, v,x) := (x− p) · pu = 0, (3.6)F2(u, v,x) := (x− p) · pv = 0.

The points satisfying these equations make a variety in R5.Using a resultant, we can eliminate one variable, and get one polynomial

defining a hypersurface in R4. If we eliminate u, this set of points is exactlythe set V ′ = {(v,x) ∈ R×R3 | ∃u ∈ R s.t. F1(u, v,x) = F2(u, v,x) = 0}, whichcorresponds to the moving surface q2.

We want to determine the degrees in v and x of the equation defining V ′.First, we write

F1(u, v,x) =2nu−1∑i=0

fi(v,x)ui, (3.7)

F2(u, v,x) =2nu∑i=0

gi(v,x)ui,

and then use the Sylvester resultant to eliminate u. By examining the Sylvestermatrix, we can determine the degrees of the equation defining V ′. The Sylvester


matrix is a square matrix of size (4nu − 1)× (4nu − 1). It looks like this:

f0 0 · · · 0 g0 · · · 0f1 f0 · · · 0 g1 · · · 0...

.... . .

......

. . ....

f2nu−1 f2nu−2 · · · f0 g2nu−1 · · · g10 f2nu−1 · · · f1 g2nu · · · g2...

.... . .

......

. . ....

0 0 · · · f2nu−1 0 · · · g2nu

There are 2nu columns to the left, and the degree in v is 2nv for each of

these entries. There are 2nu−1 columns to the right, each entry being of degree2nv − 1. The total degree in v of the resultant is thus

N = 4nunv + (2nu − 1)(2nv − 1). (3.8)

Note the symmetry of this expression with respect to nu and nv, which confirmswhat we said previously about needing only one N .

The degree in x, that is, d2, is a little trickier to work out. The polynomialsf0, . . . , fnu−1 are of degree 1 in x, but the polynomials fnu , . . . , f2nu−1 are ofdegree 0. Furthermore, the polynomials g0, . . . , gnu are of degree 1 and thepolynomials gnu+1, . . . , g2nu are of degree 0 in x. This means that the bottomnu rows are of degree 0 in x and the rest of the 3nu − 1 rows are of degree 1.The total degrees are therefore

d1 = 3nv − 1, (3.9)d2 = 3nu − 1. (3.10)

As mentioned, the degree formulas for d1,2 and N are derived for generalparametrized surfaces, and as such are upper bounds. For some surfaces thedegrees could be effectively lower. This happens, for example, if the degree ofp is artificially high, so it can be obtained from a degree elevation of a lower-degree parametrization. The degree can drop in other cases, but if the degreein v drops, then the corresponding degree in u will typically drop in the sameway. For this reason, there will still be only one N .

A similar analysis can be carried out for rational surface patches. The degreeformulas are then:

N = 9nunv + (3nu − 2)(3nv − 2)d1 = 4nv − 2 (3.11)d2 = 4nu − 2

3.4. IMPLEMENTATION OF A TEST ALGORITHM 27

Bezier Rational(1, 1) (2, 2) (3, 3) (4, 4) (1, 1) (2, 2) (3, 3) (4, 4)

d1,2 2 5 8 11 2 6 10 14N 5 26 61 113 10 52 130 244

Table 3.1: Degrees for Bezier and rational surfaces of degrees of the form (n, n).Since nu = nv we also have d1 = d2.

Examples of the degrees for Bezier and rational surfaces of degrees (n, n)with n ranging from 1 to 4 is shown in Table 3.1. As far as we know, theseresults are new2. The numbers d1 and d2 are the degrees of an algebraic surfacethat is perpendicular to a parametric surface along an entire isocurve, and thishas not been noted before.

3.4 Implementation of a test algorithm

To test our ideas, we have implemented an algorithm for computing closestpoints for tensor product Bezier surfaces. We have chosen not to use resultantsfor this. Instead we rely on solving a system of linear equations, which will beexplained below. The reason is that this is a numerically very stable method,which allows us to use the Bernstein form for all polynomials in a straightforwardway, which would not have been the case for resultant based methods. Besides,we do not get into possible problems with base points.

A central object in our implementation is the “moving ruled surface”

r(u, v, w) = p(u, v) + wn(u, v), (3.12)

where p is the given surface, n is the normal vector, and w is an additionalparameter. This can be thought of as a trivariate tensor product Bezier object.For fixed (u0, v0), the line r(u0, v0, w), w ∈ R, is orthogonal to the surface atp(u0, v0). In other words, all points on this line has p(u0, v0) as a footpoint.

Another property we have used in our implementation, is that evaluat-ing an algebraic function q(x) on an n-variate Bernstein tensor polynomialr(u1, . . . , un) yields a new n-variate Bernstein tensor polynomial. If we writeq(x) =

∑j bjx

j , where j is a multi-index, xj is a monomial in (x, y, z) in multi-

2We thank the referee for urging us to make this point.


index form, and bj are the coefficients, we have a factorization

q(r(u1, . . . , un)) = bTDTB(u1, . . . , un). (3.13)

Here, b is the coefficients bj organized in a vector, D is a matrix of numbers, andB(u1, . . . , un) is a basis of n-variate Bernstein tensor polynomials, also organizedin a vector. If q is a degree d algebraic function and r is a degree (m1, . . . ,mn)Bernstein polynomial, then q(r) is a degree (dm1, . . . , dmn) Bernstein polyno-mial. In our implementation, we use evaluation routines for algebraic functionson Bernstein polynomials in order to find such matrix factorizations.

The moving surfaces q1 and q2 are defined by an array of coefficients. Forq1(x;u) we need to determine the coefficients b1,i;j of q1,i(x) =

∑j b1,i;jx

j , seeEq. (3.2). This means that we can use numerical linear algebra to find the vectorb1 of coefficients in q1. More precisely, we need to find a vector in the null-spaceof D1. We used a technique based on Gauss elimination and back-substitutionfor this, which is faster than, say, SVD of D1. This way of using numericallinear algebra has previously been used in implicitization, see [7].

The algorithm follows the two-step structure described in Section 3.2.

Step 1. Preprocessing

Input: A parametric surface p(u, v).

1. Construct a “moving ruled surface” r(u, v, w)

2. Insert r into q1 to get the equation q1(r(u, v, w), u)) = 0. This canbe factored into the linear equation

BT (u, v, w)D1b1 = 0, (3.14)

where b1 is the vector of coefficients in q1,i. Similarly for the v-direction.

3. Solve the matrix equation D1b1 = 0 by e.g. Gauss elimination andback-substitution. Similarly for the v-direction.

Output: The vectors b1 and b2, or equivalently, the moving surfaces q1and q2.

Step 2. For each given point x0

Input: A point x0 in space.

1. Find the closest point on the boundary curves.

3.4. IMPLEMENTATION OF A TEST ALGORITHM 29

Moving surfaces Newton’s methodAverage no. of hits 413 374

Running timesFull algo. ∼ 1− 2 minJust Step 2 ∼ 1 s ∼ 5 s

Accuracy ∼ 10−7 ∼ 10−13

Table 3.2: Average results for running the two closest point algorithms on tenrandom biquadratic surfaces. For details, see the text.

2. Insert x0 in q1 and q2 to get univariate equations in u and v:

q1(x0;u) = 0, (3.15)q2(x0; v) = 0.

3. Find all roots ui and vj .4. Check each pair (ui, vj) and the closest point on the boundary to find

the closest point.

Output: The parameters (ucl, vcl) of the closest point pcl.

As an example, we tested the algorithm on a set of ten random biquadraticBezier surfaces. That is, the control points were random points in the unit cube.We expect that within the family of biquadratic surfaces such surfaces will bea challenge for any closest point algorithm. For each surface, 1000 points weregenerated randomly in the bounding box, and their closest points on the surfacewere computed. However, points whose closest points were found to lie on theboundary were discarded. For comparison, we also implemented an algorithmbased on Newton’s method. We used a PC with two Intel Pentium 4 2.8GHzprocessors to run these algorithms. The results are shown in Table 3.2.

Table 3.2 reports the average number of closest points found (hits) for eachsurface. For both algorithms, less than half of the 1000 random points gave hitsbecause a majority of the closest points were on the boundaries of the surfaces.(The surfaces had complicated geometries with lots of self-intersections.) Themoving surfaces algorithm was consistently better for getting hits – Newton’smethod produced a lot of messages for “No convergence”.

Running times were considerably longer for the full moving surfaces algo-rithm, with 1− 2 minutes. Most of this time is spent in the preprocessing stepwhere the moving surfaces are constructed. When only Step 2 of the algorithmis considered it is much faster.


Finally, the accuracy given in Table 3.2 is an average of the errors for thereported closest points. The averaging is over the order of magnitude of theerrors, i.e. it is an average of the log of the errors for each point. An error fora single point was defined in terms of the angle θ between the vector (x0 − pcl)and the normal n(ucl, vcl) at the computed closest point, see Figure 3.4. As

n(u ,v )cl cl

pcl

x0

θ

Figure 3.4: The error can be measured by the angle θ between the normal n atpcl and the vector to the point x.

we can see, Newton’s method produced much better accuracy than the movingsurfaces. Sources of error for the moving surfaces method are the building ofthe matrix D, the Gauss elimination, the insertion of x0 to get the polynomialequations, and the solving of these equations. The results in Table 3.2, however,does not include iterative refinements.

Looking into the details for each surface – not shown in Table 3.2 – it turnsout that there is a complementary property for the two algorithms: Surfacesthat had a low accuracy for moving surfaces also had a high number of hits.For example, one random surface had an accuracy of 10−5 vs. 10−13 for movingsurfaces and Newton’s method, respectively, while the hit numbers were 265 vs.193.

It is necessary to make some remarks about problems with the memory usageof our implementation of the moving surfaces algorithm. The amount of memoryneeded for the matrix D1 (or D2) was about 350 Mbytes for the biquadraticsurface. This is a lot, but does not cause any problems. For a bicubic Beziersurface, however, the corresponding memory requirement is about 4 Gbytes withdouble precision! But even going to single precision was too much to handle for

3.5. DISCUSSION 31

our PCs.

3.5 Discussion

Moving surfaces provides a new method for computing closest points to a para-metric surface. It is an alternative to the conventional algorithms based oniterations and Newton’s method, or to subdivision methods.

Compared to a Newton based closest point algorithm, it takes a long timeto set up the system of moving surfaces for each parametric surface, but a shorttime to compute the closest point once a point in space is given. This suggeststhat the potential use of the moving surfaces method is for situations where alarge number of closest points are to be computed for each given surface, andwhere long preprocessing times are acceptable.

Furthermore, the moving surfaces method is better than Newton’s methodfor actually finding the closest points for surfaces with complex geometry. Thus,if this kind of stability is desired, the moving surfaces method may also be a bet-ter choice, combined with an iterative refinement of the resulting closest points.In other words, the closest points found from the moving surfaces method couldbe used as the starting point of the Newton iterations.

However, we had problems with the implementation of the algorithm, dueto memory shortage, when applied on the realistic case of bicubic surfaces.The large amount of memory is mainly used for building the matrices D1 andD2. In contrast, the amount of memory needed for storing the moving surfacesthemselves corresponds to only (N+1)(d1,2+1)(d1,2+2)(d1,2+3)/6 doubles (i.e.the dimensions of the vectors b1 and b2, respectively. In the bicubic case thisnumber is 10230. This shows that we must find another way of implementingthe algorithm, or that we must find a way to use moving surfaces together withapproximations. But if we can afford the preprocessing of the coefficients of themoving surfaces we can get fast and accurate calculations of closest points.

Chapter 4

Solving a closest pointproblem by subdivision

4.1 Introduction

In this paper, a closest point problem is solved by using subdivision techniques,and it is shown that a considerable speed-up is possible if relatively simple ideasare used.

Closest point problems are heavily used in CAGD applications. Applicationsinclude surface smoothing, surface fitting, and curve and surface selection. It isalso important to note that in some applications, solving closest point problemsbecomes the bottleneck of the algorithm. This fact alone makes it interestingto be able to solve such problems fast.

The common way of solving closest point problems is by using iterativemethods (see [12] and references therein). These are generally fast, but theneed for an initial guess makes them error prone. Some times these kinds oferrors, even when highly uncommon, can ruin the result. In these cases, asubdivision method can be the best way of ensuring high quality output.

The closest point problem can be solved by using a numerically stable alge-braic polynomial solver, and this approach is explored in [19, Section 4.2]. Thismethod is robust, but it is much slower than what one can get from a gooditerative method or a subdivision algorithm.

Note that closest point problems for discrete sets are very different from theproblem of finding closest points on smooth surfaces. However, in a CAGD

33

34 CHAPTER 4. CLOSEST POINTS BY SUBDIVISION

system it may be interesting to find the closest point on a model consisting ofmany smooth surfaces patches. Given such a problem, the techniques concerningthe discrete case found in [37] will be useful in addition to a good understandingof the smooth case considered here.

Section 4.2 defines the problem and explains the basic way of solving it bysubdivision. Notes about how different implementations should be evaluatedand what happens in special cases are also included here.

Section 4.3 explores different changes to the basic algorithm and how thesechanges affect the speed of implementations. Section 4.3.8 contains a tablesummarizing run times for the different implementations in Section 4.3.

Section 4.4 contains an error analysis of subdivision methods. A formulagiving a bound on the error is produced, and this formula can be used whendetermining if the output is accurate enough for a given application.

The chapter ends with Section 4.5, the conclusion.

4.2 Definition of the problem

In this chapter, a surface patch ϕ of (bi)degree (d, e) is a map ϕ : [0, 1]2 → R3

given by an array of control points cij ∈ R3 and the formula

ϕ(u, v) =d∑i=0

e∑j=0

cij(d

i

)(1− u)iud−i

(e

j

)(1− v)jve−j .

This surface is called a tensor product Bezier surface with control points cij .For a point x ∈ R3, we want to find the parameter point (u0, v0) ∈ [0, 1]2

that minimizes the distance function (u, v) → ‖ϕ(u, v)− x‖.Note that if x is in the image of ϕ, then the closest point problem specializes

to the inverse problem. This problem has been studied in, for example, [21].In a CAGD system one would like to solve the same problem in a more

general setting, for example where ϕ is a tensor product spline. However, sincewe can reduce the spline case to the Bezier case by knot insertion, consideringonly the Bezier case is not a big limitation.

4.2.1 The basic method

The idea was to start by implementing a simple method for finding closestpoints, and then introduce different changes that might or might not speed

4.2. DEFINITION OF THE PROBLEM 35

up the implementation. The basic method described here will be improved inSection 4.3.

Let F : [0, 1]2 → R be defined as the distance squared function, F (u, v) =‖ϕ(u, v)−x‖2. The critical points of F are the zero set of two partial derivatives,Fu(u, v) and Fv(u, v). If we evaluate one of these we get

Fu(u, v) = 2(ϕ(u, v)− x) · ϕu(u, v),

where ϕu = ∂ϕ∂u . Note that this is a very natural orthogonality condition: when

Fu(u′, v′) and Fv(u′, v′) are both zero, the vector ϕ(u′, v′) − x is orthogonalto the tangent plane of ϕ associated to the parameter value (u′, v′), or theparameterization is degenerate in this point. Abusing language, we say that(u′, v′) is a critical point for x if Fu(u′, v′) = Fv(u′, v′) = 0. It is clear thatthe closest point (u0, v0) will be either a critical point or on the border of theparameter domain.

The basic method is straightforward. Find the closest point on the border,and then find all the critical points by applying a subdivision solver to the tensorproduct functions Fu and Fv, and compare the distances. The shortest distancewill give the closest point.

One note on the closest point on the border may be useful: To find theclosest point on one side, for example (0, 1) × {0}, it is sufficient to evaluateF (u, 0) on zeros of the univariate polynomial Fu(u, 0). Every implementationin this chapter will start by finding the closest point on the border.

4.2.2 Quality of the output and special cases

Even though the algorithms described in this chapter have been tested thor-oughly, it is the application of the output data that should determine if anyalgorithm is “accurate enough”.

When doing subdivision, one may be very unlucky, and the subdivisionwill take a lot of time. This happens when the set of critical points is one-dimensional. The possibility of a one-dimensional set of solutions is by far thebiggest problem with subdivision methods, and usually must be handled withgreat care.

When this happens in our case, all points on each connected componentof the set of critical points will give the same distance to x, so any point onone such component will do as a solution to the problem. For this reasonspecial cases are not such a big problem, at least when we can detect it. Wesolve this by counting the number of calls to the recursive function, and, if


it is called too many times, we abort and start over with a lower maximumrecursion depth. Together with iterative refinement of best points, this shouldgive us good answers in practically all cases.

When the processing of one point is completed, we know to which depth therecursion has been completed successfully. Then the error analysis in Section4.4 can be used to determine if the guaranteed accuracy is sufficient.

4.3 Improving the basic method

All implementations follow this pattern:

• Initialize (u0, v0, D) = (0, 0, F (0, 0)). This set of variables represents thebest parameter point so far with the distance squared.

• For each (u, v) ∈ {(0, 1), (1, 0), (1, 1)}, evaluate F (u, v) and, if F (u, v) <D, update (u0, v0, D).

• For each side of the unit square, solve a univariate polynomial by subdi-vision and update (u0, v0, D) if a new closest point is found.

• Recursively find candidate points in the interior of the unit square andupdate (u0, v0, D).

It is the last item that will change when improving the basic method. But beforewe discuss these changes, we need some simple notation.

Let ϕc : [0, 1]2 → Rn be given by (d+ 1)(e+ 1) control points c = (cij) andthe formula

ϕc(u, v) =d∑i=0

e∑j=0

cij(d

i

)(1− u)iud−i

(e

j

)(1− v)jve−j .

We say that c = (cij) represents ϕc on the square [0, 1]2 ⊂ R2. Furthermore,we say that b = (bij) represents ϕc on the rectangle [a1, b1]× [a2, b2] if

ϕb(u, v) = ϕc((1− u)a1 + ub1, (1− v)a2 + vb2).

Such representations can be calculated easily by using de Casteljau’s algorithm,and if 0 ≤ ai ≤ bi ≤ 1 for i = 1, 2, the calculations are stable.

The input to the basic recursive solver is a rectangle [a1, b1] × [a2, b2] andtwo sets of control points (scalars) representing Fu and Fv. If either Fu or Fv

4.3. IMPROVING THE BASIC METHOD 37

has control points of constant signs, then the solver concludes that there are nobase points in the rectangle and returns. Else it subdivides the rectangle andcalls itself with new control points representing Fu and Fv in the subdividedrectangles, once for each sub-rectangle. The algorithm also keeps track of therecursion level, and when it reaches the bottom it evaluates the midpoint andupdates (u0, v0, D) if a new closest point has been found.

Now we will focus on different changes to the implementations.

4.3.1 Changing the subdivision

The first change is to subdivide ϕ(u, v) − x, ϕu(u, v) and ϕv(u, v) instead ofsubdividing Fu and Fv. Since the cost of subdividing is roughly proportional tothe cube of the degree, subdividing these three vector valued functions is fasterthan subdividing the two scalar functions Fu and Fv.

However, this change means that the dot product must be carried out whenchecking whether the control points of Fu or Fv have constant sign. The cost ofthe multiplication of tensor product Bezier functions is roughly proportional tothe fourth power of the degree, and that indicates that changing the subdivisionis a bad idea. However, even though changing the subdivision turned out tomake the algorithm slower, it allowed us to do other optimizations (see Sections4.3.2 and 4.3.3) that let us do the full multiplication less often.

Experiments showed that only subdividing ϕ(u, v) − x and calculating thesubdivided ϕu(u, v) and ϕv(u, v) from ϕ(u, v)− x is a bad idea. The operationis unstable, and this instability slowed down the implementations considerably.On the other hand, it is conceivable that this instability can become negligiblein some cases, when the recursive level is much smaller than the precision of thecomputer arithmetic used.

4.3.2 Changing the multiplication algorithm to allow anearly exit

The conventional way to do the multiplication of two tensor product polynomi-als is to initialize the result control points to zero, then loop through the controlcoefficients of one factor and, for each such coefficient, loop through the coeffi-cients of the other factor, while accumulating the result. The multiplication inthe dot products (ϕ(u, v) − x) · ϕu and (ϕ(u, v) − x) · ϕv was changed so thateach coefficient of the result was calculated before starting on the next. Theloops became more complicated, but the change allows the constant sign test tostop early if both signs are encountered.


The implementation was further improved by first calculating the cornercoefficients. These coefficients only depend on one control point from eachfactor, so they can be calculated very fast. Also, when checking for constantsign in Fu and Fv, the corner coefficients are most likely to differ.

This change resulted in a considerable improvement in speed, but not enoughto offset the speed lost by the changes in Section 4.3.1.

4.3.3 Introducing a box test and a plane test

When we enter the recursive function, the variable D holds the best distancesquared. Let {bij} represent ϕ(u, v) − x in the rectangle [a1, b1] × [a2, b2], soϕ([a1, b1], [a2, b2]) − x = ϕb([0, 1], [0, 1]) is contained in the convex hull of thecontrol points {bij}. We can estimate the distance from x to the surface patchϕ([a1, b1], [a2, b2]) and return if it is too big. This is done by calculating thedistance from 0 to the smallest box containing {bij}. If this distance is biggerthan

√D, we can return before doing the multiplication. This is called the box

test.While calculating the smallest box containing the control points {bij}, we

can also find the control point that is closest to 0. Let this point be bαβ , andcalculate bαβ · bij for each control point bij . If all of these dot products arebigger than the constant

√D‖bαβ‖, we can return before doing the multiplica-

tion. This is called the plane test, since it checks if all control points bij are onthe other side (than the origin) of the plane X · bαβ =

√D‖bαβ‖.

For the box test and the plane test to work as well as possible we alsocalculate distances at the middle point and the midpoints of the sides beforesubdividing. This way, when entering the recursive function, we have alreadytested the corner points. This increases the chance of the box test and the planetest helping us.

The box test and the plane test resulted in a huge speed-up, making thealgorithm faster than the basic method.

4.3.4 Using the second order derivatives

For some rectangles [a1, b1]× [a2, b2] the function F may be convex, and there-fore the rectangle can contain at most one base-point. This property can bedetermined from the signs of the eigenvalues of the Hessian of F . If the productof the eigenvalues is positive in the entire rectangle and the Newton refinementconverges to a point inside the rectangle, then this point is the only base-pointin the rectangle.


A method for calculating the control coefficients of the product of the eigen-values was introduced, but the high degree made this too costly, and it onlymade the implementations slower.

4.3.5 The recursive algorithm explained

This section explains the algorithm we got after applying the changes in Sec-tions 4.3.1, 4.3.2 and 4.3.3. The algorithm follows the pattern specified in thebeginning of Section 4.3. When we enter the recursive function, it is assumedthat the corners of the square to be considered has been evaluated, and that thevariables (u0, v0) and D have been updated accordingly. We keep track of thetotal number of calls to be able to abort in special cases. The recursive functionworks as follows:

Input: The square [u1, u2]× [v1, v2], control points bij representing ϕ− x andvectors biju and bijv representing ϕu and ϕv in this square.Output: Void, the variables (u0, v0) and D will be updated if necessary.

• If the number of calls to the recursive function is bigger than a constant,in our case 16384, abort. If else, increase the variable holding the numberof calls to the recursive function.

• Calculate the distance from the origin to the smallest box containing thecontrol points bij . If this is bigger than

√D, return. Also perform the

plane test described in Section 4.3.3.

• Calculate the control points aij ∈ R representing 12Fu, one at a time,

starting with the corner points a0,0 = b0,0 · b0,0u , a2d−1,0 = bd,0 · bd−1,0

u ,a0,2e = b0,e ·b0,e

u and a2d,2e = bd,e ·bd−1,eu . If both signs are encountered,

stop calculating coefficients and skip to the next bullet point. If all signsare the same, return.

• Do the same as above for Fv.

• Let u′ = 12 (u1 + u2) and v′ = 1

2 (v1 + v2). Evaluate F (u1, v′), F (u2, v

′),F (u′, v1), F (u′, v2) and F (u′, v′) and update (u0, v0) and D accordingly.

• If the square is sufficiently small, return.

• Calculate representations of ϕ−x, ϕu and ϕv on the four squares [u1, u′]×

[v1, v′], [u1, u′] × [v′, v2], [u′, u2] × [v1, v′] and [u′, u2] × [v′, v2] using de

Casteljau’s algorithm on bij , biju and bijv . Then call the recursive function


with these values. In most cases it pays to sort the order of these callsbased on the values F (u1, v

′), F (u2, v′), F (u′, v1) and F (u′, v2). To be

precise, if F (u1, v′) < F (u2, v

′), then call the the recursive function forthe square [u1, u

′] × [v1, v′] before the square [u′, u2] × [v1, v′] and thesquare [u1, u

′] × [v′, v2] before [u′, u2] × [v′, v2]. Similarly, if F (u′, v1) <F (u′, v2), handle the square [u1, u

′] × [v1, v′] before [u1, u′] × [v′, v2] and

[u′, u2]× [v1, v′] before [u′, u2]× [v′, v2].

4.3.6 The basic method with the box and plane tests

After seeing the huge improvement from the tests in Section 4.3.3, it was naturalto try to improve the basic method using the same tests. The resulting algorithmcalculates representations of Fu, Fv and ϕ(u, v)−x in the unit square. Then therecursive function does essentially the same as the algorithm in Section 4.3.5,except that no multiplication needs to be carried out.

The result is a method that is a little slower than the algorithm in Sec-tion 4.3.5. However, the difference is so small that a strong conclusion cannotbe drawn - the result can be very different on different hardware, and, mostimportantly, on other test cases.

4.3.7 Doing a preconditioned constant sign test

General subdivision algorithms can be sped up considerably by doing a precon-ditioned constant sign test [26]. The idea is to do a special linear transformationof the equations to be solved, and then to check if any of the resulting equationshas control points of constant sign. The linear transformation is determinedby the cofactor matrix of the Jacobian matrix evaluated in the midpoint of thesquare.

Preconditioning requires the equations to be of the same degree, so we elevatethe degree of Fu and Fv to (2d, 2e), and transform the system by using thecofactor matrix of the Hessian of F at the midpoint:(

G1

G2

)=(Fvv(u′, v′) −Fuv(u′, v′)−Fuv(u′, v′) Fuu(u′, v′)

)(FuFv

)If the control points of either G1 or G2 have constant sign (different from zero)we can conclude that the rectangle contains no critical point.

For our datasets, doing preconditioning helped a lot when the box and planetests where not present, cutting processing time in half. When the box andplane tests were used, the effect ranged from a slowdown of a few percent to a


speedup of a few percent. It is highly likely that this will be different on otherdatasets.

4.3.8 Speed measurements

The implementations were tested on many surfaces and on many points persurface. To be exact, for each bi-degree, 100 surfaces with control points placedrandomly in the unit cube were tested, and for each surface 100 random pointsin the unit cube were selected. Table 4.1 shows the times for seven algorithms:

• the algorithm described in 4.3.5

• the algorithm from Section 4.3.5 without the plane test from Section 4.3.3

• the algorithm from Section 4.3.5 without the plane test and the box testfrom Section 4.3.3

• the algorithm from Section 4.3.5 without the improved calculation of theproduct 1

2Fu(u, v) = (ϕ(u, v)− x) · ϕu(u, v) from Section 4.3.2

• the basic algorithm

• the basic algorithm with a box test

• the basic algorithm with a box test and a plane test

Each of these algorithms were tested with and without preconditioning, andthe table shows the best time for each algorithm.

Degree (2, 2) (3, 3) (4, 4) (3, 9) (7, 7) (20, 20)

Algorithm from Section 4.3.5 2.31 4.78 8.37 18.2? 32.0? 652?

no plane test 3.89 9.59 20.1 43.9 98.6 2823no plane or box test 12.2 52.8 170 495 1671 204009no multiplication optimization 3.65? 9.14? 19.3? 46.2? 110? 4348?

Basic algorithm 5.80 21.1 52.9? 138 390? 23586with box test 2.73 6.21 11.5 22.3 45.9 861?

with box and plane test 2.40 5.10? 8.79? 17.9 32.0? 561?

Table 4.1: Time in seconds spent to solve the closest point problem for 100surfaces and 100 point per surface on a 2.80 GHz Pentium 4 CPU. A ? indicatesthat it did not pay to do a preconditioned constant sign test.


The maximum recursion level was set to 32, but numbers should be repre-sentative. Experiments showed that the time was roughly proportional to thelevel of recursion, at least for reasonable values.

The usual cautions apply: It is unlikely that these test cases give a very goodindication of what is fastest for less random data sets on different hardware.Because of this, it is recommended that anyone who needs to calculate closestpoints fast do their own timings.

Remark: I also experimented with different compiler optimization flags, andthe optimal set of flags was not constant for different degrees. To be specific,it was the -funroll-loops options that helped in some cases, but made thingsworse in others. If this problem is to be solved in production code, the bestcompiler available should be used. The table shows times for the best set ofcompiler settings for each case.

4.4 Error analysis

If infinite precision in the calculations is assumed, then we can develop a lowerbound L of ‖F ([0, 1], [0, 1])‖ in terms of the distance squared D returned bythe algorithm, the bi-degree (d, e) of ϕ, the depth n of the recursion, and thediameter

R = maxi,j,k,l

{‖cij − ckl‖}

of the control points. We also assume that the number of recursive calls is notconstrained in any way except in terms of depth. This means that the actualconstrained algorithm will give worse results in some rare cases, but it will beable to report what level was reached successfully, giving us a worse guaranteedaccuracy. We say that the depth of recursion is n if any square bigger than2−n × 2−n that may contain the closest point is subdivided.

We know that a square of size 2−n×2−n containing the closest point (u0, v0)will be considered by the algorithm. Let this square be denoted [u1, u2]× [v1, v2]with u1 ≤ u0 ≤ u2, v1 ≤ v0 ≤ v2 and u2 − u1 = v2 − v1 ≤ 2−n. The cornersof this small square have been evaluated as candidates for the return value ofthe algorithm, so D ≤ F (ui, vj) for i = 1, 2 and j = 1, 2. We now assumeu0 − u1 ≤ 2−n−1 and v0 − v1 ≤ 2−n−1, so that (u1, v1) is the sample pointclosest to (u0, v0).

The derivatives ϕu and ϕv have limited range:

‖ϕu(u, v)‖ ≤ dR for all (u, v) ∈ [0, 1]2 (4.1)‖ϕv(u, v)‖ ≤ eR for all (u, v) ∈ [0, 1]2 (4.2)

4.4. ERROR ANALYSIS 43

This limits the difference ϕ(u0, v0)− ϕ(u1, v1):

‖ϕ(u0, v0)− ϕ(u1, v1)‖ ≤ 2−n−1(d+ e)R (4.3)

Thus we have a lower bound for the shortest distance,

‖ϕ(u0, v0)− x‖ ≥√D − 2−n−1(d+ e)R.

This bound is not very impressive, but we can improve this bound by using thefact that Fu(u0, v0) = 0 and that the second derivatives are bounded as follows:

‖ϕuu(u, v)‖ ≤ 2d(d− 1)R for all (u, v) ∈ [0, 1]2 (4.4)‖ϕuv(u, v)‖ ≤ 2deR for all (u, v) ∈ [0, 1]2 (4.5)‖ϕvv(u, v)‖ ≤ 2e(e− 1)R for all (u, v) ∈ [0, 1]2 (4.6)

We can now make a distance preserving coordinate change such that ϕ(u0, v0) =(0, 0, 0), x = (−‖ϕ(u0, v0) − x‖, 0, 0) and ϕ(u1, v1) =: (a, b, c). From equations(4.1) and (4.2) we get

b2 + c2 ≤ (2−n−1(d+ e)R)2 =: B.

Furthermore, equations (4.4), (4.5) and (4.6) gives

|ϕu(u, v) · (1, 0, 0)| ≤ 2−nd(e+ d− 1)|ϕv(u, v) · (1, 0, 0)| ≤ 2−ne(d+ e− 1)

on the rectangle [u1, u0]× [v1, v0]. This gives

|a| ≤ 2−2n−1(d+ e)(d+ e− 1).

We can refine this a little bit: Equations (4.4), (4.5) and (4.6) give

|ϕu(u, v) · (1, 0, 0)| ≤ 2−nd((d− 1)|u0 − u|+ e|v0 − v|) and|ϕv(u, v) · (1, 0, 0)| ≤ 2−ne(d|u0 − u|+ (e− 1)|v0 − v|)

Setting u0 − u = v0 − v = t and integrating the bound on the derivatives, weget

|a| ≤∫ 2−n−1

0

2−n(d+ e)(d+ e− 1)t dt = 2−2n−2(d+ e)(d+ e− 1) =: A.


From Pythagoras we get D ≤ (‖ϕ(u0, v0)− x‖+A)2 +B. From this we getthe lower bound L of the shortest distance

L :=√D −B −A ≤ ‖ϕ(u0, v0)− x‖.

The corresponding upper bound of the error√D − ‖ϕ(u0, v0)− x‖ can be cal-

culated in a stable way:√D − ‖ϕ(u0, v0)− x‖ ≤

√D − L =

√D −

√D −B +A

=B√

D +√D −B

+A

This is better than the error bound in equation (4.3) in most cases, when Dis much bigger than B. A few examples can illustrate this pretty well. If thebi-degree is (3, 3), the diameter of the control points is 1, the distance returnedis√D = 0.01, the depth of the recursion is 26, then the error is at most

1.016 · 10−13. If the recursion level is increased to 32, then the error is at most2.48 · 10−17.

4.5 Conclusion

The closest point problem treated in this chapter is quite common in geometricapplications, and often needs to be solved by a computer algorithm in the fastestpossible way. Subdivision methods has the advantage that they can be madevery fast in almost all cases, and the guaranteed accuracy of the algorithm isknown. For special points, when the subdivision method takes too long, theapplication must decide the proper action. In some cases it is natural to fallback to a more accurate method. In other cases it is natural to simply discardthe point and move on to the next.

The result is a flexible set of algorithms that should be usable for mostapplications.

Chapter 5

Monoid hypersurfaces1

Pal Hermunn Johansen, Magnus Løberg, Ragni Piene

5.1 Introduction

A monoid hypersurface is an (affine or projective) irreducible algebraic hyper-surface which has a singularity of multiplicity one less than the degree of thehypersurface. The presence of such a singular point forces the hypersurface tobe rational: there is a rational parameterization given by (the inverse of) thelinear projection of the hypersurface from the singular point.

The existence of an explicit rational parameterization makes such hypersur-faces potentially interesting objects in computer aided design. Moreover, sincethe “space” of monoids of a given degree is much smaller than the space of allhypersurfaces of that degree, one can hope to use monoids efficiently in (approx-imate or exact) implicitization problems. These were the reasons for consideringmonoids in the paper [35]. In [28] monoid curves are used to approximate othercurves that are close to a monoid curve, and in [29] the same is done for monoidsurfaces. In both articles the error of such approximations are analyzed – foreach approximation, a bound on the distance from the monoid to the originalcurve or surface can be computed.

1This chapter has been submitted as an article for the proceedings of the conference COM-PASS II, and has been accepted by the editors of the book.

45

46 CHAPTER 5. MONOID HYPERSURFACES

In this article we shall study properties of monoid hypersurfaces and theclassification of monoid surfaces with respect to their singularities. Section 5.2explores properties of monoid hypersurfaces in arbritrary dimension and overan arbitrary base field. Section 5.3 contains results on monoid surfaces, bothover arbritrary fields and over R. The last section deals with the classification ofmonoid surfaces of degree four. Real and complex quartic monoid surfaces werefirst studied by Rohn [32], who gave a fairly complete description of all possiblecases. He also remarked [32, p. 56] that some of his results on quartic monoidshold for monoids of arbitrary degree; in particular, we believe he was awareof many of the results in Section 5.3. Takahashi, Watanabe, and Higuchi [38]classify complex quartic monoid surfaces, but do not refer to Rohn. (They citeJessop [14]; Jessop, however, only treats quartic surfaces with double points andrefers to Rohn for the monoid case.) Here we aim at giving a short descriptionof the possible singularities that can occur on quartic monoids, with specialemphasis on the real case.

5.2 Basic properties

Let k be a field, let k denote its algebraic closure and Pn := Pnk

the projectiven-space over k. Furthermore we define the set of k-rational points Pn(k) as theset of points that admit representatives (a0 : · · · : an) with each ai ∈ k.

For any homogeneous polynomial F ∈ k[x0, . . . , xn] of degree d and pointp = (p0 : p1 : · · · : pn) ∈ Pn we can define the multiplicity of Z(F ) at p. Weknow that pr 6= 0 for some r, so we can assume p0 = 1 and write

F =d∑i=0

xd−i0 fi(x1 − p1x0, x2 − p2x0, . . . , xn − pnx0)

where fi is homogeneous of degree i. Then the multiplicity of Z(F ) at p isdefined to be the smallest i such that fi 6= 0.

Let F ∈ k[x0, . . . , xn] be of degree d ≥ 3. We say that the hypersurfaceX = Z(F ) ⊂ Pn is a monoid hypersurface if X is irreducible and has a singularpoint of multiplicity d− 1.

In this article we shall only consider monoids X = Z(F ) where the singularpoint is k-rational. Modulo a projective transformation of Pn over k we may –and shall – therefore assume that the singular point is the point O = (1 : 0 :· · · : 0).

5.2. BASIC PROPERTIES 47

Hence, we shall from now on assume that X = Z(F ), and

F = x0fd−1 + fd,

where fi ∈ k[x1, . . . , xn] ⊂ k[x0, . . . , xn] is homogeneous of degree i and fd−1 6=0. Since F is irreducible, fd is not identically 0, and fd−1 and fd have nocommon (non-constant) factors.

The natural rational parameterization of the monoid X = Z(F ) is the map

θF : Pn−1 → Pn

given byθF (a) = (fd(a) : −fd−1(a)a1 : . . . : −fd−1(a)an),

for a = (a1 : · · · : an) such that fd−1(a) 6= 0 or fd(a) 6= 0.The set of lines through O form a Pn−1. For every a = (a1 : · · · : an) ∈ Pn−1,

the lineLa := {(s : ta1 : . . . : tan)|(s : t) ∈ P1} (5.1)

intersects X = Z(F ) with multiplicity at least d − 1 in O. If fd−1(a) 6= 0 orfd(a) 6= 0, then the line La also intersects X in the point

θF (a) = (fd(a) : −fd−1(a)a1 : . . . : −fd−1(a)an).

Hence the natural parameterization is the “inverse” of the projection of X fromthe point O. Note that θF maps Z(fd−1) \ Z(fd) to O. The points where theparameterization map is not defined are called base points, and these points areprecisely the common zeros of fd−1 and fd. Each such point b corresponds tothe line Lb contained in the monoid hypersurface. Additionally, every line oftype Lb contained in the monoid hypersurface corresponds to a base point.

Note that Z(fd−1) ⊂ Pn−1 is the projective tangent cone to X at O, andthat Z(fd) is the intersection of X with the hyperplane “at infinity” Z(x0).

Assume P ∈ X is another singular point on the monoid X. Then the line Lthrough P and O has intersection multiplicity at least d − 1 + 2 = d + 1 withX. Hence, according to Bezout’s theorem, L must be contained in X, so thatthis is only possible if dimX ≥ 2.

By taking the partial derivatives of F we can characterize the singular pointsof X in terms of fd and fd−1:

Lemma 5.1. Let ∇ = ( ∂∂x1

, . . . , ∂∂xn

) be the gradient operator.

(i) A point P = (p0 : p1 : · · · : pn) ∈ Pn is singular on Z(F ) if and only iffd−1(p1, . . . , pn) = 0 and p0∇fd−1(p1, . . . , pn) +∇fd(p1, . . . , pn) = 0.


(ii) All singular points of Z(F ) are on lines La where a is a base point.

(iii) Both Z(fd−1) and Z(fd) are singular in a point a ∈ Pn−1 if and only if allpoints on La are singular on X.

(iv) If not all points on La are singular, then at most one point other than Oon La is singular.

Proof. (i) follows directly from taking the derivatives of F = x0fd−1 + fd, and(ii) follows from (i) and the fact that F (P ) = 0 for any singular point P .Furthermore, a point (s : ta1 : . . . : tan) on La is, by (i), singular if and only if

s∇fd−1(ta) +∇fd(ta) = td−1(s∇fd−1(a) + t∇fd(a)) = 0.

This holds for all (s : t) ∈ P1 if and only if ∇fd−1(a) = ∇fd(a) = 0. This proves(iii). If either ∇fd−1(a) or ∇fd−1(a) are nonzero, the equation above has atmost one solution (s0 : t0) ∈ P1 in addition to t = 0, and (iv) follows.

