Enumerative Geometryof Plane Curves
Lucia Caporaso1. Spaces of Plane CurvesCounting problems are among the most basic in math-ematics. Enumerative geometry studies these problemswhen they concern geometric entities, but its interactionwith other areas has been overwhelming over the past threedecades. In this paper we focus on algebraic plane curvesand highlight the interplay between enumerative issuesand topics of a different type.
The classical ambient space for algebraic geometry is thecomplex projective space,βπ, viewed as a topological spacewith the Zariski topology. The Zariski closed subsets aredefined as the the zero loci of a given collection of homoge-neous polynomials in π+1 variables, with coefficients inβ.These closed sets are called βalgebraic varietiesβ when con-sidered with the algebraic structure induced by the polyno-mials defining them.
Plane curves are a simple, yet quite interesting, type ofalgebraic variety. As sets, they are defined as the zeroes in
Lucia Caporaso is professore ordinario di matematica at UniversitΓ Roma Tre,Italy. Her email address is [email protected].
Communicated by Notices Associate Editor Daniel Krashen.
For permission to reprint this article, please contact:[email protected].
DOI: https://doi.org/10.1090/noti2094
the plane, β2, of a nonzero homogeneous polynomial inthree variables. A homogeneous polynomial of degree πin three variables, π₯0, π₯1, π₯2, has the form
πΊπ = βπ+π+π=ππ,π,πβ₯0
ππ,π,ππ₯π0π₯π1π₯π2 , (1)
where the coefficients ππ,π,π are in β. The set of all such
polynomials is a complex vector space of dimension (π+22)
and, by definition, two nonzero polynomials determinethe same curve if and only if they are multiples of oneanother. Therefore the set of all plane curves of degree πcan be identified with the projective space of dimensionππ β (π+2
2) β 1 = π(π + 3)/2,
ππ β space of plane curves of degree π = βππ .If π is small, these spaces are well known. For π = 1
we have the space of all lines, which is a β2. For π = 2we have the space of all βconics,β a β5. This is more in-teresting as there are three different types of conics: (a)smooth conics, corresponding to irreducible polynomials;(b) unions of two distinct lines, corresponding to the prod-uct of two polynomials of degree 1 with different zeroes;and (c) double lines, corresponding to the square of a poly-nomial of degree 1. Notice that conics of type (c) form a
JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 771
space of dimension 2 and conics of type (b) form a spaceof dimension 4, as each of the two lines varies in β2. Sincethe family of all conics has dimension 5, we see that mostconics are smooth or, with a suggestive terminology, βthegeneral conic is smooth,β which is a shorthand for βtheset of smooth conics is dense and open in the space of allconics.β
The assortment of types of curves gets larger as the de-gree π gets larger, but for any π the general curve in ππ issmooth, i.e., given by a polynomial whose three partialderivatives have no common zeroes.
So, smooth curves form a Zariski open dense subset inππ. This claim is an instance of a remarkable phenomenonin algebraic geometry. Indeed, let ππ be the subset in ππparametrizing singular (i.e., nonsmooth) curves. By whatwe said, ππ is closed in ππ, hence the zero locus of somepolynomials; therefore ππ is an algebraic variety. More-over, as we shall see, the geometry of ππ is all the moreinteresting as it reflects some properties of the curves itparametrizes. The phenomenon we are witnessing is thefact that the sets parametrizing algebraic varieties of a cer-tain type have themselves a natural structure of algebraicvariety; they are usually called βmoduli spacesβ and are acentral subject in current mathematics.
In this spirit, let us go back to plane curves and give aninterpretation to the dimension of the spaces of curves weencountered so far. We introduced in (1) the general poly-nomial, πΊπ, of degree π; now we consider the projectivespace ππ with homogeneous coordinates {ππ,π,π, βπ, π, π β₯0 βΆ π+π+π = π}, and the product ππΓβ2. The polynomialπΊπ is bihomogeneous of degree 1 in the ππ,π,π, and π in theπ₯π. Therefore the locus where πΊπ vanishes is a well-definedsubset of ππ Γ β2, and it is an algebraic variety which wedenote by β±π. We view β±π as a βuniversal familyβ of planecurves of degree π. In fact we have the two projections,written π1 and π2,
β±π β ππ Γ β2π1
zztttttt
ttttt π2
%%JJJ
JJJJ
JJJ
ππ β2
(2)
and the restriction of π1 to β±π expresses it as a family ofplane curves: the preimage in β±π of a point, [π] β ππ,parametrizing a curve, π β β2, is isomorphic to π , andit is mapped to π by the projection, π2, to β2.
The fact that ππ has dimension ππ = π(π + 3)/2 tellsus that if we fix ππ points in β2 there will exist somecurve of degree π passing through them, and the curvewill be unique for a general choice of points. In fact, fixπ1, β¦ , πππ β β2; a curve passes through ππ if the poly-nomial defining it vanishes at ππ. Therefore the curvespassing through our points are determined by imposingπΊπ(ππ) = 0 for all π = 1, β¦ , ππ. This gives the following
system of ππ homogeneous linear equations in 1 + ππ un-knowns (the ππ,π,π):
πΊπ(π1) = β― = πΊπ(πππ ) = 0.The solutions of this system form a vector space of dimen-sion at least 1, with equality if and only if the equationsare linearly independent, which will happen for generalpoints π1, β¦ , πππ . Since a one-dimensional vector space ofpolynomials corresponds to a unique curve, we derive thatthere exists at least one curve through our fixed points, andthe curve will be unique for a general choice of points. Inshort
ππ=max {πβΆany π points in β2 lie in a curve of degree π} ,and we solved our first, however easy, enumerative prob-lem by showing that the number of curves of degree π pass-ing through ππ general points is equal to 1. The phraseβgeneral pointsβ means that the ππ points vary in a denseopen subset of (β2)ππ .
Let us now focus on ππ, the space of singular planecurves of degree π. It turns out that ππ is a hypersur-face in ππ, i.e., the set of zeroes of one polynomial, hencedimππ = ππβ1. Arguing as before, the dimension of ππ canbe interpreted as the maximum number of points in theplane which are always contained in some singular curveof degree π.
For instance, four points always lie in some singularconic, and it is easy to describe which. If the four pointsare general (i.e., no three are collinear), there are exactlysix lines lines passing through two of them, and our con-ics are given by all possible pairs of them. This gives a totalof three conics pictured in Figure 1.
β’
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
β’
OOOOOO
OOOOOO
OOOOOOO β’
//////////
////
β’ β’ β’
oooooo
oooooo
ooooooo
β’ β’ β’ β’ β’ β’
Figure 1. The three singular conics through four points.
If three of the fixed points are collinear, we take allconics given by the union of the line through the threepoints with any line through the fourth point; since theset of lines through a point has dimension one, we get aone-dimensional space of conics. If the four points arecollinear, we have the two-dimensional space of conicsgiven by the union of the line through the points with anarbitrary line.
Summarizing, if (and only if) the four points are gen-eral (i.e., no three are collinear), there exist finitely manysingular conics through them, and the number of such
772 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6
conics is always three, regardless of the choice of the fourpoints.
As easy as this is for conics, things get more complicatedalready for π = 3. Here dimπ3 = 8, and counting theβcubicsβ through eight points is much harder.
The key is to give this number a different interpretation1
and identify it with another invariant of ππ, its degree as asubvariety of ππ.
2. The Degree of the Severi VarietyThe degree of a subvariety in projective space is the num-ber of points of intersection with as many generically cho-sen hyperplanes as its dimension. In ππ there are hyper-planes with a special geometric meaning, parametrizingcurves passing through a fixed point. Indeed, let π be apoint in β2, and letπ»π be the locus in ππ of curves throughπ:
π»π = {[π] β ππ βΆ π β π}. (3)
Thus π»π is the zero locus in ππ of the homogeneous linearpolynomial πΊπ(π), hence π»π is a hyperplane. Therefore,deg ππ, the degree of ππ, is the number of singular curvespassing through dimππ general points, as claimed. Recall-ing that dimππ = ππ β 1, we want to solve the following.
Problem 1. Compute the number of singular plane curvesof degree π passing through ππ β 1 general points. Equiva-lently, compute the degree of ππ.
Since ππ is a hypersurface, its degree is equal to the num-ber of points of intersection with a general line. So, we fixa general line, πΏ, in ππ and notice that πΏ corresponds to afamily of curves of degree π, a so-called βpencil of curves.βMore precisely, from diagram (2) we restrict the projectionβ±π β ππ over πΏ to get a map
π βΆ π³ βΆ πΏ β ππwhose fiber over every point, β β πΏ, is the plane curveof degree π corresponding to the curve parametrized by β.The word βpencilβ indicates that the base of the family, πΏ,is a line.
We identify πΏ with β1 and denote by π‘0, π‘1 its homoge-neous coordinates. Then our pencil is given by the zeroesin β1π‘0,π‘1 Γ β2π₯0,π₯1,π₯2 of a polynomial
πΉ(π‘0, π‘1; π₯0, π₯1, π₯2),bihomogeneous of degree 1 in π‘0, π‘1 and π in π₯0, π₯1, π₯2, sothat π³ β β1 Γ β2 is set of zeroes of πΉ. Since πΏ is a gen-eral line in ππ, it intersects ππ transversally in finitely manypoints. These are the points of πΏ such that the fiber of πis singular, and our goal is to count them. We first countthe singular points of the fibers of π, which are determined
1βMathematics is the art of giving the same name to different things.ββH. Poincare
by the solutions of the following system, where πΉπ₯π is thepartial derivative with respect to π₯π,
πΉπ₯0 = πΉπ₯1 = πΉπ₯2 = 0. (4)
Since πΉ is bihomogeneous of bidegree (1, π), each πΉπ₯π isbihomogenous of bidegree (1, π β 1) and corresponds to ahypersurface in β1 Γβ2 of the same bidegree. The numberof solutions of the system (4) is thus the number of pointsof intersection inβ1Γβ2 of three hypersurfaces of bidegree(1, π β 1).
