An easy introduction to Algebraic Geometry and Rational ...folk.uio.no/torgunnk/CMA.pdfIntroduction...

CMA seminarUniversity of Oslo

11.March 2010

An easy introduction toAlgebraic Geometry

andRational Cuspidal Plane Curves

Torgunn Karoline MoeCMA/MATH

University of Oslo

Torgunn Karoline Moe

Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

The most interesting objects in the world

• How many and what kind of cusps can a rationalcuspidal plane curve have?


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

This happens today

• Basic algebraic geometry

• Plane algebraic curves

• Singularity theory

• Rational cuspidal plane curves


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Algebraic geometry in a nutshell

• The study of geometric objects using algebraicmethods.

• Can find new and surprising properties of both theobjects and the methods.

• Main tool: commutative algebra.• Rings• Ideals

• Main objects: varieties.• Curves• Surfaces


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

The worlds we work within

• Algebraically closed fields – C.

• Affine spaces of dimension n – Cn,(x1, . . . , xn).

• Projective spaces of dimension n – PnC,

(x0 : . . . : xn).

• Pn can be constructed using Cn+1, identifying pointsin the affine space lying on the same line through theorigin.

Pn ∼= (Cn+1 r {(0, . . . , 0)})/ ∼,

(a0 : . . . : an)∼(λa0 : . . . : λan), ∀ λ ∈ C∗.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

The objects we work with

• An algebraic set in Cn is the zero set V of a finiteset of polynomials in the ring C[x1, . . . , xn].

• An algebraic set in Pn is the zero set V of a finite setof homogeneous polynomials in C[x0, . . . , xn].

• In our worlds open sets are complements of algebraicsets.

• An affine variety is a closed subset of Cn which cannot be decomposed into smaller, closed subsets.

• A projective variety is an irreducible closed subset ofPn.

• An algebraic set in a space of dimension n defined bya single irreducible (homogeneous) polynomial is ahypersurface – a variety of dimension n − 1.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Let’s get it all down to earth

• The projective plane P2 has coordinates (x : y : z).

• One irreducible homogeneous polynomial F (x , y , z)defines V(F ) – a curve in P2.

• The degree of the curve is the degree of thepolynomial.

• Letting z = 1, the polynomial f (x , y) = F (x , y , 1)will define a curve V(f (x , y)) in a space isomorphicto C2.

• Letting y = 1, we get the curve V(f (x , z)) inanother affine plane.

• Letting x = 1, we get V(f (y , z)).

• These three affine curves constitute the projectivecurve V(F ).

• Technically, we have covered P2 by three open affinesets isomorphic to C2,

P2 r V(z), P2 r V(y), P2 r V(x).


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

A typical conic I - V(x2 + y2 − z2)

z = 1 y = 1 x = 1

• Remember that this is just the real picture - thecomplex world hides its secrets.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

A typical conic II - V(x2 + y2+(z − 1)2)

• All conic curves in P2 (circles, ellipses, hyperbolasand parabolas) are equivalent when we work over C.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

The nodal cubic - V(zx2 − zy2 − x3)

• This curve has one obviously interesting point in(0 : 0 : 1).


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

The cuspidal cubic - V(zy2 − x3)

• This curve also has an interesting point in (0 : 0 : 1),but it is different from the point in the previousexample.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Let’s compare

The nodal cubic The cuspidal cubic

• How and why are these interesting and different -even over C?


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Some more theory I

• A point a = (a0 : a1 : a2) on a curve V(F ) is calledsingular if it is in the zero set of all the partialderivatives of F ,

V(Fx(a),Fy (a),Fz(a)).

• A curve can only have a finite number of singularpoints.

• The other points on the curve are called smooth.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Some more theory II

• Every smooth point a on a curve has a uniqe tangentline given by

V(Fx(a)x + Fy (a)y + Fz(a)z).

• Every singular point p has one or more tangentline(s).

• For p = (0 : 0 : 1) singular,

F (x , y , 1) = fm(x , y) + fm+1(x , y) + . . . + fd(x , y).

• The tangent line(s) of C at p is given by the zeroset(s) of each reduced linear factor of fm(x , y).

• A tangent line is special because it touches the curveat the given point a bit more than other lines.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

More about the interesting points

• Unbelieveably many different kinds of singularities.

