SMS scnews item created by John Enyang at Sun 17 Feb 2013 1350
Type: Seminar
Modified: Fri 22 Feb 2013 1308; Fri 22 Feb 2013 1311
Distribution: World
Expiry: 2 Mar 2013
Calendar1: 1 Mar 2013 1205-1255
CalLoc1: Carslaw 454
Auth: enyang@penyang.pc (assumed)

# Stability for (bi)canonical curves

### Alper

###### Speaker:

Jarond Alper (Australian National University)

###### Title:

Stability for (bi)canonical curves

###### 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.

---------------------------------------------------------------------------------------

We will take the speaker to lunch after the talk.

See the Algebra Seminar web page for information about other seminars in the series.

John Enyang John.Enyang@sydney.edu.au

Actions: