Geometry of Compact Moduli Spaces

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: moduli spaces of stable pairs introduced by Kollar, Shepherd-Barron, and Alexeev. The deepest results are expected for moduli of Del Pezzo surfaces and certain classes of surfaces of general type, for moduli of stable rational curves, and for their higher-dimensional analogues: compact moduli spaces of hyperplane arrangements. New methods are based on the interaction between Mori theory and tropical algebraic geometry, which assigns to any algebraic variety an object of combinatorial nature, basically a polytope. This polytope (a tropical variety) surprisingly encodes a lot of geometric information about the variety and most importantly about its compactifications. Tropical varieties were originally introduced in the community of computational algebraic geometers who study how to read invariants of algebraic varieties from their equations. It turns out that proposed methods can also be used backwards to advance computational algebraic geometry. In particular, they can be used to find implicit equations of algebraic varieties given parametrically. 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.