Note that it is possible to construct monoids where F ∈ k[x0, . . . , xn], butwhere no points of multiplicity d− 1 are k-rational. In that case there must be(at least) two such points, and the line connecting these will be of multiplicityd − 2. Furthermore, the natural parameterization will typically not induce aparameterization of the k-rational points from Pn−1(k).

5.3 Monoid surfaces

In the case of a monoid surface, the parameterization has a finite number ofbase points. From Lemma 5.1 (ii) we know that all singularities of the monoidother than O, are on lines La corresponding to these points. In what followswe will develop the theory for singularities on monoid surfaces — most of theseresults were probably known to Rohn [32, p. 56].

We start by giving a precise definition of what we shall mean by a monoidsurface.

Definition 5.2. For an integer d ≥ 3 and a field k of characteristic 0 the poly-nomials fd−1 ∈ k[x1, x2, x3]d−1 and fd ∈ k[x1, x2, x3]d define a normalized non-degenerate monoid surface Z(F ) ⊂ P3, where F = x0fd−1+fd ∈ k[x0, x1, x2, x3]if the following hold:

(i) fd−1, fd 6= 0

5.3. MONOID SURFACES 49

(ii) gcd(fd−1, fd) = 1

(iii) The curves Z(fd−1) ⊂ P2 and Z(fd) ⊂ P2 have no common singular point.

The curves Z(fd−1) ⊂ P2 and Z(fd) ⊂ P2 are called respectively the tangentcone and the intersection with infinity.

Unless otherwise stated, a surface that satisfies the conditions of Definition5.2 shall be referred to simply as a monoid surface.

Since we have finitely many base points b and each line Lb contains at mostone singular point in addition to O, monoid surfaces will have only finitely manysingularities, so all singularities will be isolated. (Note that Rohn includedsurfaces with nonisolated singularities in his study [32].) We will show that thesingularities other than O can be classified by local intersection numbers.

Definition 5.3. Let f, g ∈ k[x1, x2, x3] be nonzero and homogeneous. Assumep = (p1 : p2 : p3) ∈ Z(f, g) ⊂ P2, and define the local intersection number

Ip(f, g) = lgk[x1, x2, x3]mp

(f, g),

where k is the algebraic closure of k, mp = (p2x1−p1x2, p3x1−p1x3, p3x2−p2x3)is the homogeneous ideal of p, and lg denotes the length of the local ring as amodule over itself.

Note that Ip(f, g) ≥ 1 if and only if f(p) = g(p) = 0. When Ip(f, g) = 1 wesay that f and g intersect transversally at p. The terminology is justified bythe following lemma:

Lemma 5.4. Let f, g ∈ k[x1, x2, x3] be nonzero and homogeneous and p ∈Z(f, g). Then the following are equivalent:

(i) Ip(f, g) > 1

(ii) f is singular at p, g is singular at p, or ∇f(p) and ∇g(p) are nonzero andparallel.

(iii) s∇f(p) + t∇g(p) = 0 for some (s, t) 6= (0, 0)

Proof. (ii) is equivalent to (iii) by a simple case study: f is singular at p if andonly if (iii) holds for (s, t) = (1, 0), g is singular at p if and only if (iii) holds for(s, t) = (0, 1), and ∇f(p) and ∇g(p) are nonzero and parallel if and only if (iii)holds for some s, t 6= 0.


We can assume that p = (0 : 0 : 1), so Ip(f, g) = lgS where

S =k[x1, x2, x3](x1,x2)

(f, g).

Furthermore, let d = deg f , e = deg g and write

f =d∑i=1

fixd−i3 and g =

e∑i=1

gixe−i3

where fi, gi are homogeneous of degree i.If f is singular at p, then f1 = 0. Choose ` = ax1 + bx2 such that ` is not

a multiple of g1. Then ` will be a nonzero non-invertible element of S, so thelength of S is greater than 1.

We have ∇f(p) = (∇f1(p), 0) and ∇g(p) = (∇g1(p), 0). If they are parallel,choose ` = ax0 + bx1 such that ` is not a multiple of f1 (or g1), and argue asabove.

Finally assume that f and g intersect transversally at p. We may assumethat f1 = x1 and g1 = x2. Then (f, g) = (x1, x2) as ideals in the local ringk[x1, x2, x3](x1,x2). This means that S is isomorphic to the field k(x3). Thelength of any field is 1, so Ip(f, g) = lgS = 1.

Now we can say which are the lines Lb, with b ∈ Z(fd−1, fd), that contain asingularity other than O:

Lemma 5.5. Let fd−1 and fd be as in Definition 5.2. The line Lb contains asingular point other than O if and only if Z(fd−1) is nonsingular at b and theintersection multiplicity Ib(fd−1, fd) > 1.

Proof. Let b = (b1 : b2 : b3) and assume that (b0 : b1 : b2 : b3) is a singularpoint of Z(F ). Then, by Lemma 5.1, fd−1(b1, b2, b3) = fd(b1, b2, b3) = 0 andb0∇fd−1(b1, b2, b3) + ∇fd(b1, b2, b3) = 0, which implies Ib(fd−1, fd) > 1. Fur-thermore, if fd−1 is singular at b, then the gradient ∇fd−1(b1, b2, b3) = 0, so fd,too, is singular at b, contrary to our assumptions.

Now assume that Z(fd−1) is nonsingular at b = (b1 : b2 : b3) and the intersec-tion multiplicity Ib(fd−1, fd) > 1. The second assumption implies fd−1(b1, b2, b3) =fd(b1, b2, b3) = 0 and s∇fd−1(b1, b2, b3) = t∇fd(b1, b2, b3) for some (s, t) 6= (0, 0).Since Z(fd−1) is nonsingular at b, we know that ∇fd−1(b1, b2, b3) 6= 0, so t 6= 0.Now (−s/t : b1 : b2 : b3) 6= (1 : 0 : 0 : 0) is a singular point of Z(F ) on the lineLb.


Recall that an An singularity is a singularity with normal form x21+x

22+x

n+13 ,

see [3, p. 184].

Proposition 5.6. Let fd−1 and fd be as in Definition 5.2, and assume P = (p0 :p1 : p2 : p3) 6= (1 : 0 : 0 : 0) is a singular point of Z(F ) with I(p1:p2:p3)(fd−1, fd) =m. Then P is an Am−1 singularity.

Proof. We may assume that P = (0 : 0 : 0 : 1) and write the local equation

g := F (x0, x1, x2, 1) = x0fd−1(x1, x2, 1) + fd(x1, x2, 1) =d∑i=2

gi (5.2)

with gi ∈ k[x0, x1, x2] homogeneous of degree i. Since Z(fd−1) is nonsingular at0 := (0 : 0 : 1), we can assume that the linear term of fd−1(x1, x2, 1) is equal tox1. The quadratic term g2 of g is then g2 = x0x1 + ax2

1 + bx1x2 + cx22 for some

a, b, c ∈ k. The Hessian matrix of g evaluated at P is

H(g)(0, 0, 0) = H(g2)(0, 0, 0) =

0 1 01 2a b0 b 2c

which has corank 0 when c 6= 0 and corank 1 when c = 0. By [3, p. 188], P isan A1 singularity when c 6= 0 and an An singularity for some n when c = 0.

The index n of the singularity is equal to the Milnor number

µ = dimk

k[x0, x1, x2](x0,x1,x2)

Jg= dimk

k[x0, x1, x2](x0,x1,x2)(∂g∂x0

, ∂g∂x1, ∂g∂x2

) .

We need to show that µ = I0(fd−1, fd)−1. From the definition of the intersectionmultiplicity, it is not hard to see that

I0(fd−1, fd) = dimk

k[x1, x2](x1,x2)

(fd−1(x1, x2, 1), fd(x1, x2, 1)).

The singularity at p is isolated, so the Milnor number is finite. Furthermore,since gcd(fd−1, fd) = 1, the intersection multiplicity is finite. Therefore bothdimensions can be calculated in the completion rings. For the rest of the proof weview fd−1 and fd as elements of the power series rings k[[x1, x2]] ⊂ k[[x0, x1, x2]],and all calculations are done in these rings.


Since Z(fd−1) is smooth at O, we can write

fd−1(x1, x2, 1) = (x1 − φ(x2))u(x1, x2)

for some power series φ(x2) and invertible power series u(x1, x2). To simplifynotation we write u = u(x1, x2) ∈ k[[x1, x2]].

The Jacobian ideal Jg is generated by the three partial derivatives:

∂g

∂x0= (x1 − φ(x2))u

∂g

∂x1= x0

(u+ (x1 − φ(x2))

∂u

∂x1

)+∂fd∂x1

(x1, x2)

∂g

∂x2= x0

(−φ′(x2)u+ (x1 − φ(x2))

∂u

∂x2

)+∂fd∂x2

(x1, x2)

By using the fact that x1−φ(x2) ∈(∂g∂x0

)we can write Jg without the symbols

∂u∂x1

and ∂u∂x2

:

Jg =(x1 − φ(x2), x0u+ ∂fd

∂x1(x1, x2),−x0uφ

′(x2) + ∂fd

∂x2(x1, x2)

)To make the following calculations clear, define the polynomials hi by writing

fd(x1, x2, 1) =∑di=0 x

i1hi(x2). Now

Jg =(x1 − φ(x2), x0u+

∑di=1 ix

i−11 hi(x2),−x0uφ

′(x2) +∑di=0 x

i1h′i(x2)

),

sok[[x0, x1, x2]]

Jg=

k[[x2]](A(x2))

where

A(x2) = φ′(x2)(∑d

i=1 iφ(x2)i−1hi(x2))

+(∑d

i=0 φ(x2)ih′i(x2)).

For the intersection multiplicity we have

k[[x1, x2]](fd−1(x1, x2, 1), fd(x1, x2, 1)

) =k[[x1, x2]](

x1 − φ(x2),∑di=0 x

i1hi(x2)

) =k[[x2]](B(x2)

)where B(x2) =

∑di=0 φ(x2)ihi(x2). Observing that B′(x2) = A(x2) gives the

result µ = I0(fd−1, fd)− 1.


Corollary 5.7. A monoid surface of degree d can have at most 12d(d − 1)

singularities in addition to O. If this number of singularities is obtained, thenall of them will be of type A1.

Proof. The sum of all local intersection numbers Ia(fd−1, fd) is given by Bezout’stheorem: ∑

a∈Z(fd−1,fd)

Ia(fd−1, fd) = d(d− 1).

The line La will contain a singularity other than O only if Ia(fd−1, fd) ≥ 2,giving a maximum of 1

2d(d − 1) singularities in addition to O. Also, if thisnumber is obtained, all local intersection numbers must be exactly 2, so allsingularities other than O will be of type A1.

Both Proposition 5.6 and Corollary 5.7 were known to Rohn, who statedthese results only in the case d = 4, but said they could be generalized toarbitrary d [32, p. 60].

For the rest of the section we will assume k = R. It turns out that we can finda real normal form for the singularities other than O. The complex singularitiesof type An come in several real types, with normal forms x2

1±x22±xn+1

3 . Varyingthe ± gives two types for n = 1 and n even, and three types for n ≥ 3 odd.The real type with normal form x2

1 − x22 + xn+1

3 is called an A−n singularity, orof type A−, and is what we find on real monoids:

Proposition 5.8. On a real monoid, all singularities other than O are of typeA−.

Proof. Assume p = (0 : 0 : 1) is a singular point on Z(F ) and set g =F (x0, x1, x2, 1) as in the proof of Proposition 5.6.

First note that u−1g = x0(x1−φ(x2))+ fd(x1, x2)u−1 is an equation for thesingularity. We will now prove that u−1g is right equivalent to ±(x2

0−x21 +xn2 ),

for some n, by constructing right equivalent functions u−1g =: g(0) ∼ g(1) ∼g(2) ∼ g(3) ∼ ±(x2

0 − x21 + xn2 ). Let

g(1)(x0, x1, x2) = g(0)(x0, x1 + φ(x2), x2)

= x0x1 + fd(x1 + φ(x2), x2)u−1(x1 + φ(x2), x2)= x0x1 + ψ(x1, x2)

where ψ(x1, x2) ∈ R[[x1, x2]]. Write ψ(x1, x2) = x1ψ1(x1, x2) + ψ2(x2) anddefine

g(2)(x0, x1, x2) = g(1)(x0 − ψ1(x1, x2), x1, x2) = x0x1 + ψ2(x2).


The power series ψ2(x2) can be written on the form

ψ2(x2) = sxn2 (a0 + a1x2 + a2x22 + . . . )

where s = ±1 and a0 > 0. We see that g(2) is right equivalent to g(3) =x0x1 + sxn2 since

g(2)(x0, x1, x2) = g(3)

(x0, x1, x2

n

√a0 + a1x2 + a2x2

2 + . . .

).

Finally we see that

g(4)(x0, x1, x2) := g(3)(sx0 − sx1, x0 + x1, x2) = s(x20 − x2

1 + xn2 )

proves that u−1g is right equivalent to s(x20 − x2

1 + xn2 ) which is an equation foran An−1 singularity with normal form x2

0 − x21 + xn2 .

Note that for d = 3, the singularity at O can be an A+1 singularity. This

happens for example when f2 = x20 + x2

1 + x22.

For a real monoid, Corollary 5.7 implies that we can have at most 12d(d− 1)

real singularities in addition to O. We can show that the bound is sharp by asimple construction:Example. To construct a monoid with the maximal number of real singular-ities, it is sufficient to construct two affine real curves in the xy-plane definedby equations fd−1 and fd of degrees d− 1 and d such that the curves intersectin d(d− 1)/2 points with multiplicity 2. Let m ∈ {d− 1, d} be odd and set

fm = ε−m∏i=1

(x sin

(2iπm

)+ y cos

(2iπm

)+ 1).

For ε > 0 sufficiently small there exist at least m+12 radii r > 0, one for each

root of the univariate polynomial fm|x=0, such that the circle x2 + y2 − r2

intersects fm in m points with multiplicity 2. Let f2d−1−m be a product of suchcircles. Now the homogenizations of fd−1 and fd define a monoid surface with1 + 1

2d(d− 1) singularities. See Figure 5.1.

Proposition 5.6 and Bezout’s theorem imply that the maximal Milnor num-ber of a singularity other than O is d(d− 1)− 1. The following example showsthat this bound can be achieved on a real monoid:Example. The surface X ⊂ P3 defined by F = x0(x1x

d−22 +xd−1

3 )+xd1 has ex-actly two singular points. The point (1 : 0 : 0 : 0) is a singularity of multiplicity


Figure 5.1: The curves fm for m = 3, 5 and corresponding circles

3 with Milnor number µ = (d2 − 3d+ 1)(d− 2), while the point (0 : 0 : 1 : 0) isan Ad(d−1)−1 singularity. A picture of this surfaces for d = 4 is shown in Figure5.2.

Figure 5.2: The surface defined by F = x0(x1xd−22 + xd−1

3 ) + xd1 for d = 4.


5.4 Quartic monoid surfaces

Every cubic surface with isolated singularities is a monoid. Both smooth andsingular cubic surfaces have been studied extensively, most notably in [33], wherereal cubic surfaces and their singularities were classifed, and more recently in[36], [4], and [16]. The site [17] contains additional pictures and references.

In this section we shall consider the case d = 4. The classification of realand complex quartic monoid surfaces was started by Rohn [32]. (In additionto considering the singularities, Rohn studied the existence of lines not passingthrough the triple point, and that of other special curves on the monoid.) In [38],Takahashi, Watanabe, and Higuchi described the singularities of such complexsurfaces. The monoid singularity of a quartic monoid is minimally elliptic [42],and minimally elliptic singularities have the same complex topological type ifand only if their dual graphs are isomorphic [18]. In [18] all possible dual graphsfor minimally elliptic singularities are listed, along with example equations.

Using Arnold’s notation for the singularities, we use and extend the approachof Takahashi, Watanabe, and Higuchi in [38].

Consider a quartic monoid surface, X = Z(F ), with F = x0f3 + f4. Thetangent cone, Z(f3), can be of one of nine (complex) types, each needing aseparate analysis.

For each type we fix f3, but any other tangent cone of the same type will beprojectively equivalent (over the complex numbers) to this fixed f3. The ninedifferent types are:

1. Nodal irreducible curve, f3 = x1x2x3 + x32 + x3

3.

2. Cuspidal curve, f3 = x31 − x2

2x3.

3. Conic and a chord, f3 = x3(x1x2 + x23)

4. Conic and a tangent line, f3 = x3(x1x3 + x22).

5. Three general lines, f3 = x1x2x3.

6. Three lines meeting in a point, f3 = x32 − x2x

23

7. A double line and another line, f3 = x2x23

8. A triple line f3 = x33

9. A smooth curve, f3 = x31 + x3

2 + x33 + 3ax0x1x3 where a3 6= −1

5.4. QUARTIC MONOID SURFACES 57

To each quartic monoid we can associate, in addition to the type, severalinteger invariants, all given as intersection numbers. From [38] we know that,for the types 1–3, 5, and 9, these invariants will determine the singularity typeof O up to right equivalence. In the other cases the singularity series, as definedby Arnol’d in [1] and [2], is determined by the type of f3. We shall use, withoutproof, the results on the singularity type of O due to [38]; however, we shall usethe notations of [1] and [2].

We complete the classification begun in [38] by supplying a complete listof the possible singularities occurring on a quartic monoid. In addition, weextend the results to the case of real monoids. Our results are summarized inthe following theorem.

Theorem 5.9. On a quartic monoid surface, singularities other than the monoidpoint can occur as given in Table 5.1. Moreover, all possibilities are realizable onreal quartic monoids with a real monoid point, and with the other singularitiesbeing real and of type A−.

Proof. The invariants listed in the “Invariants and constraints” column are allnonnegative integers, and any set of integer values satisfying the equations rep-resents one possible set of invariants, as described above. Then, for each setof invariants, (positive) intersection multiplicities, denoted mi, m′

i and m′′i , will

determine the singularities other than O. The column “Other singularities” givethese and the equations they must satisfy. Here we use the notation A0 for aline La on Z(F ) where O is the only singular point.

The analyses of the nine cases share many similarities, and we have chosennot to go into great detail when one aspect of a case differs little from theprevious one. We end the section with a discussion on the possible real formsof the tangent cone and how this affects the classification of the real quarticmonoids.

In all cases, we shall write

f4 = a1x41 + a2x

31x2 + a3x

31x3 + a4x

21x

22 + a5x

21x2x3

+ a6x21x

23 + a7x1x

32 + a8x1x

22x3 + a9x1x2x

23 + a10x1x

33

+ a11x42 + a12x

32x3 + a13x

22x

23 + a14x2x

33 + a15x

43

and we shall investigate how the coefficients a1, . . . , a15 are related to the ge-ometry of the monoid.


Case Triple point Invariants and constraints Other singularities1 T3,3,4 Ami−1,

Pmi = 12

T3,3,3+m m = 2, . . . , 12 Ami−1,P

mi = 12−m2 Q10 Ami−1,

Pmi = 12

T9+m m = 2, 3 Ami−1,P

mi = 12−m3 T3,4+r0,4+r1 r0 = max(j0, k0), r1 = max(j1, k1), Ami−1,

Pmi = 4− k0 − k1,

j0 > 0 ↔ k0 > 0, min(j0, k0) ≤ 1, Am′i−1,

Pm′

i = 8− j0 − j1

j1 > 0 ↔ k1 > 0, min(j1, k1) ≤ 14 S series j0 ≤ 8, k0 ≤ 4, min(j0, k0) ≤ 2, Ami−1,

Pmi = 4− k0,

j0 > 0 ↔ k0 > 0, j1 > 0 ↔ k0 > 1 Am′i−1,

Pm′

i = 8− j0

5 T4+jk,4+jl,4+jm m1 + l1 ≤ 4, k2 + m2 ≤ 4, Ami−1,P

mi = 4−m1 − l1,k3 + l3 ≤ 4, k2 > 0 ↔ k3 > 0, Am′

i−1,P

m′i = 4− k2 −m2,

l1 > 0 ↔ l3 > 0, m1 > 0 ↔ m2 > 0, Am′′i −1,

Pm′′

i = 4− k3 − l3

min(k2, k3) ≤ 1, min(l1, l3) ≤ 1,min(m1, m2) ≤ 1, jk = max(k2, k3),jl = max(l1, l3), jm = max(m1, m2)

6 U series j1 > 0 ↔ j2 > 0 ↔ j3 > 0, Ami−1,P

mi = 4− j1,at most one of j1, j2, j3 > 1, Am′

i−1,P

m′i = 4− j2,

j1, j2, j3 ≤ 4 Am′′i −1,

Pm′′

i = 4− j3

7 V series j0 > 0 ↔ k0 > 0, min(j0, k0) ≤ 1, Ami−1,P

mi = 4− j0,j0 ≤ 4, k0 ≤ 4

8 V ′ series None9 P8 = T3,3,3 Ami−1,

Pmi = 12

Table 5.1: Possible configurations of singularities for each case


Case 1. The tangent cone is a nodal irreducible curve, and we can assumef3(x1, x2, x3) = x1x2x3 + x3

2 + x33. The nodal curve is singular at (1 : 0 : 0).

