University of Sydney Algebra Seminar

Jarod Alper (Australian National University)

Friday 1st March, 12:05-12:55pm, Carslaw 454

Stability for (bi)canonical curves

The classical construction of the moduli space of curves, \( M_g \), via Geometric Invariant Theory (GIT) relies on the asymptotic stability result of Gieseker that the \(m\)-th Hilbert Point of a pluricanonically embedded smooth 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 begin with a basic discussion of geometric invariant theory as well as how it applies to construct \(M_g\) in order to introduce and motivate the essential stability question in which this procedure rests on. The main result of the talk is: the \( m \)-th Hilbert point of a general smooth canonically or bicanonically embedded curve of any genus is GIT-semistable for all \(m > 1\). This is joint work with Maksym Fedorchuk and David Smyth.