David Smyth
Harvard University

Title: Stability of finite Hilbert points and moduli of singular curves

Abstract: The classical construction of the moduli space of stable curves via Geometric Invariant Theory relies on the asymptotic stability result of Gieseker that the m-th Hilbert Point of a pluricanonically embedded curve is GIT-stable for all sufficiently large m. Several years ago, Hassett and Keel observed that if one could carry out the GIT construction with non-asymptotic linearizations, the resulting models could be used to run a log minimal model program for the space of stable curves. A fundamental obstacle to carrying out this program is the absence of a non-asymptotic analogue of Gieseker's stability result, i.e. how can one prove stability of the m-th Hilbert point for small values of m? In this talk, we'll discuss some new results of this type. For example, we'll give a quick proof of the following result: the m-th Hilbert point of a general smooth canonically embedded curve of odd genus is GIT-semistabe for all m>1. This talk is based on joint work-in-progress with Jarod Alper and Maksym Fedorchuk.