**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)

### Algebra Seminar

# Stability for (bi)canonical curves

### Alper

###### 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

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