Math 259B - Enumerative Geometry and Moduli Theory

Welcome to Math 259!

Course description:

The basic question of enumerative geometry can be simply stated as:

How many geometric objects of a given type satisfy given geometric conditions?

For instance, one may ask for
• the number of lines intersecting 4 fixed lines in P^3 (the answer is 2);
• the number of lines on a smooth cubic surface in P^3 (the answer is 27);
• the number of lines on a quintic threefold in P^4 (the answer is 2,875);
• the number of conics on a quintic threefold (the answer is 609,250);
• ...

In many cases, this comes down to calculating intersection numbers on moduli spaces.

In the first part of the course, we will discuss Chern classes and show some of their applications to enumerative questions.

In the second part, I will give an introduction to the moduli space of curves. In particular, I hope to include a discussion of the Quot scheme, of geometric invariant theory, outline the construction of the moduli space, and possibly calculate one or two concrete intersection numbers.

(This may be a bit too ambitious for a one quarter course, so the goals may change slightly as we go.)

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu, AP&M 6-101.

Lectures: MWF, 12pm-12:50pm, TBA.

Office hours:

Wednesday 1-2pm.

I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

Prerequisites:

I will make an attempt to be as self-contained as the topic permits. However, I will assume a solid background in algebraic topology, and in algebraic geometry at the level of Math 203, or alternatively some background in complex geometry at the level of Math 250C.

There will be no homeworks or exams for this class.

Lecture Summaries

• Lecture 1: Motivation: why we care about intersection theory on moduli spaces. Examples of moduli problems. Fine and coarse moduli spaces. Universal families.
• Lecture 2: Classifying spaces. Grassmannians. The moduli functor of genus g curves cannot be finely represented by a scheme. The Hilbert functor. Examples: Hilbert scheme of points on a curve, points on surfaces, hypersurfaces in projective space, conics and twisted cubics in P^3.
• Lecture 3: The Quot functor. Quot scheme over a curve. Cohomology and quantum cohomology of Grassmannians.
• Lecture 4: More examples of Quot schemes. Quot schemes of quotients of the trivial sheaf over P^1. I described it, showed smoothness and calculated the dimension.
• Lecture 5: Understanding the topology of moduli spaces. Torus actions. Euler characteristics of Grassmannians, Quot schemes of quotients of the trivial sheaf over P^1, Hilbert schemes of points over toric surfaces.
• Lecture 6: Remarks on the cohomology of the Hilbert scheme of points. Connections with representation theory. Heisenberg algebra and Fock space.
• Lecture 7-8: Introduction to equivariant cohomology. Restriction to fixed point locus in equivariant cohomology induces an isomorphism after inverting torus characters. Proof that the topological Euler characteristic equals that of the torus-fixed point locus.
• Lecture 9: Some intersection theory: Euler classes via vanishing loci of sections. Chern classes. Equivariant Euler and Chern classes. Examples.
• Lecture 10: Tools: the localization theorem for isolated fixed points. Proof and first examples.
• Lecture 11: Examples of rational curve counts: 27 lines on cubic surfaces. 2875 lines on quintic threefold. 609250 conics on quintic threefold.
• Lecture 12: Number of lines intersecting 2(n-1) hyperplanes in P^n. Schubert calculus via Vafa-Intriligator formula.
• Lecture 13: Counting higher genus curves. Motivation, and various parameter spaces for curves. Strategy for constructing the moduli space of curves. Review of theorems on curves: Riemann-Roch, Serre duality, Kodaira vanishing. Canonical embeddings of curves.
• Lecture 14: Strategy for constructing the Hilbert scheme. Castelnuovo-Mumford regularity and some examples.
• Lecture 15: Castelnuovo-Mumford's lemma. m-regularity implies vanishing of higher cohomology in all degrees at least m.
• Lecture 16: Uniform bound on regularity of subsheaves with fixed Hilbert polynomial.
• Lecture 17: Flattening stratifications defined. The Hilbert scheme constructed. The Hilbert scheme is projective. A discussion of flatness over nonsingular curves.
• Lectures 18-19: Proof of the flattening stratification.
• Lecture 20: Introduction to geometric invariant theory. Types of quotients: categorical, orbit spaces, good, geometric. Construction of affine quotients.
• Lecture 21: Locally universal families and coarse representability by geometric quotients. I illustrated this for the moduli of curves.
• Lecture 22: Affine quotients are good. Good quotients are categorical. If the action is closed, we obtain a geometric quotient.
• Lecture 23: Quasiprojective quotients. Semistable and stable points. Construction of the quotient.
• Lecture 24: Hilbert-Mumford criterion. Example: smooth plane cubics are stable. Explained how to conclude the construction of M_g once stability of smooth curves is proved.
• Lecture 25: Moduli space of stable curves. Cohomology for moduli of genus 0 marked curves. Psi classes.
• Lecture 26: String equation. Descendant integrals in genus 0.
• Lecture 27: Descendant integrals in genus 1 and the dilaton equation. Higher genus: Witten's conjecture. The tautological ring and Faber's conjectures.