Geometry of Moduli Spaces of Curves and Surfaces

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The principal investigator plans to study geometry of moduli spaces of stable curves of Deligne, Knudsen, and Mumford and its higher dimensional analogues, namely the moduli spaces of stable surfaces introduced by Kollar, Shepherd-Barron, and Alexeev. The first goal is to uncover birational geometry of the compact moduli space of algebraic curves. The main conjecture is inspired by Mirror Symmetry from physics and classical algebraic geometry of linear series. It describes the cone of effective divisors and (some part of) the variation of birational contractions of the moduli space of stable rational curves in terms of compactified Jacobians of reducible curves of high genus. The second goal is to develop new methods of explicitly describing the moduli spaces of stable canonically polarized surfaces and to apply these methods in several classical situations. Algebraic geometry studies algebraic varieties: shapes defined by systems of polynomial equations. Algebraic varieties have discrete characteristics that allow to classify their species: rational curves, Del Pezzo surfaces, Abelian varieties, Calabi-Yau varieties, varieties of general type, etc. Varieties of each type depend on certain continuous parameters (called moduli) and the set of all parameters has a rich structure of the so-called moduli space. One is particularly interested in compact moduli spaces that parametrize varieties with allowed mild degenerations. For example, a hyperbola xy=C on the plane can degenerate to the union of two lines xy=0 when C goes to 0. The principal investigator will study these compact moduli spaces and related problems in algebraic geometry. This centuries-old concept of pure mathematics has rich relationship with physics, and applications to computational algebraic geometry will lead to new algorithms useful in algebraic statistics and mathematical biology.