We do know how to compute this number because wecan compute intersections in projective space.
We write π»β(βπ) for the cohomology ring with β€-coefficients of the projective space βπ, whose cup productcan be interpreted as the intersection product. As a ring,π»β(βπ) is isomorphic to β€[π₯]/(π₯π+1), where π₯ is identifiedwith the cohomology class, βπ β π»2(βπ), corresponding toa hyperplane. Hence π₯π β π»2π(βπ) corresponds to a linearsubspace of complex dimension π β π and real codimen-sion 2π. The intersection product in βπ depends only onthe cohomology class, and the degree of the intersectionof π hypersurfaces is the product of their degrees; in alge-braic geometry, this is Bezoutβs theorem. This degree is theappropriate count for the number of points of intersectionof the π hypersurfaces.
One usually identifies zero-dimensional classes, like theclass of the intersection of two curves in β2, with their de-gree. This amounts to identifying the top cohomologygroup, π»2π(βπ), with β€ so that the class of a point corre-sponds to 1.
The generator, β1, for β1 is the dual of a point, and thegenerator, β2, forβ2 is the dual of a line, with β21 = 0 and β22equal to the class of a point; with the above identification,we write β22 = 1.
What about β1Γβ2? It satisfies a KΓΌnneth type formula,so that its cohomology ring is generated by the pullbacksof the generators of the two factors. Let us denote by π₯π thepullback of βπ for π = 1, 2.
By what we said, the number of solutions of the system(4) is the degree of the triple intersection of the class π₯1 +(π β 1)π₯2. The following basic relations are easily seen tohold (again identifying the top cohomology group withβ€):
π₯31 = π₯21π₯2 = π₯32 = 0, π₯1π₯22 = 1.Hence
(π₯1 + (π β 1)π₯2)3 = 3(π β 1)2.Therefore the number of singularities in the fibers of π isequal to 3(π β 1)2.
By the generality of the line πΏ, every singular fiber hasexactly one singular point. Hence the number of singularfibers of π is 3(π β 1)2, and hence
deg ππ = 3(π β 1)2 (5)
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is the answer to Problem 1. This confirms that there arethree singular conics through four general points, and ittells us, for example, that there are twelve singular cubicspassing through eight general points. Notice that, differ-ently from what happens with conics, if the eight pointsare general (no three are collinear, no six on a conic), eachof these cubics will be irreducible, i.e., not the the unionof a line and a conic.
In answering Problem 1 we mentioned that the generalcurve in ππ has exactly one singular point. Moreover, thispoint is a βnode,β the simplest type of singularity a curvecan have, whose analytic local equation has the form π₯2 =π¦2.
We now consider curves with more singular points. Wedenote by ππ,πΏ the Severi variety of plane irreducible curvesof degree π with at least πΏ nodes; see [14]. More precisely,ππ,πΏ is defined as the closure in ππ of the locus of irreduciblecurves of degree π with πΏ nodes:
ππ,πΏ β {[π] β ππ βΆ π irreducible with πΏ nodes}.
If πΏ = 0, then ππ,0 = ππ; if πΏ = 1 and π β₯ 3, we haveππ,1 = ππ.
It is clear that for ππ,πΏ to be nonempty, πΏ cannot be toobig, for example it is easy to see that πΏ must be less than(π β 1)2/2. Indeed, suppose we have an irreducible curve,π , of degree π β₯ 3 with at least (π β 1)2/2 nodes. Throughthese nodes there certainly passes a curve, π , of degree πβ2,because ππβ2 > (π β 1)2/2. Hence the degree of the inter-section of π and π is at least 2(π β 1)2/2 = (π β 1)2, butthis contradicts Bezoutβs theorem, according to which thedegree of the intersection of π and π is π(π β 2). A morerefined analysis gives (πβ1
2) as a sharp upper bound on πΏ,
and we have
Fact 2.1. If πΏ > (πβ12), then ππ,πΏ is empty. Assume πΏ β€
(πβ12), then
(a) ππ,πΏ is irreducible of dimension ππ β πΏ;(b) ππ,πΏ is smooth at points parametrizing irreducible
curves with exactly πΏ nodes, and the locus of suchpoints is open and dense in ππ,πΏ.
The irreducibility of ππ,πΏ is proved in [9]. The num-ber (πβ1
2) is the arithmetic genus of a plane curve of degree
π. There are two types of genus for a curve: the geometricgenus and the arithmetic genus, which coincide if the curveis smooth and irreducible.
The genus of a smooth curve is the topological genusof the real surface underlying the curve. For example, asmooth plane curve of degree 1 or 2 has genus 0 and itsunderlying real surface is the sphere, π2. In degree 3, thegenus is 1 and the surface underlying a smooth cubic is atorus.
contract loop
desingularize node
Figure 2. A singular specialization and its desingularization.
The geometric genus of an irreducible singular curve isdefined as the genus of its desingularization.
The arithmetic genus can be thought of as the total en-ergy of the curve, with the geometric genus being the po-tential energy. Just like the total energy of a system remainsconstant, so does the arithmetic genus in a family of curvesof fixed degree. On the other hand the potential energycan be converted, all or part of it, to a βless usefulβ energy,and indeed the geometric genus of a curve can decrease ina specialization, but never increase. In particular, a familyof curves of genus 0 specializes to a curve of genus 0, all ofwhose irreducible components must have genus 0.
What about positive genus? Consider a family ofsmooth curves of degree π β₯ 3 specializing to an irre-ducible curve with πΏ nodes. The underlying family of topo-logical surfaces has, as general fiber, a surface of genusπ = (πβ1
2), hence with π handles; see Figure 2 for a pic-
ture with π = 1 and πΏ = 1. In this family every node ofthe specialization is generated by the contraction of a looparound a handle of the general fiber. The surface underly-ing the special curve is no longer a topological manifold atthe point where the loop got contracted, where the surfacelooks locally like two disks with centers identified. Sep-arating the two disks desingularizes the curve so that theunderlying surface has one fewer handle, hence its genusgoes down by 1.
This was an informal explanation for the fact that thegeometric genus of an irreducible curve of degree π with πΏnodes is equal to (πβ1
2)βπΏ; hence if πΏ > (πβ1
2) there exist no
such curves. We refer to [1] and [2] for the general theoryof curves.
Recapitulating, ππ,πΏ can be defined as the the closure inππ of the locus of irreducible curves with geometric genusπ = (πβ1
2) β πΏ. A simple calculation gives a different
expression for its dimension:
dimππ,πΏ = 3π + π β 1.
The basic enumerative problem, generalizing Problem 1,is the following.
774 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6
Problem 2. Compute the number of irreducible planecurves of degree π and genus π passing through 3π + π β 1general points.
Or, compute the degree of ππ,πΏ. If πΏ = 0, then π = (πβ12),
and the answer is 1. If πΏ = 1, then π = (πβ12) β 1, and we
know the answer is 3(π β 1)2. For πΏ β€ 8 the answer, givenin [10], is again a polynomial in π; see also [5] and [7]. Forbigger πΏ the first general solutions were discovered as recur-sive, rather than closed, formulas, as we shall illustrate inthe rest of the paper.
3. Recursive Enumeration for Rational CurvesA rational curve is an irreducible curve of geometric genus0. By what we said earlier, a family of rational curves canspecialize only to a curve all of whose irreducible compo-nents are rational. This makes the enumeration problemin genus 0 self-contained and solvable by a recursive for-mula, which expresses the degree of the Severi variety indegree π and genus 0 in terms of the degrees of the Severivarieties in lower degrees and genus, again, 0.
We denote by π π the Severi variety of rational curves ofdegree π:
π π β ππ,(πβ12 ), ππ β dimπ π = 3πβ1, ππ β degπ π.(6)
Of course, π1 = 1. In case π = 0 the answer to Problem 2is the following.
Theorem 3.1 (Kontsevichβs formula). For π β₯ 2,
ππ = βπ1+π2=π
ππ1ππ2π1π2 [(3π β 43π1 β 2)π1π2 β ( 3π β 4
3π1 β 3)π22] .
As explained in [11], this formula was discovered ina rather different context, and it came as a beautiful sur-prise. While establishing the mathematical foundationsfor GromovβWitten theory (a theory largely inspired byideas from physics), Kontsevich and Manin gave an ax-iomatic construction of the GromovβWitten invariantsand of the quantum cohomology ring, a generalization ofthe classical cohomology ring of a projective algebraic va-riety. For β2 the quantum product on the quantum coho-mology ring was defined using our numbers ππ, which ap-peared as GromovβWitten invariants. The above formulawas found as the condition characterizing the associativityof the quantum product.
The proof we shall illustrate was not among the first tobe given (for which we refer to [11] and [12], or to [13]),but is close in spirit to our answer to Problem 1.
The shape of the formula indicates that we should usesplittings of the curve into a union of two components ofdegrees π1 and π2 with π = π1 + π2. We will do that viaa one-dimensional family of curves similar to the pencil
π1
π2
π1
π2Figure 3. Reducible specializations of rational quartics.
we used to compute the degree of ππ, but with an oppo-site point of view. In the previous case the unknown wasthe number of special curves. Now the number of specialcurves will be easy to compute and will be used to deter-mine the unknown, ππ.