If f4(1, 0, 0) 6= 0, then O is a T3,3,4 singularity [38]. We recall that (1 : 0 : 0)cannot be a singular point on Z(f4) as this would imply a singular line on themonoid, so we assume that either (1 : 0 : 0) 6∈ Z(f4) or (1 : 0 : 0) is a smoothpoint on Z(f4). Let m denote the intersection number I(1:0:0)(f3, f4). SinceZ(f3) is singular at (1 : 0 : 0) we have m 6= 1. From [38] we know that Ois a T3,3,3+m singularity for m = 2, . . . , 12. Note that some of these complexsingularities have two real forms, as illustrated in Figure 5.3.

Figure 5.3: The monoids Z(x3 + y3 + 5xyz− z3(x+ y)) and Z(x3 + y3 + 5xyz−z3(x − y)) both have a T3,3,5 singularity, but the singularities are not rightequivalent over R. (The pictures are generated by the program [8].)

Bezout’s theorem and Proposition 5.6 limit the possible configurations ofsingularities on the monoid for each m. Let θ(s, t) = (−s3 − t3, s2t, st2). Thenthe tangent cone Z(f3) is parameterized by θ as a map from P1 to P2. Whenwe need to compute the intersection numbers between the rational curve Z(f3)and the curve Z(f4), we can do that by studying the roots of the polynomial


f4(θ). Expanding the polynomial gives

f4(θ)(s, t) = a1s12 − a2s

11t+ (−a3 + a4)s10t2 + (4a1 + a5 − a7)s9t3

+ (−3a2 + a6 − a8 + a11)s8t4 + (−3a3 + 2a4 − a9 + a12)s7t5

+ (6a1 + 2a5 − a7 − a10 + a13)s6t6

+ (−3a2 + 2a6 − a8 + a14)s5t7 + (−3a3 + a4 − a9 + a15)s4t8

+ (4a1 + a5 − a10)s3t9 + (−a2 + a6)s2t10 − a3st11 + a1t

12.

This polynomial will have roots at (0 : 1) and (1 : 0) if and only if f4(1, 0, 0) =a1 = 0. When a1 = 0 we may (by symmetry) assume a3 6= 0, so that (0 : 1) isa simple root and (1 : 0) is a root of multiplicity m − 1. Other roots of f4(θ)correspond to intersections of Z(f3) and Z(f4) away from (1 : 0 : 0). The mul-tiplicity mi of each root is equal to the corresponding intersection multiplicity,giving rise to an Ami−1 singularity if mi > 0, as described by Proposition 5.6,or a line La ⊂ Z(F ) with O as the only singular point if mi = 1.

The polynomial f4(θ) defines a linear map from the coefficient space k15 of f4to the space of homogeneous polynomials of degree 12 in s and t. By elementarylinear algebra, we see that the image of this map is the set of polynomials ofthe form

b0s12 + b1s

11t+ b2s10t2 + · · ·+ b12t

12

where b0 = b12. The kernel of the map corresponds to the set of polynomials ofthe form `f3 where ` is a linear form. This means that f4(θ) ≡ 0 if and only iff3 is a factor in f4, making Z(F ) reducible and not a monoid.

For every m = 0, 2, 3, 4, . . . , 12 we can select r parameter points

p1, . . . , pr ∈ P1 \ {(0 : 1), (1 : 0)}

and positive multiplicities m1, . . . ,mr with m1 + · · ·+mr = 12−m and try todescribe the polynomials f4 such that f4(θ) has a root of multiplicity mi at pifor each i = 1, . . . , r.

Still assuming a3 6= 0 whenever a1 = 0, any such choice of parameter pointsp1, . . . , pr and multiplicities m1, . . . ,mr corresponds to a polynomial q = b0s

12+b1s

11t+ · · ·+ b12t12 that is, up to a nonzero constant, uniquely determined.

Now, q is equal to f4(θ) for some f4 if and only if b0 = b12. If m ≥ 2, then qcontains a factor stm−1, so b0 = b12 = 0, giving q = f4(θ) for some f4. In fact,when m ≥ 2 any choice of p1, . . . , pr and m1, . . . ,mr with m1+· · ·+mr = 12−mcorresponds to a four dimensional space of equations f4 that gives this set ofroots and multiplicities in f4(θ). If f ′4 is one such f4, then any other is of the


form λf ′4 + `f3 for some constant λ 6= 0 and linear form `. All of these givemonoids that are projectively equivalent.

When m = 0, we write pi = (αi : βi) for i = 1, . . . , r. The condition b0 = b12on the coefficients of q translates to

αm11 · · ·αmr

r = βm11 · · ·βmr

r . (5.3)

This means that any choice of parameter points (α1 : β1), . . . , (αr : βr) andmultiplicities m1, . . . ,mr with m1 + · · · +mr = 12 that satisfy condition (5.3)corresponds to a four dimensional family λf ′4 + `f3, giving a unique monoid upto projective equivalence.

For example, we can have an A11 singularity only if f4(θ) is of the form(αs−βt)12. Condition (5.3) implies that this can only happen for 12 parameterpoints, all of the form (1 : ω), where ω12 = 1. Each such parameter point (1 :ω) corresponds to a monoid uniquely determined up to projective equivalence.However, since there are six projective transformations of the plane that mapsZ(f3) onto itself, this correspondence is not one to one. If ω12

1 = ω122 = 1,

then ω1 and ω2 will correspond to projectively equivalent monoids if and only ifω3

1 = ω32 or ω3

1ω32 = 1. This means that there are three different quartic monoids

with one T3,3,4 singularity and one A11 singularity. One corresponds to thoseω where ω3 = 1, one to those ω where ω3 = −1, and one to those ω whereω6 = −1. The first two of these have real representatives, ω = ±1.

It easy to see that for any set of multiplicities m1 + · · · +mr = 12, we canfind real points p1, . . . , pr such that condition (5.3) is satisfied. This completelyclassifies the possible configurations of singularities when f3 is an irreduciblenodal curve.

Case 2. The tangent cone is a cuspidal curve, and we can assumef3(x1, x2, x3) = x3

1 − x22x3. The cuspidal curve is singular at (0 : 0 : 1) and can

be parameterized by θ as a map from P1 to P2 where θ(s, t) = (s2t, s3, t3). Theintersection numbers are determined by the degree 12 polynomial f4(θ). As inthe previous case, f4(θ) ≡ 0 if and only if f3 is a factor of f4, and we will assumethis is not the case. The multiplicity m of the factor s in f4(θ) determines thetype of singularity at O. If m = 0 (no factor s), then O is a Q10 singularity. Ifm = 2 or m = 3, then O is of type Q9+m. If m > 3, then (0 : 0 : 1) is a singularpoint on Z(f4), so the monoid has a singular line and is not considered in thisarticle. Also, m = 1 is not possible, since f4(θ(s, t)) = f4(s2t, s3, t3) cannotcontain st11 as a factor.

For each m = 0, 2, 3 we can analyze the possible configurations of othersingularities on the monoid. Similarly to the previous case, any choice of pa-rameter points p1, . . . , pr ∈ P1 \ {(0 : 1)} and positive multiplicities m1, . . . ,mr


with∑mi = 12−m corresponds, up to a nonzero constant, to a unique degree

12 polynomial q.When m = 2 or m = 3, for any choice of parameter values and associated

multiplicities, we can find a four dimensional family f4 = λf ′4 + `f3 with theprescribed roots in f4(θ). As before, the family gives projectively equivalentmonoids.

When m = 0, one condition must be satisfied for q to be of the form f4(θ),namely b11 = 0, where b11 is the coefficient of st11 in q.

For example, we can have an A11 singularity only if q is of the form (αs −βt)12. The condition b11 = 0 implies that either q = λs12 or q = λt12. Thefirst case gives a surface with a singular line, while the other gives a monoidwith an A11 singularity (see Figure 5.2). The line from O to the A11 singularitycorresponds to the inflection point of Z(f3).

For any set of multiplicities m1, . . . ,mr with m1 + · · · +mr = 12, it is nothard to see that there exist real points p1, . . . , pr such that the condition b11 = 0is satisfied. It suffices to take pi = (αi : 1), with

∑miαi = 0 (the condition

corresponding to b11 = 0). This completely classifies the possible configurationsof singularities when f3 is a cuspidal curve.

Case 3. The tangent cone is the product of a conic and a line that is nottangent to the conic, and we can assume f3 = x3(x1x2 + x2

3). Then Z(f3) issingular at (1 : 0 : 0) and (0 : 1 : 0), the intersections of the conic Z(x1x2 + x2

3)and the line Z(x3). For each f4 we can associate four integers:

j0 := I(1:0:0)(x1x2 + x23, f4), k0 := I(1:0:0)(x3, f4),

j1 := I(0:1:0)(x1x2 + x23, f4), k1 := I(0:1:0)(x3, f4).

We see that k0 > 0 ⇔ f4(1 : 0 : 0) = 0 ⇔ j0 > 0, and that Z(f4) is singularat (1 : 0 : 0) if and only if k0 and j0 both are bigger than one. These casesimply a singular line on the monoid, and are not considered in this article. Thesame holds for k1, j1 and the point (0 : 1 : 0).

Define ri = max(ji, ki) for i = 1, 2. Then, by [38], O will be a singularity oftype T3,4+r0,4+r1 if r0 ≤ r1, or of type T3,4+r1,4+r0 if r0 ≥ r1.

We can parameterize the line Z(x3) by θ1 where θ1(s, t) = (s, t, 0), and theconic Z(x1x2 +x2

3) by θ2 where θ2(s, t) = (s2,−t2, st). Similarly to the previouscases, roots of f4(θ1) correspond to intersections between Z(f4) and the lineZ(x3), while roots of f4(θ2) correspond to intersections between Z(f4) and theconic Z(x1x3 + x2

3).For any legal values of of j0, j1, k0 and k1, parameter points

(α1 : β1), . . . , (αmr : βmr ) ∈ P1 \ {(0 : 1), (1 : 0)},


with multiplicities m1, . . . ,mr such that m1 + · · · + mr = 4 − k0 − k1, andparameter points

(α′1 : β′1), . . . , (α′m′

r: β′m′

r) ∈ P1 \ {(0 : 1), (1 : 0)},

with multiplicities m′1, . . . ,m

′r′ such that m′

1 + · · ·+m′r′ = 8− j0 − j1, we can

fix polynomials q1 and q2 such that

• q1 is nonzero, of degree 4, and has factors sk1 , tk0 and (βis − αit)mi fori = 1, . . . , r,

• q2 is nonzero, of degree 8, and has factors sj1 , tj0 and (β′is − α′it)m′

i fori = 1, . . . , r′.

Now q1 and q2 are determined up to multiplication by nonzero constants. Writeq1 = b0s

4 + · · ·+ b4t4 and q2 = c0s

8 + · · ·+ c8t8.

The classification of singularities on the monoid consists of describing theconditions on the parameter points and nonzero constants λ1 and λ2 for thepair (λ1q1, λ2q2) to be on the form (f4(θ1), f4(θ2)) for some f4.

Similarly to the previous cases, f4(θ1) ≡ 0 if and only if x3 is a factor in f4and f4(θ2) ≡ 0 if and only if x1x2+x2

3 is a factor in f4. Since f3 = x3(x1x2+x23),

both cases will make the monoid reducible, so we only consider λ1, λ2 6= 0.We use linear algebra to study the relationship between the coefficients

a1 . . . a15 of f4 and the polynomials q1 and q2. We find (λ1q1, λ2q2) to be ofthe form (f4(θ1), f4(θ2)) if and only if λ1b0 = λ2c0 and λ1b4 = λ2c8. Further-more, the pair (λ1q1, λ2q2) will fix f4 modulo f3. Since f4 and λf4 correspondto projectively equivalent monoids for any λ 6= 0, it is the ratio λ1/λ2, and notλ1 and λ2, that is important.

Recall that k0 > 0 ⇔ j0 > 0 and k1 > 0 ⇔ j1 > 0. If k0 > 0 and k1 > 0,then b0 = c0 = b4 = c8 = 0, so for any λ1, λ2 6= 0 we have (λ1q1, λ2q2) =(f4(θ1), f4(θ2)) for some f4. Varying λ1/λ2 will give a one-parameter family ofmonoids for each choice of multiplicities and parameter points.

If k0 = 0 and k1 > 0, then b0 = c0 = 0. The condition λ1b4 = λ2c8 impliesλ1/λ2 = c8/b4. This means that any choice of multiplicities and parameterpoints will give a unique monoid up to projective equivalence. The same goesfor the case where k0 > 0 and k1 = 0.

Finally, consider the case where k0 = k1 = 0. For (λ1q1, λ2q2) to be of theform (f4(θ1), f4(θ2)) we must have λ1/λ2 = c8/b4 = c0/b0. This translates intoa condition on the parameter points, namely

(β′1)m′

1 · · · (β′r′)m′r′

βm11 · · ·βmr

r=

(α′1)m′

1 · · · (α′r′)m′r′

αm11 · · ·αmr

r. (5.4)


In other words, if condition (5.4) holds, we have a unique monoid up to projectiveequivalence.

It is easy to see that for any choice of multiplicities, it is possible to findreal parameter points such that condition (5.4) is satisfied. This completes theclassification of possible singularities when the tangent cone is a conic plus achordal line.

Case 4. The tangent cone is the product of a conic and a line tangent tothe conic, and we can assume f3 = x3(x1x3 + x2

2). Now Z(f3) is singular at(1 : 0 : 0). For each f4 we can associate two integers

j0 := I(1:0:0)(x1x3 + x22, f4) and k0 := I(1:0:0)(x3, f4).

We have j0 > 0 ⇔ k0 > 0, j0 > 1 ⇔ k0 > 1. Furthermore, j0 and k0 areboth greater than 2 if and only if Z(f4) is singular at (1 : 0 : 0), a case we haveexcluded. The singularity at O will be of the S series, from [1], [2].

We can parameterize the conic Z(x1x3 + x22) by θ2 and the line Z(x3) by

θ1 where θ2(s, t) = (s2, st,−t2) and θ1(s, t) = (s, t, 0). As in the previous case,the monoid is reducible if and only if f4(θ1) ≡ 0 or f4(θ2) ≡ 0. Consider twononzero polynomials

q1 = b0s4 + b1s

3t+ b2s2t2 + b3st

3 + b4t4

q2 = c0s8 + c1s

7t+ · · ·+ c7st7 + c8t

8.

Now (λ1q1, λ2q2) = (f4(θ1), f4(θ2)) for some f4 if and only if λ1b0 = λ2c0 andλ1b1 = λ2c1. As before, only the cases where λ1, λ2 6= 0 are interesting. Wesee that (λ1q1, λ2q2) = (f4(θ1), f4(θ2)) for some λ1, λ2 6= 0 if and only if thefollowing hold:

• b0 = 0 ↔ c0 = 0 and b1 = 0 ↔ c1 = 0

• b0c1 = b1c0.

The classification of other singularities (than O) is very similar to the previ-ous case. Roots of f4(θ1) and f4(θ2) away from (1 : 0) correspond to intersectionsof Z(f3) and Z(f4) away from the singular point of Z(f3), and when one suchintersection is multiple, there is a corresponding singularity on the monoid.

Now assume (λ1q1, λ2q2) = (f4(θ1), f4(θ2)) for some λ1, λ2 6= 0 and somef4. If b0 6= 0 (equivalent to c0 6= 0) then j0 = k0 = 0 and λ1/λ2 = c0/b0.If b0 = c0 = 0 and b1 6= 0 (equivalent to c1 6= 0), then j0 = k0 = 1, andλ1/λ2 = c1/b1. If b0 = b1 = c0 = c1 = 0, then j0, k0 > 1 and any value of λ1/λ2


will give (λ1q1, λ2q2) of the form (f4(θ1), f4(θ2)) for some f4. Thus we get aone-dimensional family of monoids for this choice of q1 and q2.

Now consider the possible configurations of other singularities on the monoid.Assume that j′0 ≤ 8 and k′0 ≤ 4 are nonnegative integers such that j0 > 0 ↔k0 > 0 and j0 > 1 ↔ k0 > 1. For any set of multiplicities m1, . . . ,mr withm1 + · · · + mr = 4 − k′0 and m′

1, . . . ,m′r′ with m′

1 + · · · + m′r′ = 8 − j′0, there

exists a polynomial f4 with real coefficients such that f4(θ1) has real roots awayfrom (1 : 0) with multiplicities m1, . . . ,mr, and f4(θ2) has real roots away from(1 : 0) with multiplicities m′

1, . . . ,m′r′ . Furthermore, for this f4 we have k0 = k′0

and j0 = j′0. Proposition 5.6 will give the singularities that occur in addition toO.

This completes the classification of the singularities on a quartic monoid(other than O) when the tangent cone is a conic plus a tangent.

Case 5. The tangent cone is three general lines, and we assume f3 = x1x2x3.For each f4 we associate six integers,

k2 := I(1,0,0)(f4, x2), l1 := I(0,1,0)(f4, x1), m1 := I(0,0,1)(f4, x1),k3 := I(1,0,0)(f4, x3), l3 := I(0,1,0)(f4, x3), m2 := I(0,0,1)(f4, x2).

Now k2 > 0 ⇔ k3 > 0, l1 > 0 ⇔ l3 > 0, and m1 > 0 ⇔ m2 > 0. If both k2

and k3 are greater than 1, then the monoid has a singular line, a case we haveexcluded. The same goes for the pairs (l1, l3) and (m1,m2).

When the monoid does not have a singular line, we define jk = max(k2, k3),jl = max(l1, l3) and jm = max(m1,m2). If jk ≤ jl ≤ jm, then [38] gives that Ois a T4+jk,4+jl,4+jm singularity.

The three lines Z(x1), Z(x2) and Z(x3) are parameterized by θ1, θ2 and θ3where θ1(s, t) = (0, s, t), θ2(s, t) = (s, 0, t) and θ3(s, t) = (s, t, 0). Roots of thepolynomial f4(θi) away from (1 : 0) and (0 : 1) correspond to intersectionsbetween Z(f4) and Z(xi) away from the singular points of Z(f3).

As before, we are only interested in the cases where none of f4(θi) ≡ 0 fori = 1, 2, 3, as this would make the monoid reducible.

For the study of other singularities on the monoid we consider nonzero poly-nomials

q1 = b0s4 + b1s

3t+ b2s2t2 + b3st

3 + b4t4,

q2 = c0s4 + c1s

3t+ c2s2t2 + c3st

3 + c4t4,

q3 = d0s4 + d1s

3t+ d2s2t2 + d3st

3 + d4t4.

Linear algebra shows that (λ1q1, λ2q2, λ3q3) = (f4(θ1), f4(θ2), f4(θ3)) for somef4 if and only if λ1b0 = λ3d4, λ1b4 = λ2c4, and λ2c0 = λ3d0. A simple analysis


shows the following: There exist λ1, λ2, λ3 6= 0 such that

(λ1q1, λ2q2, λ3q3) = (f4(θ1), f4(θ2), f4(θ3))

for some f4, and such that Z(f4) and Z(f3) have no common singular point ifand only if all of the following hold:

• b0 = 0 ↔ d4 = 0 and b0 = d4 = 0 → (b1 6= 0 or d3 6= 0),

• b4 = 0 ↔ c4 = 0 and b4 = c4 = 0 → (b3 6= 0 or c3 6= 0),

• c0 = 4 ↔ d0 = 0 and c0 = d0 = 0 → (c1 6= 0 or d1 6= 0),

• b0c4d0 = b4c0d4.

Similarly to the previous cases we can classify the possible configurations ofother singularities by varying the multiplicities of the roots of the polynomialsq1, q2 and q3. Only the multiplicities of the roots (0 : 1) and (1 : 0) affectthe first three bullet points above. Then, for any set of multiplicities of therest of the roots, we can find q1, q2 and q3 such that the last bullet point issatisfied. This completes the classification when Z(f3) is the product of threegeneral lines.