• Can classify singularities using invariants:• Branches – counting the number of times the curve

passes through the point.• A singularity with more than one branch is called a

multiple point.• A singularity with only one branch is called a cusp.

• Multiplicity – the amount of intersection between ageneral line and the curve at the point.

• Is equal to the m in fm(x , y) for p = (0 : 0 : 1).

• Tangent intersection – the intersection multiplicityof the tangent line and the curve at the point.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Multiplicity 2


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Detonating the algebraic bomb

• Can investigate the inside of a singularity by blowingit up.

• Replace the singularitiy with a projective line.

• In an affine neighbourhood of the singularity, look atall the lines through the point.

• Lift each line to a height corresponding to the slopeof the line.

• Observe that the curve is practically unchangedoutside the singularity.

• Close the curve and get a new curve.

• Look at the point(s) of the new curve correspondingto the blown up singularity.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Blowing up the cuspidal cubic


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

• Yes, the blown up space is strange and funny. And it doesn’t reallylook like that. But don’t worry.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Useful properties of a cusp

• When a cusp is blown up, we have only one pointcorresponding to the singularitiy.

• This point might still be singular.

• Then we blow up again.

• Let mi denote the multiplicity of the remainingsingularity after i blowing-ups.

• For a cusp we define the multiplicity sequence m• m = (m,m1, . . . ,ms).• Have m ≥ m1 ≥ . . . ≥ ms .• There are more restrictions here.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Let’s go back to start

• How many and what kind of cusps can a rationalcuspidal curve have?

• A curve is called cuspidal if all its singular points arecusps.

• A curve of degree d is rational ⇐⇒(d − 1)(d − 2)

2=

∑singular points

(∑

i

mi (mi − 1)

2).

• A rational curve can be given by a parametrization.

• By the formula, the cuspidal cubic is the onlyrational cuspidal curve of degree 3.

• A rational cuspidal plane curve of degree d must alsosatisfy

• Bezout: mp + mq ≤ d .• Matsuoka–Sakai: d < 3 ·m,

where m is the highest multiplicity of the cusps.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Rational cuspidal curves of degree 4

(2), (2), (2) (22), (2) (23)

(3)


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Rational cuspidal curves of degree 5

# Cusps Curve Cuspidal configuration # Curves

1C1 (4) 3 – ABCC2 (26) 1

2C3 (3, 2), (22) 2 – ABC4 (3), (23) 1C5 (24), (22) 1

3C6 (3), (22), (2) 1C7 (22), (22), (22) 1

4 C8 (23), (2), (2), (2) 1


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Conjecture [Piontkowski (2007)]

• There is only one rational cuspidal plane curve withmore than three cusps – the curve of degree 5 withcuspidal configuration [(23), (2), (2), (2)].

• The only tricuspidal curves are• [Fenske, Flenner & Zaidenberg (1996-1999)]

Series d mp mq mr For dI d (d − 2) (2a) (2d−2−a) d ≥ 4II 2a + 3 (d − 3, 2a) (3a) (2) d ≥ 5III 3a + 4 (d − 4, 3a) (4a, 22) (2) d ≥ 7

• The curve of degree 5 with cuspidal configuration[(22), (22), (22)].

Result [Tono (2005)]

• A rational cuspidal curve has ≤ 8 cusps.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

A world of opportunities for me

• Use Cremona transformations to give newrestrictions.

• Construct cuspidal curves by projecting a rationalsmooth curve in Pn.

• There is a connection between the number of cuspsand the centre of projection that is used.

• This is linked to the tangents of the smooth curve inPn.

• Try to interpret the problem in other worlds; i.e.toric geometry or tropical geometry.


Cuspidal curves

Introduction

Algebraicgeometry

Algebraiccurves

Singularitytheory

Cuspidal curves

References

Useful literature

T. FenskeRational cuspidal plane curves of type (d , d − 4) withχ(ΘV 〈D〉) ≤ 0.

H. Flenner, M. ZaidenbergOn a class of rational cuspidal plane curves.

H. Flenner, M. ZaidenbergRational cuspidal plane curves of type (d , d − 3).

R. HartshorneAlgebraic Geometry.

M. Namba.Geometry of projective algebraic curves.

J. Piontkowski.On the Number of the Cusps of Rational CuspidalPlane Curves.

I hope there’s more cake!

An easy introduction to Algebraic Geometry and Rational ...folk.uio.no/torgunnk/CMA.pdfIntroduction...

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