To get our one-dimensional family we intersect π π withgeneral hyperplanes until we get a curve. Since intersect-ing with a general hyperplane decreases the dimension by1, we need to intersect with ππ β 1 hyperplanes. We usehyperplanes of type π»π, defined in (3). So, fix π1, β¦ , πππβ1general points in β2 and set
πΆ = π π β© π»π1 β©β― β©π»πππβ1 .Now πΆ is the curve in ππ parametrizing the family
of rational curves of degree π through the basepointsπ1, β¦ , πππβ1, which we write as follows
πΆ Γ β2 β π³ βΆ πΆ.The basic idea is that our degree,ππ, is equal to the num-
ber of points of intersection between πΆ and one more gen-eral hyperplaneπ»π, as this is equal to the number of curvesin π π passing through π1, β¦ , πππβ1, π. To put this idea towork, we need to study the geometry of the family π³ β πΆ.
By Fact 2.1 and Bertiniβs theorems, πΆ is irreducible andthe subset of points parametrizing irreducible curves with(πβ1
2) nodes and no other singularities is open, dense, and
contained in the smooth locus of πΆ. Let us look at the re-ducible curves parametrized by πΆ. There are finitely manyof them and, by what we said earlier, they must be unionsof two rational curves of smaller degrees.
Example 3.2. Let π = 4, hence πΏ = 3 and π4 = 11. OurπΆ parametrizes quartics with three nodes passing throughten points. The reducible curves parametrized by πΆ, writ-ten π1 βͺ π2, are of two types, drawn in Figure 3. First type:for any partition of the basepoints into two subsets, π΅1 andπ΅2, of five points, let ππ be the conic through π΅π for π = 1, 2(drawn on the left in the picture). Second type: for anypartition of the basepoints into a subset, π΅1, of cardinal-ity 2 and a subset, π΅2, of cardinality 8, let π1 be the linethrough π΅1, and let π2 be one of the twelve nodal cubicsthrough π΅2.
We use the following notation: π1 βͺ π2 denotes areducible curve of our family, with π1 containing the
JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 775
basepoint π1. The degree of π1 will be π1, and π2 has de-gree π2 = πβπ1. We say that such curves are of type (π1, π2).Let us count them; we have
ππ1 + ππ2 = ππ β 1,which is the number of basepoints of our family. There-fore for every partition of the basepoints into two subsets,π΅1 and π΅2, of respective cardinalities, ππ1 and ππ2 , there existπππ rational curves of degree ππ passing through π΅π. Sincewe are assuming that the basepoint π1 lies on π1, the num-ber of curves of type (π1, π2) is equal to ππ1ππ2 times thenumber of partitions of the basepoints π2, β¦ , πππβ1 intotwo subsets of cardinalities ππ1 β 1 and ππ2 , hence equal to
ππ1ππ2(ππ β 2ππ1 β 1). (7)
Now, is πΆ singular at such special points? As πΆ is ageneral linear section of the Severi variety π π, its singular-ity at any point reflects the local geometry of π π, whichdepends on the singularities of the corresponding planecurve. Let us count the singular points of a curve of type(π1, π2). This amounts to counting the nodes of each com-ponent and the π1π2 nodes in which the two componentsintersect, which gives a total of
(π1 β 12 ) + (π2 β 1
2 ) + π1π2 = (π β 12 ) + 1.
Now, when rational curves of degree π specialize to acurve of type (π1, π2) each of the (πβ1
2) nodes of the gen-
eral fiber specializes to a node of the special fiber. By theabove computation, there is exactly one node of the spe-cialization which is not the limit of a βgeneralβ node. Inother words, the special curve has exactly one node thatgets βsmoothed.β But can all the nodes of the special curvebe smoothed in this way? No, only the π1π2 nodes lyingin the intersection of the two components can.
To see why, suppose we have a curve, π = π1 βͺ π2, oftype (π1, π2) occurring as the specialization of a family asparametrized above by a smooth curve, π. Write π΅ β πfor this family, and let π’0 β π be the point parametrizingπ . Up to shrinkingπ near π’0, we can assume that all fibersaway from π’0 are irreducible with (πβ1
2) nodes. The surface
π΅ is necessarily singular along the nodes of the irreduciblefibers, hence it has (πβ1
2) singular curves, which we resolve
by desingularizing π΅. In Figure 4 we have the example ofrational quartics specializing to a reducible curve (see alsoFigure 3 and Example 3.2); the dotted red curves representthe singular curves of π΅. Denote by π΅β² the desingulariza-tion of π΅ so that we have a chain of maps
π βΆ π΅β² βΆπ΅βΆπwhose composition is a new family of curves. This opera-tion has the effect of desingularizing the general fibers, so
π΅ π΅β²Desingularize
Figure 4. Reducible specialization and desingularization.
that π βΆ π΅β² β π is a family whose fibers away from π’0 is asmooth rational curve.
The curve π1βͺπ2 will be desingularized at all nodes butthe onewhich is smoothed inπ΅ (i.e., the nodewhich is nota limit of nodes, marked by a circle in Figure 4), hence thefiber of π over π’0 is reducible with exactly one node. If, bycontradiction, all points in π1 β© π2 were limits of generalnodes, then they will all be desingularized when passingto π΅β², and the fiber of π over π’0 will be disconnected. Butthis contradicts the connectedness principle, as the fibersof π away from π’0 are all connected.
In conclusion, the local geometry of π π at a curve, π ,of type (π1, π2) reflects that in a general deformation thereis exactly one node of π which gets smoothed, so thatπ π is the intersection of smooth branches, each of whichcorresponds to the node of π which gets smoothed alongthat branch. By what we said, only the π1π2 βintersectionβnodes can be smoothed, and the fact that all such nodesare actually smoothable follows by a symmetry argument.Concluding, π π is, locally at π , the transverse intersectionof π1π2 smooth branches. Hence the curve πΆ has an ordi-nary π1π2-fold point and, under π΅ β πΆ, the preimage of apoint of type (π1, π2) is made of π1π2 distinct points.
We want to compute ππ using, as for Problem 1, in-tersection theory. We start from the family of curvesparametrized by πΆ and let π΅ β πΆ be its desingularization.We pull back to π΅ the original family π³ β πΆ but, as wenoted above, the so-obtained surface is singular along thenodes of its irreducible fibers, hence we replace it by itsdesingularization, π΄. We have a commutative diagram
π΄ //
π
((
ποΏ½οΏ½
π³
οΏ½οΏ½
οΏ½ οΏ½ // β2 Γ π π
οΏ½οΏ½
// β2
π΅ // πΆ οΏ½ οΏ½ // π π
where π is a family of rational curves with finitely manyreducible fibers.
Our π΄ is a ruled surface, birational to β1 Γ π΅, and itsintersection product is something we can handle. First, weneed a section of π. Every basepoint of the family deter-mines such a section; let π be the section corresponding
776 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6
to π1. Thus π is the curve in π΄ intersecting each fiber of πin one point and such that π(π) = π1.
We denote by π β π΅ the set of points over which thefiber of π is reducible. For every π β π, we denote its fiberby ππ,1 βͺ ππ,2 with the convention that the map π sendsππ,π to a plane curve of degree ππ, and the curve of degree π1contains π1; so ππ,2 does not intersectπ. We write π(π1, π2)for the set of points parametrizing a curve of type (π1, π2),and π‘(π1, π2) for its cardinality.
We computed in (7) the number of points in πΆparametrizing curves of type (π1, π2). Over each such pointthere are π1π2 points of π΅, hence
π‘(π1, π2) = π1π2ππ1ππ2(ππ β 2ππ1 β 1). (8)
As π΄ is a ruled surface, its intersection ring is generatedby the curves
{π, π, ππ,2 βπ β π},and we will use the same symbols for curves and their co-homology classes. We have the obvious relations, for everyβπ β π,
π β π =1, π β ππ,2=0, π β ππ,2=0, π2=0, π2π,2=β1,
identifying, as before, the top cohomology group ofπ΄ withβ€. To complete the intersection table, we need π2. Wecompute it using a second section from the basepoints, solet πβ² be the section corresponding to, say, πππβ1. We haveπ β πβ² = 0 and (πβ²)2 = π2, therefore π2 = (π β πβ²)2/2.
Now, the intersection numbers of π and πβ² differ onlyon the generatorsππ,2 for which the basepoint πππβ1 lies onπ(ππ,2), in which case we haveπβ² β ππ,2 = 1. Write π β π forthe set of such points, so that π β πβ² and βπβπ ππ,2 havethe same class in the intersection ring of π΄. Hence
π2 = 12(π β πβ²)2 = 1
2(βπβπππ,2)2
= β#π2 = β12 βπ1+π2=π
π (π1, π2),
where π (π1, π2) is the number of points in π of curves oftype (π1, π2). Arguing similarly as for π‘(π1, π2), we obtain
π (π1, π2) = π1π2ππ1ππ2(ππ β 3ππ1 β 1).
Now, the preimage of a general point, π, in β2 un-der the map π is the set of curves of our family passingthrough π, whose cardinality is, of course, degπ. Hencedegπ is the number of rational curves of degree π passingthrough π1, β¦ , πππβ1, π, that is our unknown, ππ. On theother hand, let β2 be the class of a line in β2, then degπis equal to (πββ2)2, an intersection number on π΄. Hencecomputing ππ is the same as computing (πββ2)2. In the
intersection ring of π΄, we can write
πββ2 = πππ + πππ + βπβπ
ππππ,2 (9)
for some coefficients ππ, ππ , ππ. We can compute these co-efficients by intersecting both sides of (9) with the threetypes of generators. We have
πββ2 β π = 0, πββ2 β π = π, πββ2 β ππ,2 = π2by construction. These give linear relations which, withthe intersection table, enable us to determine the coeffi-cients and obtain
πββ2 = ππ β (ππ2)π β βπ1+π2=π
( βπβπ(π1,π2)
π2ππ,2) .
