ragni piene centre of mathematics for applications ...introduction curves surfaces curves on...

Introduction Curves Surfaces Curves on surfaces

Curves and surfaces

Ragni PieneCentre of Mathematics for Applications,

University of Oslo, Norway

“What is algebraic geometry?”IMA, April 13, 2007


Outline

Introduction

Curves

Surfaces

Curves on surfaces


Introduction

Algebraic geometry =study of common solutions of a given set of polynomial equations.

The equations 3x + 5y = 1 and 2x− 9y = 13have the common solution x = 2 and y = −1


What is so fascinating about solving equations?

The interplay between algebraic manipulations and geometricinterpretation.

The above solution can be found by (linear) algebra, or bygeometry (the intersection of two lines).

ALGEBRA +GEOMETRY = TRUE


What is a curve?

Intuitively, a curve is a geometric object of dimension one:a moving particle traces a curve.

For example, the leminiscate is given as the set of points (x, y) inthe plane such that

x = sin t1+cos2 t

and y = sin t cos t1+cos2 t

Is this an algebraic curve?


Yes: as a point set, it is equal to the plane curve given by theequation

(x2 + y2)2 = x2 − y2

and it has the rational parameterization

x =u + u3

1 + u4and y =

u− u3

1 + u4

The parameterized curve x = t, y = sin t is not algebraic:it intersects the x-axis in infinitely many points.


Projective curves

An algebraic variety V = V(f1, . . . , fr) ⊂ Pn(k) is the set ofcommon zeros of finitely many homogeneous polynomials.

A rational function on V is the restriction to V of a rationalfunction f

g on Pn(k), where f and g are polynomials of the samedegree.

The function field K = K(V ) is the set of rational functions on V .

The dimension of V is equal to the transcendence degree of Kover k.

V is a curve if tr degk(K) = 1.

Nonsingular projective curves with the same function field areisomorphic.

Any algebraic curve has a unique nonsingular model.


The arithmetic genus

Recall that the Hilbert polynomial P (t) of a variety V ⊂ Pn(k)satisfies (for m big enough) P (m) =

(n+m

n

)− dimk I(V )m.

The arithmetic genus pa of V is defined to be

pa = (−1)dim V (P (0)− 1)

If V is a curve, P (t) = dt + 1− pa, where d is the degree of V .

Example. Let V ⊂ P3 be the twisted cubic, given byu 7→ (1 : u : u2 : u3).

Then the ideal I(V ) is generated by the 2-minors of the matrix[x0 x1 x2

x1 x2 x3

]We find P (t) = 3t + 1 and pa = 0.


Plane curves

A plane curve: V = V(f) ⊂ P2(k), where f is a homogeneouspolynomial.

The degree d of V is the degree of f .

P (t) =(t+22

)−

(t−d+2

2

)= dt + 1−

(d−12

)The arithmetic genus of a plane curve of degree d is

pa = (d− 1)(d− 2)/2

Here are real (k = R) curves of arithmetic genus 1 and 0:


Topological genus

A complex (k = C) algebraic curve can be viewed as atwo-dimensional real manifold, a Riemann surface.

Topological genus: g=# holes in the Riemann surface

Euler number: e = # vertices −# edges +# faces in atriangulation

Euler–Poincare formula: e = b0 − b1 + b2 = 1− 2g + 1 = 2− 2g(The bi are the Betti numbers.)

The first curve has topological genus 1, the second curve hastopological genus 0.


Hirzebruch–Riemann–Roch:

Topological genus = Arithmetic genus

Proof. Project V ⊂ Pn to a plane curve V ⊂ P2 with δ doublepoints.

Algebra. Compare the arithmetic genus of V with that of V :

pa(V ) = pa(V )− δ =(d− 1)(d− 2)

2− δ

Geometry. Compute the number d∗ of tangents to V through agiven point P :

d∗ = d(d− 1)− 2δ


Topology. Project V from P to get a map V → P1.

The map is a topological d-fold cover, with d∗ ramification points.Compare the topological genus of V with that of P1:

Triangulate the sphere P1 and lift the triangulation to V :

e(V ) = (d#{vertices of P1}−d∗)−d#{edges of P1}+d#{faces of P1}

e(V ) = d · e(P1)− d∗

2− 2g = d(2− 2 · 0)− d∗ = 2d− d∗


Algebra:

pa =12(d− 1)(d− 2)− δ

Geometry:d∗ = d(d− 1)− 2δ

Topology:2− 2g = 2d− d∗

Algebra + Geometry + Topology ⇒ 1− pa = 1− g


Classification of curves

Have noted that all nonsingular projective curves with the samefunction field are isomorphic and that any curve has a uniquenonsingular model.

Classification problem: describe the “moduli space” of all curves ofgenus g.

A curve of genus 0 is called rational. The nonsingular model of anyrational curve is P1.

All tori are topologically the same, but: There is a one-dimensionalfamily of algebraic curves of genus 1.

Indeed, all Riemann surfaces of the same genus are topologicallythe same, but: The moduli space of algebraic curves of genus ghas dimension 3g − 3, for g ≥ 2.


Classification of curves in a fixed space

The Hilbert scheme parameterizes the set of curves of given degreeand given arithmetic genus, in a given projective space.

• The Hilbert scheme of degree d curves in P2 is the projectivespace Pd(d+3)/2.

Indeed, a curve of degree d is given by a homogenous polynomialin three variables, of degree d. The space of such polynomials hasdimension (

d + 22

)=

d(d + 3)2

+ 1.

• The Hilbert scheme of degree 3 curves of genus 0 in P3 is theunion of a nonsingular 12-dimensional and a nonsingular15-dimensional algebraic variety, intersecting along anonsingular 11-dimensional variety.


Surfaces

A surface is a variety V ⊂ Pn(k) of dimension 2.

Dimension 2 means that the transcendence degree over k of thefunction field K(V ) is 2.

The arithmetic genus: pa = P (0)− 1

Topologically, a surface is a four-dimensional real variety.

The Euler number e =∑

(−1)ibi is a topological invariant.

The genus p of a “canonical curve” on V is another topologicalinvariant.


Noether’s formula

1 + pa =112

(p− 1 + e)

Proof. Project V to a surface V ⊂ P3 with generic singularities (adouble curve and finitely many triple points and pinch points).


Algebra. Compute the arithmetic genus of V in terms of thearithmetic genera of V , the double curve, and the inverse image ofthe double curve (“conductor square”).

Geometry. Compute the class of V : the number of tangent planesto V through a given line L. Compute the rank of V : the numberof tangent planes to V at points in H ∩ V through a given pointP .

Topology. Compute the ramification curve of the projectionV → P2 in terms of topological invariants of V and P2.

Algebra + Geometry + Topology = OK


Classification of surfaces

Given a function field K of transcendence degree 2 over k, thereare infinitely many surfaces V such that K(V ) = K.

The reason for this is that one can blow up a point on a surface:replace the point by the set of all the tangent directions throughthat point. This set is a P1, and is called an exceptional curve.

A surface which contains no exceptional curves is called minimal.

Almost all surfaces have a unique minimal model.

Enriques–Kodaira classification:

• κ = −∞ : rational surfaces and ruled surfaces over curves ofgenus > 0

• κ = 0 : Enriques surfaces, K3 sufaces, and elliptic surfaces

• κ = 1 : elliptic fibrations over a curve of genus ≥ 2• κ = 2 : surfaces of general type


Curves on surfaces

Try to describe a variety by describing its subvarieties.

Given a surface V , what kind of curves does it contain?

For example, a surface V ⊂ Pn that contains infinitely many linesis a ruled surface (κ = −∞).

• Can any surface V be “covered” by rational curves?

• Are there surfaces containing no rational curves?

• If there are only finitely many rational curves, can one countthem?


Rational curves on P2

A curve in P2 of degree d with only δ double points as singularitiesis rational (has genus 0) if and only if

δ = (d− 1)(d− 2)/2.

The set of plane curves of degree d is Pd(d+3)/2.

The curves with (d− 1)(d− 2)/2 double points is a subvariety ofdimension d(d + 3)/2− (d− 1)(d− 2)/2 = 3d− 1.

Let Nd denote the number of plane rational curves of degree dpassing through 3d− 1 given points.

N1 = 1, N2 = 1, N3 = 12, N4 = 620, . . .


Kontsevich’s recursion formula:

Nd =∑

d1+d2=d

Nd1Nd2

(d2

1d22

(3d− 43d1 − 2

)− d3

1d2

(3d− 43d1 − 1

))

If we set nd := Nd(3d−1)! , then

nd =∑

d1+d2=d

nd1nd2

d1d2((3d1 − 2)(3d2 − 2)(d + 2) + 8(d− 1))6(3d− 1)(3d− 2)(3d− 3)

Kontsevich’s proof used a degeneration argument: mapping theone-dimensional family of curves through 3d− 2 points to P1

using the cross-ratio, and counting degrees above 0 and 1.


