Math 259B - Enumerative Geometry and Moduli
Welcome to Math 259!
The basic question of enumerative geometry can be simply stated
How many geometric objects of a given type satisfy given geometric
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
In many cases, this comes down to calculating intersection
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
(This may be a bit too ambitious for a one quarter course, so the
may change slightly as we go.)
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: MWF, 12pm-12:50pm, TBA.
- Wednesday 1-2pm.
lecture or by appointment. Also, feel free to drop in if you see me in my
- 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
complex geometry at the
level of Math 250C.
- There will be no homeworks or exams for this
- Lecture 1:
Motivation: why we care about intersection theory on moduli
Examples of moduli problems. Fine and coarse moduli spaces.
- 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
on surfaces, hypersurfaces in projective space, conics and twisted cubics
- 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
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
isomorphism after inverting torus characters. Proof that the topological
equals that of the torus-fixed point locus.
- Lecture 9: Some intersection theory:
via vanishing loci of sections. Chern
Equivariant Euler and Chern classes. Examples.
- Lecture 10: Tools: the localization theorem for
points. Proof and first examples.
- Lecture 11: Examples of rational curve counts: 27
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
- Lecture 14: Strategy for constructing the Hilbert
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
- 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
moduli of genus 0 marked curves. Psi classes.
- Lecture 26: String equation. Descendant integrals in
- Lecture 27: Descendant integrals in genus 1 and the dilaton
equation. Higher genus: Witten's conjecture. The tautological ring and