Case 6. The tangent cone is three lines meeting in a point, and we canassume that f3 = x3

2 − x2x23. We write f3 = `1`2`3 where `1 = x2, `2 = x2 − x3

and `3 = x2 + x3, representing the three lines going through the singular point(1 : 0 : 0). For each f4 we associate three integers j1, j2 and j3 defined as theintersection numbers ji = I(1:0:0)(f4, `i). We see that j1 = 0 ⇔ j2 = 0 ⇔ j3 = 0,and that Z(f4) is singular at (1 : 0 : 0) if and only if two of the integers j1,j2, j3 are greater then one. (Then all of them will be greater than one.) Thesingularity will be of the U series [1], [2].

The three lines Z(`1), Z(`2) and Z(`3) can be parameterized by θ1, θ2, andθ3 where θ1(s, t) = (s, 0, t), θ2(s, t) = (s, t, t) and θ2(s, t) = (s, t,−t).

For the study of other singularities on the monoid we consider nonzero poly-nomials

q1 = b0s4 + b1s

3t+ b2s2t2 + b3st

3 + b4t4,

q2 = c0s4 + c1s

3t+ c2s2t2 + c3st

3 + c4t4,

q3 = d0s4 + d1s

3t+ d2s2t2 + d3st

3 + d4t4.

Linear algebra shows that (λ1q1, λ2q2, λ3q3) = (f4(θ1), f4(θ2), f4(θ3)) for somef4 if and only if λ1b0 = λ2c4 = λ3d0, and 2λ1b1 = λ2c1 + λ3d1. There exist


λ1, λ2, λ3 6= 0 such that (λ1q1, λ2q2, λ3q3) = (f4(θ1), f4(θ2), f4(θ3)) for some f4and such that Z(f4) and Z(f3) have no common singular point if and only if allof the following hold:

• b0 = 0 ↔ c0 = 0 ↔ d0 = 0,

• if b0 = c0 = d0 = 0, then at least two of b1, c1, and d1 are different fromzero,

• 2b1c0d0 = b0c1d0 + b0c0d1.

As in all the previous cases we can classify the possible configurations ofother singularities for all possible j1, j2, j3. As before, the first bullet pointonly affect the multiplicity of the factor t in q1, q2 and q3. For any set ofmultiplicities for the rest of the roots, we can find q1, q2, q3 with real roots ofthe given multiplicities such that the last bullet point is satisfied. This completesthe classification of the singularities (other than O) when Z(f3) is three linesmeeting in a point.

Case 7. The tangent cone is a double line plus a line, and we can assumef3 = x2x

23. The tangent cone is singular along the line Z(x3). The line Z(x2) is

parameterized by θ1 and the line Z(x3) is parameterized by θ2 where θ1(s, t) =(s, 0, t) and θ2(s, t) = (s, t, 0). The monoid is reducible if and only if f4(θ1)or f4(θ2) is identically zero, so we assume that neither is identically zero. Foreach f4 we associate two integers, j0 := I(1:0:0)(f4, x2) and k0 := I(1:0:0)(f4, x3).Furthermore, we write f4(θ2) as a product of linear factors

f4(θ2) = λsk0r∏i=0

(αis− t)mi .

Now the singularity at O will be of the V series and depends on j0, k0 andm1, . . . ,mr.

Other singularities on the monoid correspond to intersections of Z(f4) andthe line Z(x2) away from (1 : 0 : 0). Each such intersection corresponds to aroot in the polynomial f4(θ1) different from (1 : 0). Let j′0 ≤ 4 and k′0 ≤ 4 beintegers such that j0 > 0 ↔ k0 > 0. Then, for any homogeneous polynomialsq1, q2 in s, t of degree 4 such that s is a factor of multiplicity j′0 in q1 and ofmultiplicity k′0 in q2, there is a polynomial f4 and nonzero constants λ1 and λ2

such that k0 = k′0, j0 = j′0 and (λ1q1, λ2q2) = (f4(θ1), f4(θ2)). Furthermore,if q1 and q2 have real coefficients, then f4 can be selected with real coeficients.This follows from an analysis similar to case 5 and completes the classificationof singularities when the tangent cone is a product of a line and a double line.


Case 8. The tangent cone is a triple line, and we assume that f3 = x33.

The line Z(x3) is parameterized by θ where θ(s, t) = (s, t, 0). Assume thatthe polynomial f4(θ) has r distinct roots with multiplicities m1, . . . ,mr. (Asbefore f4(θ) ≡ 0 if and only if the monoid is reducible.) Then the type of thesingularity at O will be of the V ′ series [3, p. 267]. The integers m1, . . . ,mr areconstant under right equivalence over C. Note that one can construct examplesof monoids that are right equivalent over C, but not over R (see Figure 5.4).

Figure 5.4: The monoids Z(z3 + xy3 + x3y) and Z(z3 + xy3 − x3y) are rightequivalent over C but not over R.

The tangent cone is singular everywhere, so there can be no other singular-ities on the monoid.

Case 9. The tangent cone is a smooth cubic curve, and we write f3 =x3

1 + x32 + x3

3 + 3ax1x2x3 where a3 6= −1. This is a one-parameter family ofelliptic curves, so we cannot use the parameterization technique of the othercases. The singularity at O will be a P8 singularity (cf. [3, p. 185]), and othersingularities correspond to intersections between Z(f3) and Z(f4), as describedby Proposition 5.6.

To classify the possible configurations of singularities on a monoid with anonsingular (projective) tangent cone, we need to answer the following ques-tion: For any positive integers m1, . . . ,mr such that

∑ri=1mi = 12, does there,

for some a ∈ R \ {−1}, exist a polynomial f4 with real coefficients such thatZ(f3, f4) = {p1, . . . , pr} ∈ P2(R) and Ipi(f3, f4) = mi for i = 1, . . . , r? Rohn


[32, p. 63] says that one can always find curves Z(f3), Z(f4) with this property.Here we shall show that for any a ∈ R \ {−1} we can find a suitable f4.

In fact, in almost all cases f4 can be constructed as a product of linear andquadratic terms in a simple way. The difficult cases are (m1,m2) = (11, 1),(m1,m2,m3) = (8, 3, 1), and (m1,m2) = (5, 7). For example, the case where(m1,m2,m3) = (3, 4, 5) can be constructed as follows: Let f4 = `1`2`

23 where `1

and `2 define tangent lines at inflection points p1 and p3 of Z(f3). Let `3 definea line that intersects Z(f3) once at p3 and twice at another point p2. Note thatthe points p1, p2 and p3 can be found for any a ∈ R \ {−1}.

The case (m1,m2) = (11, 1) is also possible for every a ∈ R \ {−1}. For anypoint p on Z(f3) there exists an f4 such that Ip(f3, f4) ≥ 11. For all except afinite number of points, we have equality [25], so the case (m1,m2) = (11, 1) ispossible for any a ∈ R \ {−1}. The case (m1,m2,m3) = (8, 3, 1) is similar, butwe need to let f4 be a product of the tangent at an inflection point with anothercubic.

The case (m1,m2) = (5, 7) is harder. Let a = 0. Then we can construct aconic C that intersects Z(f3) with multiplicity five in one point and multiplicityone in an inflection point, and choosing Z(f4) as the union of C and twice thetangent line through the inflection point will give the desired example. The samecan be done for a = −4/3. By using the computer algebra system Singular[11] we can show that these constructions can be continuously extended to anya ∈ R \ {−1}. This completes the classification of singularities on a monoidwhen the tangent cone is smooth.

In the Cases 3, 5, and 6, not all real equations of a given type can betransformed to the chosen forms by a real transformation.

In Case 3 the conic may not intersect the line in two real points, but ratherin two complex conjugate points. Then we can assume f3 = x3(x1x3 +x2

1 +x22),

and the singular points are (1 : ±i : 0). For any real f4, we must have

I(1:i:0)(x1x3 + x21 + x2

2, f4) = I(1:−i:0)(x1x3 + x21 + x2

2, f4)

andI(1:i:0)(x3, f4) = I(1:−i:0)(x3, f4),

so only the cases where j0 = j1 and k0 = k1 are possible. Apart from that, noother restrictions apply.

In Case 5, two of the lines can be complex conjugate, and we assume f3 =x3(x2

1 + x22). A configuration from the previous analysis is possible for real

coefficients of f4 if and only if m1 = m2, k2 = l1, and k3 = l3. Furthermore,only the singularities that correspond to the line Z(x3) will be real.


In Case 6, two of the lines can be complex conjugate, and then we mayassume f3 = x3

2 + x33. Now, if j3 denotes the intersection number of Z(f4) with

the real line Z(x2 +x3), precisely the cases where j1 = j2 are possible, and onlyintersections with the line Z(x2 + x3) may contribute to real singularities.

This concludes the classification of real and complex singularities on realmonoids of degree 4.

Remark. In order to describe the various monoid singularities, Rohn [32]computes the “class reduction” due to the presence of the singularity, in (almost)all cases. (The class is the degree of the dual surface [30, p. 262].) The classreduction is equal to the local intersection multiplicity of the surface with twogeneral polar surfaces. This intersection multiplicity is equal to the sum of theMilnor number and the Milnor number of a general plane section through thesingular point [39, Cor. 1.5, p. 320]. It is not hard to see that a general planesection has either a D4 (Cases 1–6, 9), D5 (Case 7), or E6 (Case 8) singularity.Therefore one can retrieve the Milnor number of each monoid singularity fromRohn’s work.

Chapter 6

The strata of quarticmonoids

In this chapter we will define the strata of a quartic monoid surfaces in P3 witha fixed triple point, O = (1 : 0 : 0 : 0). Then we will calculate the dimensionof all strata and the number of components of each stratum where the tangentcone is not of generic type. We use Definition 5.2 as the definition of a monoid- monoids have only isolated singularities.

6.1 Definition of the strata

Let X be a quartic monoid surface (with only isolated singular points) whereO ∈ X is the point of multiplicity 3. Furthermore, let F be an equation of X,and write F = x0f3 + f4 where f3, f4 ∈ k[x1, x2, x3], as before.

The plane curves Z(f3) and Z(f4) will be named as in Definition 5.2, Z(f3)is the tangent cone and Z(f4) is the intersection with infinity.

We will now define the set of invariants for a given monoid X = Z(F ). Foreach quartic monoid surface X = Z(F ) in P3 there is an associated type j of thetangent cone, as given by the list in Section 5.4 on page 56. Depending on thetype j, the tangent cone has a number of irreducible components with associatedmultiplicities. Also, the tangent cone has a set of special points, denoted σX .The number and types of these points depend only on the type j. For the typeswhere the tangent cone has only isolated singularities, σX is defined as the setof singular points. For type 7, the set σX is defined as the intersection of the

71

72 CHAPTER 6. THE STRATA OF QUARTIC MONOIDS

two irreducible components of the tangent cone. Furthermore, for type 8 we setσX = ∅, the triple line is considered to have no special points.

In addition to the type j we associate to X one invariant for every pair (p, C)where p ∈ σX is a special point and C = Z(h) is an irreducible component ofZ(f3) containing p. This invariant is defined as ip,C := Ip(h, f4). Note that theinvariants correspond to integers in Table 5.1.

Furthermore, for each component C = Z(h) of the tangent cone, the monoidX has an associated partition m1, . . . ,mr of

4 deg(h)−∑p∈σX

Ip(f4, h) (6.1)

determined by the intersection multiplicities between C and Z(f4) away fromσX . For all partitions we assume m1 ≥ m2 ≥ · · · ≥ mr ≥ 1, and we allow r = 0(then m1, . . . ,mr is empty and a partition of 0). Thus every partition has aunique representation m1, . . . ,mr.

The stratum of X is defined as the set of quartic monoid hypersurfaces thathave the same type, the same invariants ip,C and the same partition(s) as X.To be precise, X and Y are in the same stratum if and only if the tangent conesare of the same type and there is a bijection Φ that associates components ofX to components of Y of the same type and multiplicity and special points ofX to special points of Y , such that

• the point p ∈ σX is on a component C ⊆ X if and only if the point Φ(p)is on the component Φ(C)

• ip,C = iΦ(p),Φ(C) for all points p ∈ σX and all components C ⊆ X

• for each component C ⊆ X the associated partition of C is equal to theassociated partition of Φ(C)

The set of monoid quartic surfaces is a subset of the set of quartic surfaces,usually identified with the coefficient space P34. It is not hard to see that thedimension of the space of monoids where the triple point is fixed at O is 24, andthe space of monoids where the triple point can vary is of dimension 27.

We define the space S as the space of quartic monoids with a triple point atO and only isolated singularities. The set S is viewed as a subset of the space(A10 \ {0})× (A15 \ {0})/ ∼ ⊂ P24, where the set A10 \ {0} corresponds to thecoefficients of f3, the set A15 \ {0} corresponds to the coefficients of f4 and theequivalence relation ∼ is defined in the usual way – F is equivalent to F ′ if and

6.2. TYPES 1 TO 8 73

only if F = λF ′ for some λ 6= 0. We will write [F ] for the monoid in S definedby the equation F = x0f3 + f4 throughout this chapter.

Furthermore, S is an open subset of P24. Indeed, the complement of S is theunion of the following closed sets:

• the set of monoids Z(F ) where Z(fi) for i = 3, 4 has a common singularpoint. (Let X ⊂ P2×P24 be the set of pairs (x, F ) such that x is a singularpoint of both Z(f3) and Z(f4). This set is defined by equations on the form∂fi

∂xjfor i = 3, 4 and j = 1, 2, 3, and is thus closed. The projection to the

second factor is also closed.)

• the set corresponding to equations F where f3 and f4 has a common linearfactor (equal to the image of a closed set by a projective morphism, asabove)

• the set corresponding to equations F where f3 and f4 has a commonquadratic factor (equal to the image of a closed set by a projective mor-phism, as above)

• the set corresponding to equations F where f4 = `f3 where ` is a linearform or zero

• the set corresponding to f3 ≡ 0

6.2 Types 1 to 8

For the types 1 to 8, each stratum S can be characterized as the image of acertain morphism γS . In this section the morphisms γS will be constructedand used to calculate the dimension and the number of components of S foreach stratum. Before doing a systematic analysis of the strata, we will describethe general strategy and carry out some preliminary calculations. The purposeof these calculations is to obtain a formula for the dimension of each S. Thisformula will be correct for a stratum S if some properties of γS are proved.

For every S we will define γS as the composition γS = ψ ◦ (ϕS × id):

γS : BS × G(ϕS×id)−−−−−→ S× G ψ−−−−→ S

Each BS will be defined to be equal to, or a hypersurface in, a product ofone or more of the following:

• Ar∆ := Ar \⋃i 6=j Z(ui − uj)


• Ar∆,0 := Ar \⋃i 6=j Z(ui − uj) \

⋃i Z(ui)

• Ar0 := Ar \⋃i Z(ui)

• kr := Ar

where u1, . . . , ur are the coordinate functions on Ar. Thus BS will be a locallyclosed, not necessarily irreducible variety in some affine space. Furthermore,since BS can be defined by not more than one equation, we see the that BS isof pure dimension.

We define G to be the group of projective transformations of P3 fixing O,and let ψ be the action of G on S. We consider G as a subgroup of PGL(4) andsee that G is the set of elements in PGL(4) that can be written as a matrix onthe form

1 a1,1 a1,2 a1,3

0 a2,1 a2,2 a2,3

0 a3,1 a3,2 a3,3

0 a4,1 a4,2 a4,3

.

Since such a matrix representative is unique as an element of PGL(4) we seethat G is of dimension 12.

One possible way of constructing the variety BS and the map γS is to let BSparameterize, through ϕS , the monoids in S up to projective equivalence, andlet G correspond to projective equivalence. The construction of BS and ϕS willindeed be such that all monoids of a stratum S will be projectively equivalentto a monoid in ϕS(BS), but it will typically not be true that all the monoids inϕS(BS) will be projectively different. The construction of BS and ϕS will beexplained more thoroughly later. With this in mind, we will state some basicproperties that will allow us to write down a formula for the dimension of S.

The morphisms ϕS will be defined such that BS and ϕS(BS) have the samenumber of components. Furthermore, for types 1 to 7 the fibers of ϕS will allbe finite, so the dimension of ϕS(BS) will be equal to the dimension of BS . Forstrata of type 8 the dimension of the fibers of ϕS will vary.

Consider the case that BS is reducible and has c components. Let B(i)S for

i = 1, . . . , c denote the components of BS . These are all hypersurfaces, onefor each factor of the equation defining BS , and the intersection of two suchhypersurfaces will always be empty. This will be easy to see from the equationsof the components. We define γ(i)

S as the restriction of γS to B(i)S for i = 1, . . . , c.

The dimension of S will then be the maximum of the dimension of the imagesof γ(i)

S .

6.2. TYPES 1 TO 8 75

In fact, each image γ(i)S (B(i)

S ) will be pure dimensional of the same dimen-sion. To see this, it is sufficient (and slightly stronger) to check that the fiberγ−1S (γS(b, g)) is pure dimensional and of the same dimension for any b ∈ BS

and g ∈ G: If b ∈ B(i)S , then (γ(i)

S )−1(γ(i)S (b, g)) = γS

−1(γS(b, g)) ∩ B(i)S . Since

the components of BS are disjoint, each of the components of γ−1S (γS(b, g))

will either not intersect, or be contained in, B(i)S . Now, if γ−1

S (γS(b, g)) is puredimensional, it follows that the fiber (γ(i)

S )−1(γ(i)S (b, g)) is pure dimensional and

of the same dimension.For any irreducible component B(i)

S ×G ⊂ BS×G we know that the dimensionof the image is equal to the dimension of BS × G minus the dimension of thefibers of γ(i)

S . From this we can give a formula for the dimension of the imagesof γ(i)

S and therefore also for S:

dim γ(i)S (B(i)

S ) = dimS = dimBS + 12− dim γ−1S γS(b, g) (6.2)

where b ∈ BS and g ∈ G.Note that γS may identify some of the components B(i)

S . So, to find thenumber of components of S, we need to check when γS(B(i)

S ) = γS(B(j)S ) for

i 6= j.

When studying the map γS to find the dimension and components of S, wewill need to calculate the fiber γ−1

S (γS(b, g)) for fixed (b, g) ∈ BS × G. This isequivalent to solving the equation

γS(b, g) = γS(b, g) (6.3)

for fixed (b, g) ∈ BS × G.We will now describe the strategy for constructing the sets BS and mor-

phisms ϕS(BS), and then carry out some of the calculations needed to solve(6.3).

In the classification of the quartic monoids we fix the tangent cone for eachtype j = 1, . . . , 8. This can be done up to projective transformation. Thus,we can, and will, for each type j = 1, . . . , 8, let the tangent cone of everymonoid in the image ϕS(BS) be the same, and be described by one polynomialf

(j)3 . This polynomial will be equal to f3 in the corresponding case in Chapter 5.

Furthermore, for each type j we will select one monomial xe11 xe22 x

e33 with nonzero

coefficient in f (j)3 and, for each stratum S of type j, define ϕS such that the image

ϕS(BS) contains only monoids that can be described by a polynomial F whose


coefficients of xe1+11 xe22 x

e33 , xe11 x

e2+12 xe33 and xe11 x

e22 x

e3+13 are all zero. Note that

these choices can be made, as any monoid is projectively equivalent to a monoid[F ] where the coefficients of xe1+1

1 xe22 xe33 , xe11 x


e22 x

e3+13 are all

zero. Indeed, assuming that the coefficient of xe11 xe22 x

e33 in f3 is 1, the monoid

x0f3 + f4 where the coefficients of xe1+11 xe22 x

e33 , xe11 x


e22 x

e3+13

in f4 are c1, c2 and c3, respectively, is projectively equivalent to the monoid(x0 − c1x1 − c2x2 − c3x3)f3 + f4 having the special coefficients equal to zero.

For types 1 to 7 we also define ϕS such that if the (projectively equivalent)monoids defined by x0f3 + f4 and x0f3 + λf4 are both in the image ϕS(BS),then λ = 1 (so they are the same). For type 8 we define ϕS such that [x0f3 +f4]is in the image of ϕS if and only if [x0f3 + λf4] is in the image for all λ ∈ k∗.

We will now explain the idea of the construction of BS and ϕS . Assume thatthe tangent cone Z(f (j)

3 ) decomposes into r ≥ 1 components C1, . . . , Cr and letθi for n = 1, . . . , r be as in the case-by-case proof of Theorem 5.9. Recall thatθn, viewed as a map from P1 to P2, parameterizes Cn.

