Stability for (bi)canonical curves
by Jarod Alper
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.