Quadric surfaces

A quadric surface V ⊂ P3 is isomorphic to P1 ×P1, embedded viathe Segre map:

(s : t)× (u : v) 7→ (su : sv : tu : tv)

There are two families of lines on V . Through each point of Vpass two lines, one from each family.


Lines on a cubic surface

A cubic surface V ⊂ P3 contains 27 lines!

Every cubic surface is isomorphic to P2 blown up in 6 points. The6 exceptional curves are six of the 27 lines. The transforms of the15 lines through two of the six points are also among the 27 lines.The last 6 lines are the transforms of the conics through 5 of the 6points.

6 +(

62

)+

(65

)= 6 + 15 + 6 = 27


Combinatorial sidestep: partitions

Let p denote the partition function: p(n) is the number of ways ofwriting n = n1 + . . . + nk, where n1 ≥ · · · ≥ nk ≥ 1

The generating function for p,

ϕ(q) :=∑n≥0

p(n)qn,

is equal to the formal power series

Πm≥1(1− qm)−1 = 1 + q + 2q2 + 3q3 + 5q4 + . . . .


Rational curves on a quartic surface

Let V ⊂ P3 be a surface of degree 4.

For each integer r, let C ⊂ V be a curve such that C2/2 = r − 1,and setNr := #{ rational curves with r double points, “linearlyequivalent” to C}

The generating function for the integers Nr is

f(q) =∑

r

Nrqr

Surprise (Yau–Zaslow):

f(q) = Π(1− qm)−24 = ϕ(q)24

where ϕ is the generating function of the partition function.


Proof. (Bryan–Leung) Pass to symplectic geometry!

Degenerate V to a fibration V ′ → P1 where a general fiber hasgenus 1, with 24 fibers that have one double point and hence arerational, and C to C ′ = S + rF , where S = s(P1) is a section andF is a fiber.

Want to count curves that degenerate to S +∑24

i=1 aiFi,where F1, . . . , F24 are the fibers with one double point, and where∑24

i=1 ai = r.

There are p(ai) ways to degenerate to aiFi, hence p(a1) · · · p(a24)ways to degenerate to S +

∑24i=1 aiFi.

Therefore Nr is the coefficient of qr in 24th power of thegenerating function ϕ(q).


String theory and enumerative geometry

A quartic surface in P3 is an example of a Calabi–Yau variety.

Calabi–Yau varieties appear “naturally” in the superstring model ofthe universe.

Rational curves on a (three-dimensional) Calabi–Yau variety areinterpreted as instantons. Physicists predicted the generatingfunction for instantons on a three-dimensional variety in P4 ofdegree 5.

This lead to a boom for classical enumerative geometry, using newtools coming from physics — mirror symmetry, quantumcohomology, and Gromov–Witten theory.

The interaction between algebraic geometers, symplecticgeometers, and theoretical physicists has been intense andsurprising, with mutual benefits.


References

1. D. Mumford, Algebraic geometry I – Complex ProjectiveVarieties. Springer-Verlag, Berlin-Heidelberg-New York 1976.

2. J. Harris, I. Morrison, Moduli of curves. Graduate Texts inMathematics, 187. Springer-Verlag, New York, 1998.

3. R. Piene, M. Schlessinger, On the Hilbert schemecompactification of the space of twisted cubics, Amer. J.Math. 107 (1985), no. 4, 761–774.

4. R. Piene, A proof of Noether’s formula for the arithmeticgenus of an algebraic surface, Compositio Math. 38 (1979),no. 1, 113–119.

5. A. Beauville, Complex algebraic surfaces. LondonMathematical Society Student Texts, 34. CambridgeUniversity Press, Cambridge, 1996.


6. M. Kontsevich, Yu. Manin, Gromov–Witten classes, quantumcohomology, and enumerative geometry, Comm. Math. Phys.164 (1994), 525–562.

7. M. Kontsevich, Enumeration of rational curves via torusactions, in “The moduli space of curves (Texel Island, 1994),”335–368, Progr. Math. 129, Birkhauser Boston, Boston, MA,1995.

8. J. Bryan, N. C. Leung, The enumerative geometry of K3surfaces and modular forms, J. Amer. Math. Soc. 13 (2000),no. 2, 371–410.

9. D. Cox, S. Katz, Mirror symmetry and algebraic geometry.Mathematical Surveys and Monographs, 68. AmericanMathematical Society, Providence, RI, 1999.

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