A stratum S of type j is defined as the set of monoids having a specific set ofinvariants ip,Cn (equal to intersection numbers) and partitions associated to thecomponents C1, . . . , Cr. Each invariant ip,Cn

corresponds to to the multiplicityof a specific root (depending on p) of f4(θn(s, t)). The partition associated toCn corresponds to other roots of f4(θn(s, t)).

If two monoids [x0f(j)3 + f4] and [x0f

(j)3 + f4] are such that the polynomials

f4(θn(s, t)) = f4(θn(s, t)) for each n, then f4 − f4 is a multiple of g(j), whereg(j) = f

(j)3 for j = 1, . . . , 6, g(7) = x2x3 and g(8) = x3.

The classification in Chapter 5 (Theorem 5.9) tells us which set of polyno-mials {qn} are on the form {f4(θi(s, t))}. This enables us to define BS and ϕSsuch that the image of γS is exactly S.

When f (j)3 = g(j) the maps ϕS assign, to an element b ∈ BS , a unique monoid

[x0f(j)3 + f4] satisfying the properties above (specific coefficients equal to zero)

such that f4(θn) = qn(b) for each component Cn. The set BS and morphismϕS is constructed such that {qi(b)}b∈BS

runs through all possible polynomialson the form {f4(θi(s, t))} for the given stratum S, up to multiplication by anonzero constant and up to symmetry in the variables x1, x2 and x3. The roleof the symmetries will be clear later, in the cases where {qi(b)}b∈BS

does notrun through all possible polynomials on the form {f4(θi(s, t))}.

Still assuming f (j)3 = g(j), any two monoids [x0f

(j)3 + f4] and [x0f

(j)3 + f4]

such that f4(θi(s, t)) = λf4(θi(s, t)) for i = 1, . . . , r are projectively equivalent:f4 − λf4 is a multiple of g(j) = f

(j)3 . Thus the image of γS is equal to S.

6.2. TYPES 1 TO 8 77

When f(j)3 6= g(j) we use a similar construction, where we can easily check

that every monoid with any possible set of {f4(θn(s, t))} is projectively equiva-lent to a monoid in ϕS(BS).

As we will se in the systematic analysis of the strata, some of coordinatesin BS will correspond to roots of the polynomials {qn(b)}. The conditions on{qn(b)} to be on the form {f4(θn(s, t))} are linear conditions, but the pullbackof these will typically not be linear conditions on BS . Still, the sets BS can andwill be constructed to be (at most) hypersurfaces, and the equations of thesecan be easily factorized, giving the components of BS .

For each type j, define Mj as the group of invertible 3 × 3 matrices Msuch that f3(x) = f3(Mx). Here we let x be the column vector (x1, x2, x3)T

and abuse the notation of f3 a little. Note that Mj can be considered as anot necessarily irreducible locally closed variety in A9, where f3(x) = f3(Mx)represents the closed condition, while detM 6= 0 represents the open conditionon Mj .

Now, (b, g) ∈ γ−1S (γS(b, g)) is equivalent to γS(b, g) = γS(b, g), which is

equivalent to ϕS(b) = ψ(ϕ(b), g−1g). We write ϕ(b) = [x0f3(x) + f4(x)] ∈ Sand ϕ(b) = [x0f3(x) + f4(x)] ∈ S for the rest of this section.

Define h = g−1g and write h on the form1 a1,1 a1,2 a1,3

00 A0

(6.4)

By definition we have

ψ(ϕ(b), h) = [(x0 + a1,1x1 + a1,2x2 + a1,3x3)f3(Ax) + f4(Ax)]

= [x0f3(Ax) + (a1,1x1 + a1,2x2 + a1,3x3)f3(Ax) + f4(Ax)].

If ϕS(b) = ψ(ϕ(b), h), then it is clear that A = λM for some λ ∈ k∗ andsome M ∈M. We write ` = a1,1x1 + a1,2x2 + a1,3x3 and see that

ψ(ϕ(b), h) = [x0f3(λMx) + `f3(λMx) + f4(λMx)]

= [λ3x0f3(Mx) + λ3`f3(Mx) + λ4f4(Mx)]

= [x0f3(Mx) + `f3(Mx) + λf4(Mx)]

= [x0f3(x) + `f3(Mx) + λf4(Mx)].


Now assume that ψ(ϕ(b), h) = ϕ(b) for some b ∈ BS . From the conditionthat the coefficients of xe1+1

1 xe22 xe33 , xe11 x


e22 x

e3+13 in ϕ(b) are

all zero we see that ` is uniquely determined by λ and M , and can be thewritten on the form λ`1(M) where `1(M) only depends on f4 and M . This givesψ(ϕ(b), h) = [x0f3(x) + λ(`1(M)f3(Mx) + f4(Mx))]. Uniqueness of elementson the form [x0f3 + λf4] for strata of type 1 to 7 implies that only one λ ispossible for each M for types 1 to 7.

We define MS,b,g as the set of matrices M ∈ M such that there exists(b, g) ∈ BS × G such that γS(b, g) = γS(b, g) and h := g−1g can be written onthe form (6.4) where A = λM for some λ ∈ k∗. This set is clearly independentof g, so we define MS,b := MS,b,g for any g ∈ G. A priori, the set MS,b

is not necessarily a subgroup of Mj . However, we will check that for anystratum S and any b ∈ BS , MS,b is both a subgroup and a subvariety of Mj

of codimension zero.Note that if M ∈ Mj and ηM ∈ Mj for some η ∈ k, then f

(j)3 (x) =

f(j)3 (ηMx) = η3f

(j)3 (Mx) = η3f

(j)3 (x), so η3 = 1. Furthermore, we see that

M ∈ Mj,b if and only if ω3M ∈ Mj,b, where ω3 is a primitive third root ofunity.

This means that when S is a stratum of type 1 to 7, then for each (b, g) ∈γ−1S (γ(b, g)) there are exactly 3 corresponding elements in MS,b. If M is one

such element, then ω3M and ω23M are the other two. It follows that the dimen-

sion of the fiber γ−1S (γS(b, g)) is equal to the dimension of MS,b. This is again

equal to the dimension of Mj , so

dim γ−1S (γS(b, g)) = dimMj

for every stratum S of type 1, . . . , 7.Given this dimension formula we can write down a formula for the dimension

of S (of type 1 to 7):

dimS = dim(BS) + dim(G)− dim γ−1S γ = dimBS + 12− dimMj (6.5)

We will now consider the strata of S. For each type j we will calculate Mj

by using the computer algebra system Singular [11]. (In fact, we will just listthe elements of Mj or it generators, but the calculations are carried out by asimple Singular script.) Then, for each stratum S of type j, we will define BSand ϕS , and calculate the components of BS . We will calculate the set MS,b

6.2. TYPES 1 TO 8 79

for each b ∈ BS (and check that MS,b is in fact both a subgroup and subvarietyof Mj of codimension zero) and check if γS identifies any of the components ofBS . This will give the number of components of S, and we can safely calculatethe dimension of S using (6.5).

Primitive roots of unity will be used and we will write ωi for a primitive ithroot of unity. This means that ωji = 1 if and only if j is a multiple of i.

In cases j = 1, . . . , 6 the polynomial f (j)3 is square free, and we have parame-

terizations (s, t) → θi(s, t) for i = 1, . . . , rj of the components of Z(f (j)3 ). Then,

as we have seen in the classification of quartic monoids, f4 = f4 + `f(j)3 for some

linear form ` if and only if f4(θi(s, t)) = f4(θi(s, t)) for all i = 1, . . . , rj . Thiswill be used throughout the case-by-case analysis of the strata.

Before studying the different strata, we summarize the results in the follow-ing theorem/table:

Theorem 6.1. In the space S of quartic monoid hypersurfaces with a triplepoint at O = (1 : 0 : 0 : 0) and only isolated singularities, each stratum withtype 1 to 8 has dimension and number of components as given by Table 6.1.

Case 1. The tangent cone is a nodal cubic, and we set f3 := f(1)3 = x1x2x3 +

x32 + x3

3. The group M1 is isomorphic to the group A3 × Z3 of 18 elements,where A3 denotes the permutation group of 3 elements. M1 is generated by thefollowing three matrices:

M1 =

1 0 00 0 10 1 0

,M2 =

1 0 00 ω3 00 0 ω2

3

,M3 =

ω3 0 00 ω3 00 0 ω3

.

For the study of the different strata we have to distinguish between m = 0and m > 0 where m is defined as in Chapter 5. Recall that we define θ(s, t) =(−s3 − t3, s2t, st2) in this case.Subcase 1a. Assume m = 0 and let m1, . . . ,mr be a partition of 12. Thisdefines a stratum S. We define BS ⊂ A12

∆,0 by the equation um11 · · ·umr

r − 1,where we use u1, . . . , ur as the coordinate functions on A12.

For a point b = (b1, . . . , br) ∈ BS there is a unique polynomial f4 such thatf4(θ(s, t)) =

∏ri=1(s+ bit)mi and the coefficients of x1x

33, x2x

33 and x4

3 in f4 areall zero. The map ϕS is then defined as ϕS(b) = x0f3 + f4. The map ϕS isclearly a morphism by the analysis in Chapter 5, and the image of γS is S ⊂ S.

Let c := gcd(m1, . . . ,mr). Now BS is reducible if and only if c > 1, and thenBS has c components, each given by um1/c

1 · · ·umr/cr −ωic. This follows from the


Type Invariants dim S Components

1 m = 0, m1 + · · ·+ mr = 12 11 + r 1 + e1

2e13e2 := gcd(m1, . . . , mr)m = 2, . . . , 12, m1 + · · ·+ mr = 12−m 12 + r 1

2 m = 0, m1 + · · ·+ mr = 12 10 + r 1m = 2, 3, m1 + · · ·+ mr = 12−m 11 + r 1

3 j0 = j1 = k0 = k1 = 0 r + r′ + 10 1 + em1 + · · ·+ mr = 4, m′

1 + · · ·+ m′r′ = 8

2e := gcd(m1, . . . , mr, m′1, . . . , m′

r′ )j0 = k0 = 0, j1, k1 > 0 r + r′ + 11 1m1 + · · ·+ mr = 4− k1

m′1 + · · ·+ m′

r′ = 8− j1j0, k0, j1, k1 > 0 r + r′ + 12 1

m1 + · · ·+ mr = 4− k0 − k1

m′1 + · · ·+ m′

r′ = 8− j0 − j14 j0 = k0 = 0, m1 + · · ·+ mr = 4 r + r′ + 9 1

m′1 + · · ·+ m′

r′ = 8j0 = k0 = 1, m1 + · · ·+ mr = 4− k0 r + r′ + 10 1

m′1 + · · ·+ m′

r′ = 8− j0j0, k0 ≥ 2, m1 + · · ·+ mr = 4− k0 r + r′ + 11 1

m′1 + · · ·+ m′

r′ = 8− j05 k1,2 = k2,1 = k1,3 = k3,1 = k2,3 = k3,2 = 0 r + r′ + r′′ + 9 1 + e

m1 + · · ·+ mr = m′1 + · · ·+ m′

r′ = 4m′′

1 + · · ·+ m′′r′′ = 4

2e := gcd(m1,...,r, m′1,...,r′ , m

′′1,...,r′′ )

k1,2, k1,3 > 0, k2,1 = k3,1 = k2,3 = k3,2 = 0 r + r′ + r′′ + 10 1m1 + · · ·+ mr = 4

m′1 + · · ·+ m′

r′ = 4− k1,2

m′′1 + · · ·+ m′′

r′′ = 4− k1,3

k1,2, k1,3, k2,1, k2,3 > 0, k3,1 = k3,2 = 0 r + r′ + r′′ + 11 1m1 + · · ·+ mr = 4− k2,1

m′1 + · · ·+ m′

r′ = 4− k1,2

m′′1 + · · ·+ m′′

r′′ = 4− k1,3 − k2,3

k1,2, k1,3, k2,1, k2,3, k3,1, k3,2 > 0 r + r′ + r′′ + 12 1m1 + · · ·+ mr = 4− k2,1 − k3,1

m′1 + · · ·+ m′

r′ = 4− k1,2 − k3,2

m′′1 + · · ·+ m′′

r′′ = 4− k1,3 − k2,3

6 j1 = j2 = j3 = 0, m1 + · · ·+ mr = 4 r + r′ + r′′ + 8 1m′

1 + · · ·+ m′r′ = m′′

1 + · · ·+ m′′r′′ = 4

j1 = j2 = j3 = 1, m1 + · · ·+ mr = 3 r + r′ + r′′ + 9 1m′

1 + · · ·+ m′r′ = m′′

1 + · · ·+ m′′r′′ = 3

j1 ≥ 2, j2 = j3 = 1, r + r′ + r′′ + 10 1m1 + · · ·+ mr = 4− j1

m′1 + · · ·+ m′

r′ = m′′1 + · · ·+ m′′

r′′ = 3

7 j0 = k0 = 0 r + r′ + 11 1m1 + · · ·+ mr = m′

1 + · · ·+ m′r′ = 4

j0, k0 > 0, m1 + · · ·+ mr = 4− j0 r + r′ + 12 1m′

1 + · · ·+ m′r′ = 4− k0

8 m1 + · · ·+ mr = 4 r + 13 1

Table 6.1: The strata of S of type 1 to 8

6.2. TYPES 1 TO 8 81

fact that the polynomial um1/c1 · · ·umr/c

r − ωic has no monomial factors and aNewton polytope which has only two lattice points.

We will now study the groupMS,b through the equation γS(b, g) = γS(b, g).Write ` = a1,1x1+a1,2x2+a1,3x3 and h = g−1g on the form (6.4) where A = λM .Then the equation γS(b, g) = γS(b, g) is equivalent to f4(x) = `f3 + λf4(Mx).

Now f4(x) = `f3 + λf4(Mx) for some ` if and only if

f4(θ(s, t)) = λf4(Mθ(s, t))

where θ(s, t) is considered a column vector. The degree 12 polynomial f4(θ(s, t))is by definition

∏ri=1(s + bit)mi , while λf4(Mθ(s, t)) needs to be considered

further.Note that Miθ(s, t) for i = 1, 2, 3 are all parameterizations of Z(f3). By

analyzing these parameterizations we can find which components of ϕS(BS) areidentified by ψ and also calculate the group Ms,b. First note that M1θ(s, t) =θ(t, s), M2(θ(s, t)) = θ(s, ω3t) and M3(θ(s, t)) = ω3θ(s, t). Now we have

f4(M1θ(s, t)) = f4(θ(t, s)) =r∏i=1

(t+ bis)mi =r∏i=1

(s+1bit)mi .

Here we have used∏ri=1 (bi)

mi = 1. From this we see that if M = M1 thenλ = 1, and γS(b, g) = γS(b, g) for some g whenever b−1

i = bi for all i = 1, . . . , r.Similarly, we have f4(M2θ(s, t)) =

∏ri=1(s+ (ω3bi)t)mi , so M = M2 implies

λ = 1, and γS(b, g) = γS(b, g) for some g whenever bi = ω3bi for all i = 1, . . . , r.Finally, f4(M3θ(s, t)) = ω3

∏ri=1(s+ bit)mi , so M = M3 implies λ = ω2

3 , givingA = ω2

3M3 = I, the identity matrix, as expected. In summary, we see thatMS,b = Mj for any b ∈ BS .

To count the number of components in S we need to check five cases, namelyc = 2, 3, 4, 6, 12.

• If c = 2, then ϕS(BS) has two components, defined by∏ri=1 b

mi/2i = ±1.

Then the elements b = (b1, . . . , br), ( 1b1, . . . , 1

br) and (ω3b1, . . . , ω3br) all

belong to the same component, so S has two components.

• If c = 3, then ϕS(BS) has three components, defined by∏ri=1 b

mi/3i = ±ω3

and∏ri=1 b

mi/3i = 1. These will all be identified by ψ (where A = M2), so

S has one component.


• If c = 4, then ϕS(BS) has four components, defined by∏ri=1 b

mi/4i = ±1

and∏ri=1 b

mi/4i = ±ω4. The two components where

∏ri=1 b

mi/4i = ±ω4

will be identified by ψ (where A = M1), so S has three components.

• If c = 6, then ϕS(BS) has six components, defined by∏ri=1 b

mi/6i = ωi6

for i = 1, . . . , 6. Here ψ identifies the components where i = 1, 3, 5 andthe components where i = 2, 4, 6, so S has two components.

• If c = 12, then r = 1 and ϕS(BS) has 12 components, defined by b1 = ωi12for i = 1, . . . , 12. Here ψ identifies the components where i = 2, 6, 10, thecomponents where i = 4, 8, 12 and the components where i is odd, so Shas three components.

To sum up we see that if c = 2e13e2 , then S has 1+e1 components. Furthermore,the dimension of each component is the dimension of BS plus 12, or dimS =12 + r − 1 = 11 + r.Subcase 1b. Assume m ∈ {2, . . . , 12} and let m1, . . . ,mr be a partition of12−m. This defines a stratum S, and we define BS := Ar∆,0.

For any point b ∈ BS there is a unique polynomial f4 such that

f4(θ(s, t)) = stm−1r∏i=1

(s+ bit)

and the coefficients of x1x33, x2x

33 and x4

3 in f4 are all zero. The morphism ϕSis then defined by ϕS(b) = x0f3 + f4.

Observe that f4(M1θ(s, t)) = sm−1t∏ri=1(bis + t). When m > 2 the set

ϕS(BS) does not contain monoids with all possible values of {f4(θ(s, t))} (up tomultiplication with a nonzero constant) for the given stratum, but the projectivetransformation corresponding to M1 ensures that the image γS(BS) containsmonoids with all possible values of {f4(θ(s, t))}. Then we also know that γS(BS)is the whole stratum S.

We can say that the element M1 ∈M1 correspond to the symmetry switch-ing x2 with x3 and thereby switching the last to coordinates of θ(s, t). Thisfulfills the promise that ϕS(BS) would contain monoids with all possible val-ues of {f4(θ(s, t))} up to a multiplication with a nonzero constant and up tosymmetry.

Furthermore, we see that if m = 1, then MS,b = M1 for any b ∈ BS . Ifm > 1 then MS,b is the subgroup of M1 generated by M2 and M3.

Since BS is open in Ar we know that BS , and thus S, is irreducible. Fur-thermore, equation (6.5) gives dimS = r + 12.

6.2. TYPES 1 TO 8 83

Case 2. The tangent cone is a cuspidal cubic, and we set f3 := f(2)3 = x3

1−x22x3.

The group M2 is isomorphic to the group k∗×Z3 and is generated by matriceson the form

M1(a) =

1 0 00 a 00 0 1/a2

and M2 =

ω3 0 00 ω3 00 0 ω3

where a ∈ k∗. We see that M2 is of dimension 1 and has three components.

As in the first case, we have to analyze the casesm = 0 andm > 0 separately.Recall that θ(s, t) = (s2t, s3, t3).

Subcase 2a. Assume m = 0 and let m1, . . . ,mr be a partition of 12. Thisdefines a stratum S. Let

BS := {b = (b1, . . . , br) ∈ Ar∆ |r∑i=1

mibi = 0}.

For any b ∈ BS there is a unique polynomial f4 whose coefficients of x41,

x31x2 and x3

1x3 in f4 are all zero and f4(θ(s, t)) =∏ri=1(bis + t)mi . Define ϕS

by writing ϕS(b) = x0f3 + f4. Now, by the analysis in Chapter 5, γS is amorphism and its image is S. We see that the closure of BS as a subset of Aris isomorphic to Ar−1, so S must have exactly one component. Similarly tothe previous cases, M ∈ MS,b if and only if there exist a b ∈ BS and λ ∈ k∗

such that f4(θ(s, t)) = λf4(Mθ(s, t)). Now M1(a)θ(s, t) = θ(αs, t/α2) whereα3 = a. This gives λf4(M1(a)θ(s, t)) = λa4

∏ri=1(s + (bi/a)t)m1 , so b = b/a

and λ = a−4 is an example showing that M1(a) ∈ MS,b for any b ∈ BS . Itfollows that MS,b = M2 for any b ∈ BS .

This proves that (6.5) is valid, and dimS = r + 10.

Subcase 2b. Assume m ∈ {2, 3} and let m1, . . . ,mr be a partition of 12−m.This defines a stratum S. Let

BS := {(b1, . . . , br) ∈ Ar | bi 6= bj for all i 6= j}.

