Studies in moduli theory and birational geometry

Brown University, Providence RI

Investigators

Abstract

Abramovich will continue studying problems in moduli theory, in particular (1) the moduli stacks of twisted stable maps, reductions of moduli of principal bundles in characteristic p, Gromov-Witten theory of stacks, and (2) moduli spaces of Bridgeland-Douglas semistable objects in the derived category of a variety. Abramovich will continue studying problems in birational geometry, in particular the strong factorization conjecture and the toroidalization conjecture. The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. While algebraic geometry has contributed applications in coding, industrial control, and computation, the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as components of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not a just a metaphor but a rigorous and quite useful fact. Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety. Such stacks are a central object of study of this project. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry.