# Stability for (bi)canonical curves

*Algebra/Geometry/Topology Seminar*

*by Jarod Alper*

*Institution:*Australian National University

*Date: Fri 12th April 2013*

*Time: 3:15 PM*

*Location: 213 Richard Berry*

*Abstract*: 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.