Minimal Models of Moduli Spaces

University Of Texas At Austin, Austin TX

Investigators

Abstract

DMS-0354994 Sean M. Keel The proposed research centers on the study of fundamental objects in mathematics and theoretic physics: moduli spaces of hyperplane arrangements, moduli spaces of curves, and moduli spaces of Abelian varieties. The research involves the application of basic ideas of Mori theory. For example the largest portion of the proposal, and the portion with the largest potential payoff has to do with finding a natural compactification of the moduli of hyperplane arrangements (i.e. moduli of ordered n-tuples of hyperplanes in projective space modulo automorphisms of the projective space) with reasonable singularities. This is of interest for example because the obvious compactifications -those that come from geometric invariant theory, have boundary with literally arbitrary singularities (all possible affine schemes occur as boundary strata), so a natural desingularisation would include a desingularisation of all singularities at once. The researcher makes the elementary observation that the moduli space of hyperplane arrangements is (log) minimal, and thus the main conjectures of Mori theory predict the existence of a canonical compactification, with reasonable boundary. The main question is to find this compactification. The researcher proposes to consider the geometry of some of the basic parameterising spaces in mathematics -- the so-called moduli spaces of hyperplane arrangements, Riemann surfaces, and Abelian varieties. The unifying theme of the proposal is the application of fundamental ideas from one branch of geometry, Mori theory, to other branches, where the ideas have not yet been widely exploited. The simplest and potentially deepest example is in the first and main part of the proposal, where the researcher observes that Mori theory predicts the existence of an absolutely canonical object for the resolution of singularities -- one of the central problems in geometry.