Now, since (πββ2)2 = ππ, we have
ππ = βπ2π2 + βπ1+π2=π
πβπ(π1,π2)
(π2ππ,2)2
= βπ1+π2=π
[π2
2 π (π1, π2) β π22π‘(π1, π2)] .
Hence
ππ = βπ1+π2=π
ππ1ππ2π1π2 [π22 (
ππ β 3ππ1 β 1) β π22(
ππ β 2ππ1 β 1)] .
To see that this formula gives Theorem 3.1, set π = ππβ3 =3π β 4 and π = ππ1 β 1 = 3π1 β 2. For the term in squarebrackets we have
π22 (
ππ) β π22(
π + 1π )
= π21 + π222 (ππ) + π1π2(
ππ) β π22(
ππ) β π22(
ππ β 1)
= π1π2(ππ) β π22(
ππ β 1) β
π22 β π212 (ππ).
Summing up, for π1+π2 = π the third summand vanishes,and we are done.
For details we refer to [3], where this technique is usedto obtain other recursions enumerating rational curves onrational surfaces.
4. Curves of Positive GenusWe now look at Problem 2 for π > 0. A complete answer isprovided by means of a recursion, the precise descriptionof which would require too many new technical details.We thus limit ourselves to illustrating the main idea andthe comparison with the previous formulas.
If we consider, as we did for π = 0, the family of curvesof degree π with πΏ nodes passing through a number ofpoints equal to dimππ,πΏβ1, we will get a one-dimensionalfamily which, in contrast with the case of rational curves,will parametrize no reducible curve, in general.
JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 777
Example 4.1. Let π = 5 and πΏ = 2 so that the geometricgenus is 4 and dimπ5,2 = 18. Let πΆ be the linear sectionof π5,2 parametrizing all quintics with two nodes passingthrough seventeen points. There exists no reducible quin-tic passing through seventeen general points, as π1 + π4 =2+14 = 16 and π2+π3 = 5+9 = 14. Hence πΆ parametrizesonly irreducible curves.
To set up a recursive approach we need some furtherconstraint to force reducible curves to appear. Ourmethodis to impose that the basepoints lie all on a fixed line inthe plane. Since an irreducible curve cannot meet a linein more points than its degree, imposing a high enoughnumber of basepoints on a line will certainly cause the oc-currence of reducible curves.
For this idea to work, we must impose one basepointat a time, in order to be able to tell at which step re-ducible curves appear, and to be able to describe them.This will occur at various steps, and we will have to han-dle sections of our Severi variety having arbitrary dimen-sion. Therefore we cannot limit our study to families overa one-dimensional base, as we did for the earlier formulas.
More precisely, the procedure starts by fixing fix a line, πΏ,in β2 and a certain number of general points π1, π2, β¦ , onπΏ. Now we intersect the Severi variety with the hyperplaneπ»π1 , then with π»π2 , and so on, by keeping track of howthe intersection behaves at every step. After a certain num-ber of steps the intersection splits into irreducible compo-nents, some of which parametrize curves of type πΏ βͺ π , sothat π has lower degree and the recursion kicks in.
The price of this recursive method is that we must con-sider a new version of Severi variety, where the novelty isin the prescription of certain orders of contact with πΏ atsome of the basepoints ππ, and at arbitrary points. In otherwords, we need to introduce the Severi variety parametriz-ing plane curves of fixed (degree, genus, and) intersectionprofile with the line πΏ.
We set πΌ = (π1, β¦ , ππ) and π½ = (π1, β¦ , ππ), with ππ andππ nonnegative integers such that βπππ + β πππ = π. Onthe line πΏ we fix βππ general points, {ππ,π} with π = 1, β¦ , πand π = 1, β¦ , ππ. We define ππ,πΏ(πΌ, π½) to be the closure inππ of the locus of irreducible curves with πΏ nodes, having
(a) a point of contact of order π with πΏ at ππ,π for everyπ = 1, β¦ , π and π = 1, β¦ , ππ;
(b) ππ points of contact of order π with πΏ for every π =1, β¦ ,π, different from those in (a).
For πΌ = (0, β¦ , 0) and π½ = (π, 0, β¦ , 0) we recover the clas-sical Severi variety ππ,πΏ.
One final point: in this setup it is natural to drop thecondition that the general curve be irreducible. So, next toππ,πΏ(πΌ, π½) we consider the generalized Severi variety, definedas before but omitting the irreducibility requirement onthe general curve.
Example 4.2. Consider the generalized Severi variety withπ = 4 and πΏ = 3, defined as the closure in π4 of theset of quartics with three nodes (so πΌ = (0, β¦ , 0) andπ½ = (4, 0, β¦ , 0)). This variety is the union of two irreduciblecomponents, both of dimension 11. One component isπ4,3, whose general point parametrizes irreducible ratio-nal quartics (the same considered in Example 3.2). Thesecond component parametrizes reducible quartics givenby the union of a line and a cubic. Since π1 = 2 and π3 = 9,this component has dimension 11.
The degree of the generalized Severi variety is computedin [4] by a recursive formula from which the (more intri-cate) recursion for the degree of ππ,πΏ(πΌ, π½) follows. Theproof is based on a thorough analysis of the geometryof the Severi variety, a subject of its own interest. Morerecently, a different proof of the same formula has beengiven in [6] using tropical geometry.
In [15] an approach similar to [4] is used to enumer-ate curves of any genus in general rational surfaces. As forthe enumerative geometry of curves on different surfaces,some results are known and some interesting conjecturesare under investigation bymeans of diverse techniques. Aswe cannot, for lack of space, give here an exhaustive list ofreferences, we refer to [8] and to its bibliography.
References[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,Ge-
ometry of algebraic curves. Vol. I, Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Math-ematical Sciences], vol. 267, Springer-Verlag, New York,1985. MR770932
[2] Enrico Arbarello, Maurizio Cornalba, and Phillip A. Grif-fiths, Geometry of algebraic curves. Volume II, Grundlehrender Mathematischen Wissenschaften [Fundamental Princi-ples of Mathematical Sciences], vol. 268, Springer, Heidel-berg, 2011. With a contribution by Joseph Daniel Harris.MR2807457
[3] Lucia Caporaso and Joe Harris, Enumerating rationalcurves: the rational fibration method, Compositio Math. 113(1998), no. 2, 209β236, DOI 10.1023/A:1000446404010.MR1639187
[4] Lucia Caporaso and Joe Harris, Counting plane curves ofany genus, Invent. Math. 131 (1998), no. 2, 345β392, DOI10.1007/s002220050208. MR1608583
[5] P. Di Francesco and C. Itzykson, Quantum intersectionrings, The moduli space of curves (Texel Island, 1994),Progr. Math., vol. 129, BirkhΓ€user Boston, Boston, MA,1995, pp. 81β148, DOI 10.1007/978-1-4612-4264-2_4.MR1363054
[6] Andreas Gathmann and Hannah Markwig, The Caporaso-Harris formula and plane relative GromovβWitten invariants intropical geometry, Math. Ann. 338 (2007), no. 4, 845β868,DOI 10.1007/s00208-007-0092-4. MR2317753
778 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6
[7] Lothar GΓΆttsche, A conjectural generating function fornumbers of curves on surfaces, Comm. Math. Phys. 196(1998), no. 3, 523β533, DOI 10.1007/s002200050434.MR1645204
[8] Lothar GΓΆttsche and Vivek Shende, Refined curve countingon complex surfaces, Geom. Topol. 18 (2014), no. 4, 2245β2307, DOI 10.2140/gt.2014.18.2245. MR3268777
[9] Joe Harris,On the Severi problem, Invent. Math. 84 (1986),no. 3, 445β461, DOI 10.1007/BF01388741. MR837522
[10] Steven Kleiman and Ragni Piene, Enumerating singu-lar curves on surfaces, Algebraic geometry: Hirzebruch70 (Warsaw, 1998), Contemp. Math., vol. 241, Amer.Math. Soc., Providence, RI, 1999, pp. 209β238, DOI10.1090/conm/241/03637. MR1718146
[11] M. Kontsevich and Yu. Manin, GromovβWitten classes,quantum cohomology, and enumerative geometry, Comm.Math. Phys. 164 (1994), no. 3, 525β562. MR1291244
[12] Maxim Kontsevich, Enumeration of rational curves viatorus actions, The moduli space of curves (Texel Island,1994), Progr. Math., vol. 129, BirkhΓ€user Boston, Boston,MA, 1995, pp. 335β368, DOI 10.1007/978-1-4612-4264-2_12. MR1363062
[13] Yongbin Ruan and Gang Tian, A mathematical theory ofquantum cohomology, J. Differential Geom. 42 (1995), no. 2,259β367. MR1366548
[14] Francesco Severi, Vorlesungen ΓΌber algebraische Geometrie:Geometrie auf einer Kurve, Riemannsche FlΓ€chen, Abelsche In-tegrale (German), Berechtigte Deutsche Γbersetzung vonEugen LΓΆffler. Mit einem EinfΓΌhrungswort von A. Brill.Begleitwort zum Neudruck von Beniamino Segre. Bib-liotheca Mathematica Teubneriana, Band 32, JohnsonReprint Corp., New York-London, 1968. MR0245574
[15] Ravi Vakil, Counting curves on rational surfaces,Manuscripta Math. 102 (2000), no. 1, 53β84, DOI10.1007/s002291020053. MR1771228
Lucia Caporaso
Credits
Opener image is courtesy of Getty.Figures 1β4 are courtesy of Lucia Caporaso.Author photo is courtesy of Dario Birindelli.
JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 779
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