For any b ∈ BS there is a unique polynomial f4 whose coefficients of x41, x

31x2 and

x31x3 in f4 are all zero and f4(θ(s, t)) = sm

∏ri=1(bis+ t)mi . Define ϕS(b) = f4.

Since BS is open in Ar we know that the stratum S has only one component.Now M ∈ MS,b if and only there exist a b ∈ BS and λ ∈ k∗ such thatf4(θ(s, t)) = λf4(Mθ(s, t)). We see that λf4(M1(a)θ(s, t)) = λa4sm

∏ri=1(s +


(bi/a)t)m1 . As in Subcase 2a, b = b/a and λ = a−4 is and example showingthat M1(a) ∈MS,b, and 6.5 gives its dimension, dimS = r + 11.

Case 3. The tangent cone is a conic and a line, and we set f3 := f(3)3 =

x3(x1x2 + x23). The group M3 has six one dimensional components and is

generated by

M1(a) =

a 0 00 1/a 00 0 1

,M2 =

0 1 01 0 00 0 1

and M3 =

ω3 0 00 ω3 00 0 ω3

where a ∈ k∗.

In this case we have to study three subcases separately. Recall that θ1(s, t) =(s, t, 0) and θ2(s, t) = (s2,−t2, st).Subcase 3a. Assume j0 = j1 = k0 = k1 = 0, m1, . . . ,mr is a partition of 4 andm′i, . . . ,m

′r′ is a partition of 8. This defines a stratum S. Define

BS := {(b, b′) ∈ Ar∆,0 × Ar′

∆,0 | bm11 · · · bmr

r = (b′1)m′

1 · · · (b′r′)m′r′}.

For each b := (b, b′) ∈ BS there is a unique polynomial f4 whose coefficientsof x1x

33, x2x

33 and x4

3 are all zero, such that f4(θ1(s, t)) =∏ri=1(s+ bit)mi and

f4(θ2(s, t)) =∏r′

i=1(s+ b′it)m′

i , and we define ϕS by writing ϕS(b) = x0f3 + f4.Let u1, . . . , ur be the coordinate functions on Ar and let u′1, . . . , u

′r′ be co-

ordinate functions on Ar′ and let

c = gcd(m1, . . . ,mr,m′1, . . . ,m

′r′).

Then BS has c components, each defined by

um1/c1 · · ·umr/c

r = ωic(u′1)m′

1/c · · · (u′r′)m′r′/c

for i = 1, . . . , c. Since ui 6= 0 in BS we see that the components are disjoint.Now M ∈MS,b if and only if there exist an λ ∈ k∗ such that f4(θ1(s, t)) =

λf4(Mθ1(s, t)) and f4(θ2(s, t)) = λf4(Mθ2(s, t)). SinceM1(a)θ1(s, t) = θ1(as, t/a)and M1(a)θ2(s, t) = θ2(αs, t/α) where α2 = a we get

f4(M1(a)θ1(s, t)) = f4(θ1(as,t

a)) =

r∏i=1

(as+biat)mi = a4

r∏i=1

(s+bia2t)mi

f4(M1(a)θ2(s, t)) = f4(θ2(αs,t

α)) =

r′∏i=1

(αs+b′iαt)m

′i = a4

r′∏i=1

(s+b′iat)m

′i .

6.2. TYPES 1 TO 8 85

We see that M1(a) ∈ MS,b for any b. This follows from b = a2b andb′ = ab′ with λ = a−4, and we see that b = (b, b′) and b = (b, b′) are in thesame component of BS .

Using M2θ1(s, t) = θ1(t, s) and M2θ2(s, t) = −θ2(t,−s) we get

λf4(M2θ1(s, t)) = λ

(r∏i=1

(bi)mi

)r∏i=1

(s+1bit)mi

λf4(M2θ2(s, t)) = λ

r′∏i=1

(b′i)m′

i

r′∏i=1

(s− 1b′it)m

′i .

If M = M2, then λ =∏ri=1(bi)

−mi =∏r′

i=1(b′i)−m′

i , and setting bi = 1/bi andb′i = −1/b′i shows that and M2 ∈ MS,b for any b ∈ BS . Furthermore, if (b, b′)

is in the component of BS given by um1/c1 · · ·umr/c

r = ωicum′

1/cr+1 · · ·um

′r′/c

r+r′ , then

(b, b′) is in the “inverse” component given by um1/c1 · · ·umr/c

r = ω−ic um′

1/cr+1 · · ·um

′r′/c

r+r′ .We can safely ignore M3θ1(s, t): The identity is in MS,b, so M3 must also befor any b ∈ BS . It follows that MS,b = M3 for all b ∈ BS , so (6.5) givesdimS = r + r′ + 10.

We can now count the number of components of S. We see that γS only iden-tify components when c = 4, and then exactly two components are identified.Write c = 2e. Then S has 1 + e components.Subcase 3b. Now assume that k0 = j0 = 0, k1, j1 > 0, m1, . . . ,mr is apartition of 4 − k1 and m1, . . . ,m

′r′ is a partition of 8 − j1, defining a stratum

S. (If we switch k0 with k1 and j0 with j1 we get the same stratum.) DefineBS = Ar∆,0 ×Ar′∆,0 and ϕS(b, b′) = x0f3 + f4 where f4 is the unique polynomialwhose coefficients of x1x

33, x2x

33 and x4

3 are all zero such that f4(θ1(s, t)) =sk1∏ri=1(s + bit)mi and f4(θ2(s, t)) = sj1

∏r′

i=1(s + b′it)m′

i . Since BS is open inAr+r′ , S must have one component.

Observe that f4(M2θ1(s, t)) = tk1∏ri=1(bis + t)mi and there is no b ∈ BS

such that f4(θ(s, t)) = λf4(M2θ1(s, t)) for any λ ∈ k∗. We can check that MS,b

is the subgroup generated by M1 and M3 for all b ∈ BS , so dimS = r+ r′+11.The fact that M2 6∈ MS,b is connected with the fact that ϕS(BS) does not

contain monoids for all possible values of {f4(θn(s, t))} for the given stratum.Indeed, monoids that corresponds to switching j0 with j1 and k0 with k1 haveno monoid in ϕS(BS) with the same values of {f4(θn(s, t))}. However, these areall projectively equivalent to monoids in ϕS(BS) by a projective transformation


on the form (6.4) where A = M2. This is because M2 corresponds to switchingx1 and x2, which also corresponds to switching j0 with j1 and k0 with k1.Subcase 3c. Now assume that k0, j0, k1, j1 > 0, m1, . . . ,mr is a partition of4 − k0 − k1 and m1, . . . ,m

′r′ is a partition of 8 − j0 − j1, defining a stratum

S. Let BS := Ar∆,0 × Ar′∆,0 × A10 and define ϕS by writing ϕ(b, b′, b′′) = x0f3 +

f4 where f4 is the unique polynomial whose coefficients of x1x33, x2x

33 and x4

3

are all zero such that f4(θ1(s, t)) = sk1tk0∏ri=1(s + bit)mi and f4(θ2(s, t)) =

b′′sj1tj0∏r′

i=1(s+ b′it)m′

i . We see that, since BS has one component, S has onecomponent. Since f4(M2θ1(s, t)) = sk0tk1

∏ri=1(sbi + t)mi and f4(M2θ2(s, t)) =

b′′sj0tj1∏r′

i=1(b′is + t)m

′i we see that M2 ∈ MS,f4

if and only if j0 = j1 andk0 = k1. Note that switching j0 with j1 and k0 with k1 will give the samestratum, so we have two different definitions for BS when j0 6= j1 or k0 6= k1.However, these are symmetric, and the symmetry corresponds to switching x1

with x2, which again corresponds to M2. Thus, any of the two definitions of BSwill do for the construction of the stratum S. With this in mind we can checkthat MS,f4

= M3 if j0 = j1 and k0 = k1 (when BS is well defined), and thatMS,f4

is generated by M1 and M3 for the rest of the strata (when we have twosymmetric definitions of BS).

In every case MS,f4is a subgroup of M3 of codimension zero, so (6.5) gives

dim(S) = r + r′ + 12.

Case 4. The tangent cone is a conic plus a tangent line, and we set f3 = f(4)3 =

x3(x1x3 + x22). The group M4 is the set of matrices on the form

M(a1, a2) =

a41 a2 − a2

24a4

1

0 a1 − a22a3

1

0 0 1a21

where a1 ∈ k∗ and a2 ∈ k. We see that M4 has dimension 2. We haveto distinguish between three cases, (a) j0 = k0 = 0, (b) j0 = k0 = 1 and(c) j0, k0 ≥ 2. For each case we let m1, . . . ,mr be a partition of 4 − k0 andm′

1, . . . ,m′r′ be a partition of 8− j0, defining a stratum S. For (a) we define BS

as the set of b = (b, b′) ∈ Ar∆ × Ar′∆ such that∑ri=1mibi =

∑r′

i=1m′ib′i. For (b)

BS is defined as the set of b = (b, b′) ∈ Ar∆×Ar′∆, and for (c) BS is defined as theset of b = (b, b′, b′′) ∈ Ar∆×Ar′∆×A1

0. For all of the cases we define ϕS by writingϕS(b) = [x0f3+f4] where f4 is the unique polynomial whose coefficients of x2

1x23,

x1x2x23 and x1x

33 in f4 are all zero such that f4(θ1(s, t)) = tk0

∏ri=1(s + bit)mi

and, in subcases (a) and (b) f4(θ2(s, t)) = tj0∏r′

i=1(s + b′it)m′

i and, in subcase

6.2. TYPES 1 TO 8 87

(c) f4(θ2(s, t) = b′′tj0∏r′

i=1(s + br+it)m′i . For all strata S the set BS will be

open in some affine space, so S will have one component.

Recall that θ1(s, t) = (s, t, 0) and θ2(s, t) = (s2, st,−t2). Now M(a1, a2) ∈MS,b if and only if f4(θ1(s, t)) = λf4(M(a1, a2)θ1(s, t)) and f4(θ2(s, t)) =λf4(M(a1, a2)θ1(s, t)) for some λ ∈ k∗. We have

M(a1, a2)θ1(s, t) = θ(a41s+ a2t, a1t), M(a1, a2)θ2(s, t) = θ2(a2

1s+a2t

2a21

,t

a1),

giving

f4(M(a1, a2)θ1(s, t)) = (a1t)k0r∏i=1

(a41s+ a2t+ bia1t)mi

= tk0a16−3k01

r∏i=1

(s+

(a2

a41

+bia31

)t

)mi

f4(M(a1, a2)θ2(s, t)) = b′′(t

a1

)j0 r′∏i=1

(a21s+

a2t

2a21

+br+it

a1

)m′i

= b′′tj0a116−3j0

r′∏i=1

(s+

(a2

2a41

+br+ia31

)t

)m′i

where we set b′′ = 1 for (a) and (b).

In every case we see that λ = a3k0−161 . In case (a) we we have

∑r′

i=1m′i =

2∑ri=1mi, so

∑ri=1mi

(a2a41

+ bi

a31

)=∑r′

i=1m′i

(a22a4

1+ br+i

a31

). In case (c) we see

that b′′ = a3j0−3k01 . To sum up, we see that M(a1, a2) ∈ MS,b for all a1 ∈ k∗,

a2 ∈ k and b ∈ BS . This gives dimS = r + r′ + 9 in subcase (a), dimS =r + r′ + 10 in subcase (b) and dimS = r + r′ + 11 in subcase (c).

Case 5. The tangent cone is three lines, and we set f3 = f(5)3 = x1x2x3. The


group M5 consist of matrices on the form

M1(a1, a2) =

a1 0 00 a2 00 0 1

a1a2

, M2(a1, a2) =

0 a1 0a2 0 00 0 1

a1a2

M3(a1, a2) =

a1 0 00 0 a2

0 1a1a2

0

, M4(a1, a2) =

0 0 a1

0 a2 01

a1a20 0

M5(a1, a2) =

0 0 a1

a2 0 00 1

a1a20

, M6(a1, a2) =

0 a1 00 0 a21

a1a20 0

where a1, a2 ∈ k, and is of dimension 2. We see that M5 has 6 disjoint compo-nents, each parameterized by Mi for i = 1, . . . , 6.

Recall that θ1(s, t) = (0, s, t), θ2(s, t) = (s, 0, t) and θ3(s, t) = (s, t, 0). Wesee that M1(a1, a2)θ1(s, t) = θ1(a2s,

ta1,a2

), M1(a1, a2)θ2(s, t) = θ2(a1s,t

a1,a2)

and M1(a1, a2)θ3(s, t) = θ3(a2s, a1t), so M1(a1, a2)θi(s, t) is a parameteriza-tion of Z(x1). However, M2(s, t)θ1(s, t) = (a1s, 0, t

a1a2) = θ2(a1s,

ta1a2

), soM2(s, t)θ1(s, t) is a parameterization of Z(x2), not of Z(x1). This comes fromthe fact that M2(a1, a2) sends Z(x1) to Z(x2) and vice versa. The six compo-nents of M corresponds to the permutation group of the 3 lines Z(x1), Z(x2)and Z(x3).

To avoid notation confusion, we rewrite the intersection numbers:

k1,2 := I(1,0,0)(f4, x2), k2,1 := I(0,1,0)(f4, x1), k3,1 := I(0,0,1)(f4, x1),k1,3 := I(1,0,0)(f4, x3), k2,3 := I(0,1,0)(f4, x3), k3,2 := I(0,0,1)(f4, x2).

We have to consider 4 cases, depending on how many of the pairs (k1,2, k1,3),(k2,1, k2,3) and (k3,1, k3,2) are zero. In each of the subcases we will assume thatm1, . . . ,mr is a partition of 4− k2,1 − k3,1 associated to the component x1 = 0,m′

1, . . . ,m′r′ is a partition of 4− k1,2 − k3,2 associated to the component x2 = 0

and m′′1 , . . . ,m

′′r′′ is a partition of 4 − k1,3 − k2,3 associated to the component

x3 = 0.

Subcase 5a. Assume k1,2 = k2,1 = k3,1 = k1,3 = k2,3 = k3,2 = 0. Define BS asthe set of b = (b, b′, b′′) ∈ Ar∆,0×Ar′∆,0×Ar′′∆,0 such that (

∏ri=1 b

mii )(

∏r′′

i=1(b′′i )m′′

i ) =∏r′

i=1(b′i)m′

i . Then, for each b ∈ BS there is a unique polynomial f4 whose co-

6.2. TYPES 1 TO 8 89

efficients of x21x2x3, x1x

22x3 and x1x2x

23 are all zero such that

f4(θ1(s, t)) =r∏i=1

(bis+ t)mi ,

f4(θ2(s, t)) =r′∏i=1

(b′is+ t)m′i and

f4(θ3(s, t)) =

(r∏i=1

bmii

)r′′∏i=1

(b′′i s+ t)m′′i .

Define ϕS by writing ϕ(b) = [x0f3 + f4]. Now we see that BS has

c := gcd(m1, . . . ,mr,m′1, . . . ,m

′r′ ,m

′′1 , . . . ,m

′′r′′)

components, each given by(r∏i=1

umi/ci

) r′′∏i=1

um′′

i /cr+r′+i

= ωjc

r′∏i=1

um′

i/cr+i

for j = 1, . . . , c.We know that M ∈ MS,b if and only if there exist an b ∈ BS and λ ∈ k∗

such that f4(θi(s, t)) = λf4(Mθi(s, t)) for i = 1, 2, 3. Looking at M1(a1, a2) wesee that

f4(M1(a1, a2)θ1(s, t)) = f4

(θ1

(a2s,

t

a1a2

))=(

1a1a2

)4 r∏i=1

(a1a22bis+ t)mi

f4(M1(a1, a2)θ2(s, t)) = f4

(θ2

(a1s,

t

a1a2

))=(

1a1a2

)4 r′∏i=1

(a21a2b

′is+ t)m

′i

f4(M1(a1, a2)θ3(s, t)) = f4 (θ3 (a1s, a2t)) =

(r∏i=1

bmii

)a42

r′′∏i=1

(a1b

′′i

a2s+ t)m

′′i ,

so b = (b, b′, b′′) = (a1a22b, a

21a2b

′, a1a2b′′) with λ = a4

1a42 is an example showing

that M1(a1, a2) ∈MS,b for every b ∈ BS . Furthermore, we see that b and b isin the same component of BS .


Considering M2(a1, a2) we see that

M2(a1, a2)θ1(s, t) = θ2(a1s,t

a1a2),

M2(a1, a2)θ2(s, t) = θ1(a2s,t

a1a2),

M2θ3(s, t) = θ3(a1t, a2s).

This gives

f4(M2(a1, a2)θ1(s, t)) = f4

(θ2

(a1s,

t

a1a2

))=(

1a1a2

)4 r′∏i=1

(a21a2b

′is+ t)m

′i

f4(M1(a1, a2)θ2(s, t)) = f4

(θ1

(a2s,

t

a1a2

))=(

1a1a2

)4 r∏i=1

(a1a22bis+ t)mi

f4(M1(a1, a2)θ3(s, t)) =

(r∏i=1

bmii

)(r∏i=1

(b′′i

)mi

)a41

r′′∏i=1

(a2

a1b′′is+ t)m

′′i .

Now M2(a1, a2) ∈MS,b only if there exist a b ∈ BS such that

r∏i=1

(bis+ t)mi =r′∏i=1

(a21a2b

′is+ t)m

′i ,

r′∏i=1

(b′is+ t)m′i =

r∏i=1

(a1a22bis+ t)mi and

r′′∏i=1

(b′′i s+ t)m′′i =

r′′∏i=1

(a2

a1b′′is+ t)m

′′i .

This can only happen if the partitionsm1, . . . ,mr andm′1, . . . ,m

′r′ of 4 are equal.

This is in fact a sufficient condition: Assuming mi = m′i for i = 1, . . . , r = r′

we see that b = (b, b′, b′′) given by bi = a21a2b

′i, b

′i = a1a

22bi and b′′i = a2

a1b′′is

an element of BS , and f4(θi(s, t)) = (a1a2)4f4(θi(s, t)) for i = 1, 2, 3. In otherwords, each M2(a1, a2) ∈MS,b if and only if m1, . . . ,mr and m′

1, . . . ,m′r′ equal.

6.2. TYPES 1 TO 8 91

Furthermore, b and b are in “inverse” components of S:(∏ri=1 b

mii

)(∏r′′

i=1(b′′i )m′′

i

)(∏r′

i=1(b′i)m′

i

) =

(∏ri=1(a

21a2b

′i)mi

)(∏r′′

i=1

(a2/(a1b

′′i ))m′′

i)

(∏r′

i=1(a1a22bi)m

′i

)=

(∏r′

i=1(b′i)m′

i

)(∏r

i=1 bmii

)(∏r′′

i=1(b′′i )m′′

i

)In a very similar way we can see that M3(a1, a2) ∈ MS,b if and only if

m′1, . . . ,m

′r′ and m′′

1 , . . . ,m′′r′′ are equal partitions of 4, and M4(a1, a2) ∈MS,b

if and only if m1, . . . ,mr and m′′1 , . . . ,m

′′r′′ are equal partitions of 4.

The elements of M5 on the forms M5(a1, a2) and M6(a1, a2) are membersof MS,b if and only if all three partitions are equal, and we can easily checkthat γS does not identify identify additional components of BS .

Similarly to the previous Subcases we get the same stratum if two of thepartitions of four are switched. Such a switch corresponds to the symmetry ofswitching two of the variables x1, x2 and x3. The symmetry also correspondsto one of the elements M2(1, 1), M3(1, 1) and M3(1, 1) in M5. The result is thesame as before: Any of the definitions of BS and ϕS will do for the constructionof the stratum S.

Note that if c = 4, then r = r′ = r′′ = 1, m1 = m′1 = m′′

1 , and twocomponents are identified by γS . In all other cases c = 1 or 2 and no componentsare identified. We se that as in previous cases, if c = 2e, then S has 1 + ecomponents. Furthermore, formula (6.5) gives dimS = r + r′ + r′′ + 9.Subcase 5bcd. In these subcases BS will be open in some affine space. InSubcase 5b we assume k1,2, k1,3 > 0 and k2,1 = k2,3 = k3,1 = k3,1 = 0, inSubcase 5c we assume k1,2, k1,3, k2,1, k2,3 > 0 and k3,1 = k3,1 = 0, and inSubcase 5d we assume k1,2, k1,3, k2,1, k2,3, k3,1, k3,1 > 0. For each stratum S weassume m1, . . . ,mr is a partition of 4− k2,1 − k3,1, m′

1, . . . ,m′r′ is a partition of

4− k1,2 − k3,2 and m′′1 , . . . ,m

′′r′′ is a partition of 4− k1,3 − k2,3. As in previous

cases, the following construction is not unique for some strata, but in each suchcase one can select any definition of BS and ϕS .

