Math 245a - Topics in Algebraic Geometry

Dragos Oprea, TuTh 2:40-3:55, 381T.

The topic for this quarter is the geometry of the moduli space of curves. The rough plan is to spend some time constructing the Hilbert scheme, and then the moduli space. The last part of the course will introduce the tautological classes, and present examples and computations in low genus, as well as some intersection theory on the moduli space of stable curves.

• Week 1: The Grassmannian. Schubert calculus: Pieri and Giambelli's formulas. Vafa-Intriligator on the Grassmannian. Applications: the number of lines in projective space intersecting codimension 2 hyperplanes.
• Week 2: The Quot functor. Example: the Quot scheme on P^1 can be used to compactify spaces of morphisms to Grassmannians. Smoothness, rationality and irreducibility of Quot on P^1. Quantum cohomology of Grassmannians defined. Hilbert scheme of points on a surface. Euler characteristic computed in terms of modular forms (eta functions). Connection with representation theory.
• Week 3: Regularity of sheaves. Examples, properties and conjectures about regularity of sheaves. Uniform bounds on regularity of subsheaves. The embedding of Quot into a Grassmannian.
• Week 4: Flattening stratifications. Quot scheme constructed. Projectivity. Lehn's theorem: the Quot scheme as a zero locus in a product of grassmannians.
• Week5: Tangent space and tangent sheaf to the Quot scheme (two ways: via deformation theory and via Lehn's theorem). Started the construction of the moduli space of curves. Showed the functor cannot be representable. Showed the strategy of the construction. Showed some low genus examples (g=2, 3, 4). Introduced the DM compactification.
• Week 6-7: Quotients: categorical, good, geometric. Affine quotients. The quasiprojective case. Linearizations. Construction of the quotient. The Hilbert-Mumford criterion and proof. Examples: smooth hypersurfaces, and others. Stability of curves (just stated the results, not too many proofs).
• Week 8: M_{0,n} bar. The Chow is tautological. Relations from M_{0,4} bar. The Picard group. Fulton's conjectures. Psi classes. Kapranov's description of M_{0,n} bar. Integrals of psi classes via the string equation. Proof of the string equation.
• Weeks 9+epsilon: Genus 1 and the dilaton equation. The Witten conjectures. The lowest terms predictions: psi_1^{top} on M_{g,1} bar. Direct proof of these intersections due to Faber and Pandharipande. Mumford relations. Tautological rings. Tautological rings of the moduli of abelian varieties.