Stability for (bi)canonical curves

Friday 1st March, 12:05-12:55pm, Carslaw 454 - Please note the revised location

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