For Subcase 5b we define BS = Ar∆,0 × Ar′∆,0 × Ar′′∆,0, for Subcase 3c wedefine BS = Ar∆,0 × Ar′∆,0 × Ar′′∆,0 × A1

0 and for Subcase 5d we define BS =Ar∆,0 × Ar′∆,0 × Ar′′∆,0 × A2

0.We can define ϕS for Subcases 5b, 5b and 5c in one go by writing b =

(b, b′, b′′) for 5b, b = (b, b′, b′′, b′′′) for 5c and 5d, letting b′′′1 =∏ri=1 b

mii in


subcase 5b, and, for Subcase 5b and 5c, letting b′′′2 = 1. Then we define ϕS bywriting ϕ(b) = x0f3 + f4 where f4 is the unique polynomial whose coefficientsof x2

1x2x3, x1x22x3 and x1x2x

23 are all zero and such that

f4(θ1(s, t)) = sk3,1tk2,1

r∏i=1

(bis+ t)mi ,

f4(θ2(s, t)) = b′′′2 sk3,2tk1,2

r′∏i=1

(b′is+ t)m′i and

f4(θ3(s, t)) = b′′′1 sk2,3tk1,3

r′′∏i=1

(b′′i s+ t)m′′i .

In all subcases M1(a1, a2) ∈ MS,b for all a1, a2 ∈ k∗ and b. Furthermore,M2 ∈MS,b if and only if k1,2 = k2,1, k1,3 = k2,3, k3,1 = k3,2 andm1, . . . ,mr andm′

1, . . . ,m′r′ are equivalents partitions. This comes from the fact that M2(a1, a2)

sends Z(x1) to Z(x2) and vice versa. For the other components of M5, similarconditions apply: For each j = 2, . . . , 6, wether Mj(a1, a2) is in MS,b dependsonly on the stratum S. We can easily check that (actually, it follows) that MS,b

is both a subgroup and a subvariety of M5, and we can use (6.5) to calculatethe dimension of S. In subcase 5b dimS = r + r′ + r′′ + 10, in subcase 5cdimS = r + r′ + r′′ + 11 and in subcase 5d dimS = r + r′ + r′′ + 12.Case 6. The tangent cone is three lines meeting in a point, and we set f3 =f

(6)3 = x3

2 − x2x23. The group M6 consist, up to multiplication of third roots of

unity, of matrices on the form

M1(a1, a2, a3) =

a1 a2 a3

0 1 00 0 1

,M2(a1, a2, a3) =

a1 a2 a3

0 1 00 0 −1

,

M3(a1, a2, a3) =

a1 a2 a3

0 −1/2 1/20 3/2 1/2

,M4(a1, a2, a3) =

a1 a2 a3

0 −1/2 1/20 −3/2 −1/2

,

M5(a1, a2, a3) =

a1 a2 a3

0 −1/2 −1/20 −3/2 1/2

,M6(a1, a2, a3) =

a1 a2 a3

0 −1/2 −1/20 3/2 −1/2

,

where a1 ∈ k∗ and a2, a3 ∈ k, so M6 has 18 disjoint components of dimension3.

6.2. TYPES 1 TO 8 93

We have to distinguish between 3 subcases, (a) j1 = j2 = j3 = 0, (b)j1 = j2 = j3 = 1 and (c) j1 > 1 and j2 = j3 = 1. Let m1, . . . ,mr be a partitionof 4 − j1 associated to the component Z(x2), let m′

1, . . . ,m′r′ be a partition of

4− j2 associated to the component Z(x2 − x3) and let m′′1 , . . . ,m

′′r′′

For (a) we define BS as the set of b = (b, b′, b′′) ∈ Ar∆×Ar′∆ ×Ar′′∆ such that2∑ri=1mibi = (

∑r′

i=1m′ib′i)(∑r′′

i=1m′′i b′′i ), and ϕ(b) = [x0f3 +f4] where f4 is the

unique polynomial such that

f4(θ1(s, t)) =r∏i=1

(s+ bit)mi ,

f4(θ1(s, t)) =r′∏i=1

(s+ b′it)m′

i and

f4(θ1(s, t)) =r′′∏i=1

(s+ b′′i t)m′′

i .

For (b) we define BS = Ar∆×Ar′∆×Ar′′∆ , and ϕ(b) = ϕ(b, b′, b′′) = [x0f3 +f4]where f4 is the unique polynomial such that

f4(θ1(s, t)) = 2tr∏i=1

(s+ bit)mi ,

f4(θ1(s, t)) = t

r′∏i=1

(s+ b′it)m′

i and

f4(θ1(s, t)) = t

r′′∏i=1

(s+ b′′i t)m′′

i .

For (c) we define BS = Ar∆ × Ar′∆ × Ar′′∆ × k∗, and ϕ(b) = ϕ(b, b′, b′′, b′′′) =


[x0f3 + f4] where f4 is the unique polynomial such that

f4(θ1(s, t)) = b′′′tj1r∏i=1

(s+ bit)mi ,

f4(θ1(s, t)) = t

r′∏i=1

(s+ b′it)m′

i and

f4(θ1(s, t)) = −tr′′∏i=1

(s+ b′′i t)m′′

i .

For each stratum S we see that BS is open in some affine space, so each Swill be irreducible.

As in case 6, the families M1, . . . ,M6 represent permutations of the threelines of Z(f3). Furthermore, wether Mi(a, b) ∈ MS,b is only dependent on thethe stratum S: Mi(a, b) ∈ MS,b if and only if the partitions are compatiblewith the action of Mi on the lines. It follows that (6.5) applies and we see thatdimS = r + r′ + r′′ + 8 in subcase (a), dimS = r + r′ + r′′ + 9 in subcase (b)and dimS = r + r′ + r′′ + 10 in subcase (c).

Case 7. The tangent cone is a double line plus a line, and we set f3 = f(7)3 =

x2x23. The group M7 is generated by matrices on the forma1 a2 a3

0 1/a24 0

0 0 a4

where a1, a4 ∈ k∗ and a2, a3 ∈ k, so M is of dimension 4.

We distinguish between two subcases, (a) where j0 = k0 = 0 and (b) wherej0, k0 > 0. Let m1, . . . ,mr be a partition of 4− j0 associated to the componentZ(x2) and m′

1, . . . ,m′r′ be a partition of 4 − k0 associated to the component

Z(x2)For Subcase (a) we define BS = Ar∆×Ar′∆×A3 and for Subcase (b) we define

BS = Ar∆ ×Ar′∆ ×A3 × k∗. We write b = (b, b′, b′′, b′′′) for Subcase (b) and b =(b, b′, b′′) with b′′′ = 1 for Subcase (a). We then define ϕ(b) = [x0f3 + f4] wheref4 is the unique polynomial whose coefficients of x1x2x

23, x

22x

33 and x2x

33 are all

zero, the coefficients of x21x2x3, x1x

22x3 and x3

2x3 are b′′1 , b′′2 and b′′3 , respectively,f4(θ1(s, t)) = tj0

∏ri=0(s+ b′1t)

mi and f4(θ2(s, t)) = b′′′tk0∏r′

i=0(s+ b′it)m′

i .We easily check that γS(BS) = S and that MS,b = M7 for all S. Since

all BS are open in some affine space S we know that S has one component.

6.3. TYPE 9 - THE TANGENT CONE IS SMOOTH 95

Formula (6.5) gives the dimension of S, in case (a) dimS = r + r′ + 11 and incase (b) dimS = r + r′ + 12.

Case 8. The tangent cone is a triple line, and we set f3 = f(8)3 = x3

3. Thegroup M8 is the set of invertible matrices on the forma1 a2 a3

a4 a5 a6

0 0 ωi3

where a1, . . . , a6 ∈ k and i = 0, 1, 2.

In this case the tangent cone has no special points, and we choose a slightlydifferent approach to describing the strata. Let m1, . . . ,mr be a partition of 4.This defines a stratum S. Let BS be the set of

b = (b1,1, b1,2, . . . , br,1, br,2, b′1, . . . , b′7) ∈ (A2 \ (0, 0))r × A7

such that bi,1bj,2 6= bi,2bj,1 for each i 6= j. Define the map ϕS : BS → S asfollows:

ϕ(b) = [x0x33 + (b1,1x1 + b1,2x2)m1 · · · (br,1x1 + br,2x2)mr + b′1x

31x3

+ b′2x21x2x3 + b′3x1x

22x3 + b′4x

32x3 + b′5x

21x

23 + b′6x1x2x

23 + b′7x

22x

23]

We see that the fibers of ϕS are of dimension r−1. As in cases 1 to 7, γS(b, g) =γS(b, g) only if g−1g′ is on the form (6.4) where A = λM for some M ∈ M8

and some λ ∈ k∗. In fact, for any M ∈M8, λ ∈ k∗ and b ∈ BS there is exactlyone element h ∈ G on the form (6.4) such that ψ(ϕ(b), h) ∈ ϕ(BS). Thus thefibers of ψ|ϕS(BS)×G are of dimension 1 + dimM8 = 7, and the fibers of γS areall of dimension (r − 1) + 7 = r + 6. From (6.2) we get

dimS = dimBS + 12− dim γ−1S γS(b, g) = (2r + 7) + 12− (r + 6) = r + 13.

6.3 Type 9 - the tangent cone is smooth

In this section we will use a lemma and linear algebra to calculate the dimensionof the strata of type 9.

Recall that each stratum of type 9 is characterized by a partition of 12: Twomonoids where the tangent cone is smooth (viewed as a curve in P2) are in thesame stratum if and only if their associated partitions of 12 are the same.

For any fixed f3 of type 9 and any point p ∈ Z(f3) we can find a localpower series parameterization ψ : (k, 0) → (A2, p) of Z(f3), where A2 ⊂ P2


is a standard affine space containing p. The intersection multiplicity Ip(f3, f4)is then the multiplicity of the factor t in the power series g := f4(ψ(t)), andthe intersection multiplicity Iψ(t0)(f3, f4) is the multiplicity of (t − t0) in g.Furthermore, we see that Iψ(t0)(f3, f4) ≤ m if and only if g(j)(t0) = 0 forj = 0, . . . ,m− 1, where g(j) denotes the jth derivative of g.

Let e1, . . . , e12 be a basis of the vector space k[x1,x2,x3]4(f3)

, where k[x1, x2, x3]4denotes the set of homogeneous polynomials in x1, x2, x3 over k of degree 4.If the coefficient of x3

1 in f3 is different from zero we can, for example, choosee1, . . . , e12 as the set of monomials in x1, x2 and x3 of degree four that are nota multiple of x3

1.The following lemma will be helpful in determining the dimensions of the

strata of type 9.

Lemma 6.2. Let f3 ∈ k[x1, x2, x3] define a smooth cubic curve and let e1, . . . , e12be defined as above. For any local parameterization ψ of Z(f3), the power seriesgi(t) := ei(ψ(t)) for i = 1, . . . , 12 are linearly independent and the Wronskiandeterminant of g1(t), . . . , g12(t), equal to

det

g1(t) g2(t) · · · g12(t)g′1(t) g′2(t) · · · g′12(t)

......

. . ....

g(11)1 (t) g

(11)2 (t) · · · g

(11)12 (t)

,

does not vanish identically.

Proof. The power series g1(t), . . . , g12(t) are linearly independent by construc-tion: If

∑12i=1 cigi(t) = 0 where c1, . . . , c12 ∈ k, then g =

∑12i=1 ciei ∈

k[x1,x2,x3]4(f3)

is such that g(ψ(t)) ≡ 0. Since f3 is indecomposable, it follows that g is amultiple of f3 and thus equal to 0 in k[x1,x2,x3]4

(f3). Since e1, . . . , e12 is a basis, it

follows that c1 = · · · = c12 = 0.The functions g1, . . . , g12 are elements of a the differentiable field k((t)) with

constants k. Since g1, . . . , g12 are linearly independent over k, Proposition 2.8of [22] applies, so the Wronskian determinant of g1, . . . , g12 is not identicallyzero.

The lemma above follows from a specialization of [25, Theorem 2] (q = 4, p =3 and F := f3) and its proof. Indeed, the proof presented above is practicallyidentical to the proof of [25, Theorem 2].


Let X ∈ S be defined by F = x0f3 +f4, and S be characterized by the parti-tion of 12 equal tom1, . . . ,mr. Furthermore, write {p1, . . . , pr} = Z(f3, f4) ⊂ P2

such that Ipj (f3, f4) = mj for j = 1, . . . , r.We will now show that in an open neighborhood of X the dimension of S is

equal to 12 + r. Write Sf3 = π−1(f3) ∩ S, where π : S → P9 is the morphisminduced by the linear projection P24 99K P9. Since type 9 corresponds to anopen subset of P9 it is sufficient to show that Sf3 is of dimension 3 + r.

We assume that pi 6∈ Z(x3) for i = 1, . . . , r and write x3 = 1 for the restof the section. Thus f3, f4 define curves in A2. Let ψi : (k, 0) → (A2, pi) fori = 1, . . . , r be local parameterizations of Z(f3) and let ei for i = 1, . . . , 12 bedefined as above. Furthermore, let g(j)

i denote the row vector(∂j

∂tjie1(ψi(ti)), . . . ,

∂j

∂tjie12(ψi(ti))

)

of power series in ti.DefineM(t1, . . . , tr) to be the 12×12 matrix with rows g(j)

i where i = 1, . . . , rand j = 0, . . . ,mi − 1. We will see that this matrix completely describes Sf3 ina neighborhood of X.

For any nonzero column vector c = (c1, . . . , c12)T ∈ k12 let Scf3

denote thefour dimensional set of monoids that can be written on the form Z(x0f3 + f4)where f4 is congruent to λ

∑12i=1 ciei modulo f3 for some λ 6= 0. Note that Sc

f3is not necessarily a subset of Sf3 (contrary to what the notation may suggest).

If M(τ1, . . . , τr)c = 0 for some τ1, . . . , τr ∈ k and nonzero column vec-tor c = (c1, . . . , c12) ∈ k12, then it is clear that Sc

f3⊂ Sf3 and that for

any monoid Z(x0f3 + f4) ∈ Scf3

we have Z(f3, f4) = {ψ1(τ1), . . . , ψr(τr)} andIψi(τi)(f3, f4) = mi for i = 1, . . . , r. In other words, the column vector c givesus a four dimensional family of monoids in Sf3 that share intersection points(with multiplicites) between f3 and f4. These intersection points correspond toτ1, . . . , τr (by the local parameterizations ψ1, . . . , ψr).

We will now see that the family of monoids with these intersection pointsand associated multiplicities is exactly the family Sc

f3. We need to show that

if f4 is such that Z(f3, f4) = {ψ1(τ1), . . . , ψr(τr)} and Iψi(Ti)(f3, f4) = mi fori = 1, . . . , r, then Z(x0f3 + f4) ∈ Sc

f3. We show this by a proof of contradiction:

Assume the opposite and fix any f ′4 such that Z(x0f3 + f ′4) ∈ Scf3

and any pointp ∈ A2 such that f4(p), f ′4(p) 6= 0. Consider the polynomial f ′′4 = f4(p)f ′4 −f ′4(p)f4. Now we see that Iψi(τi)(f3, f

′′4 ) ≥ mi for i = 1, . . . , r and Ip(f3, f ′′4 ) ≥ 1,

so, by Bezout’s theorem, f ′′4 must be a multiple of f3. However, this implies


Z(x0f3 + f4) ∈ Scf3

, a contradiction. It follows that for any (τ1, . . . , τr) suchthat the equation M(τ1, . . . , τr)c = 0 there exists exactly a four dimensionalfamily of monoids in Sf3 with intersection points ψi(τi) and multiplicities mi.Furthermore, we see that if M(τ1, . . . , τr)c = 0 has no nonzero solution in cthere is no monoid in Sf3 with these intersection points and multiplicities.

Note that the equation M(τ1, . . . , τr)c = 0 has a solution (in c) if and onlyif detM(τ1, . . . , τr) = 0. By the remarks above, we see that the set defined by

detM(t1, . . . , tr) = 0 (6.6)

is of dimension four less than Sf3 . Thus, we need to show that the set definedby equation (6.6) is of dimension r − 1. This is equivalent to showing thatdetM(t1, . . . , tr) is not identically zero. The case r = 1, m1 = 12 is alreadyproven by Lemma 6.2, and this case is in fact a special case of the main result of[25]. The rest of the cases follow from Lemma 6.2 and the following proposition.Note that the proposition deals with column vectors to ease the notation, butthat we need the equivalent result for row vectors.

Proposition 6.3. Let n ∈ N and m1, . . . ,mr be a partition of n. If the columnvectors gi(ti) ∈ k((ti))n for i = 1, . . . , r are such that

det(gi(ti),g′i(ti), . . . ,g(n−1)i (ti)) 6= 0,

then the matrix with columns g(j)i (ti) where i = 1, . . . , r and j = 0, . . . ,mi − 1

is also of full rank.

Proof. It is sufficient to show that if the proposition is true for the partitionm1, . . . ,mr, then it is also true for (a) the partition m1,m2, . . . ,mr − 1, 1 and(b) the partition m1 − 1,m2, . . . ,mr−1,mr + 1. Let

e1, . . . , en = g1(t1), . . . ,g(m1−1)1 (t1),g2(t2), . . . ,g(mr−1)

r (tr).

By assumption, this is a basis of k{t1, . . . , tr}n and thus also of k{t1, . . . , tr+1}n.For (a), let

Ψ := det (e1, . . . , en−1,gr+1(tr+1))

and assume Ψ = 0 (with the intention of reaching a contradiction). It followsthat gr+1(tr+1) is in the span of e1, . . . , en−1.

The power series Ψ is a determinant function. Recall that the derivative,with respect to some variable, of a determinant function is the sum of determi-nants of the n matrices where one column is replaced with its derivative. Thisfollows from the product formula for derivation.


The jth derivative of Ψ (with respect to tr+1) is equal to

∂jΨ∂tjr+1

= det(e1, . . . , en−1,g

(j)r+1(tr+1)

)= 0

for all j. Thus g(j)r+1(tr+1) is in the span of e1, . . . , en−1 for j = 0, . . . , n − 1.

Since gr+1(tr+1), . . . ,g(n−1)r+1 (tr+1) are linearly independent, this implies that a

space of dimension n is contained in a space of dimension n−1, a contradiction.Thus Ψ 6= 0, so the proposition is true for the partition m1,m2, . . . ,mr − 1, 1.

For (b), letΨj := det

(e1, . . . , em1 , . . . , en,g

(j)r (tr)

)where em1 = g(m1−1)

1 (t1) is removed from the determinant. It is clear thatΨj = 0 for j = 1, . . . ,mr − 1 since these are determinants of matrices withrepeated columns. Assume Ψmr = 0 with the intention of getting a contradic-tion. It follows that g(mr)

r (tr) is in the span of e1, . . . , em1 , . . . , en. Furthermore,∂Ψmr

∂tr= Ψmr+1, so g(mr+1)

r (tr) is also in the span of e1, . . . , em1 , . . . , en. Con-tinuing, we see that for j > mr we have

∂Ψj

∂tr= Ψj+1 + det

(e1, . . . , em1 , . . . , en−1,g(mr)

r (tr),g(j)r (tr)

)If Ψi = 0 for i = 1, . . . , j > mr, then g(i)

r (tr) is in the span of e1, . . . , em1 , . . . , enfor i = 1, . . . , j > mr, and thus all the columns of the determinant above arecontained in the span of e1, . . . , em1 , . . . , en. The determinant is therefore equalto zero, giving Ψj+1 = Ψj = 0.

By induction Ψmi = 0 implies Ψi = 0 for all i, so the linearly independentvectors g(j)

r (tr) for j = 1, . . . , n are all in the span of e1, . . . , em1 , . . . , en. Thiscontradiction implies Ψmr 6= 0, so the proposition is true for the partitionm1 − 1,m2, . . . ,mr−1,mr + 1.

With this we have proved that each determinant detM(t1, . . . , tr) does notvanish identically, that dimSf3 = r+3 and that dimS = 12+r. This concludesthe description of the strata of S in terms of dimension. However, the techniquesin this section are not sufficient to determine the number of components in anyof the strata of type 9. Some brute force calculations have been attempted, butit seems that some additional insight is needed.

Bibliography

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