Geometry of Moduli Spaces of Curves

Explore the geometry of moduli spaces of curves through this comprehensive summer school program covering two fundamental areas of algebraic geometry. Delve into vector bundles of coinvariants and conformal blocks, which serve as invariants of curves attached to Lie groups and are canonically isomorphic to global sections of ample line bundles on moduli stacks of G-bundles. Study how these bundles generalize theta functions for smooth curves and examine their global generation properties in the case of genus g=0, where their first Chern classes provide semi-ample line bundles that illuminate the birational geometry of moduli spaces. Investigate the cohomology classes on moduli spaces of curves and abelian varieties, focusing on tautological classes and cohomology classes related to spaces of modular forms. Learn about the properties of moduli spaces and their compactifications, natural cohomology classes and their relationships, and the challenging problem of determining relationships between tautological classes. Examine how moduli spaces of curves can be used to understand generalized theta functions and explore the connections between different types of cohomology classes. Cover advanced topics including Gromov-Witten invariants of complex Lagrangian Grassmannians, the dimension of moduli spaces of monomial curves through Deligne formula variations, the average size of 2-Selmer groups over function fields, d-elliptic loci in genus 2, birational geometry of moduli spaces of points on lines, and computation of Gopakumar-Vafa invariants. Gain insights into open problems such as the F-conjecture and develop understanding of nef divisors in any genus, Chern classes of bundles of coinvariants, and global sections of ample line bundles on Bun_G(C) in both smooth and nodal cases.