Dave Swinarski (Fordham University)
-
PDL C-38
Preseminar Title: Introduction to Geometric Invariant Theory
Abstract: We introduce some of the basic concepts of geometric invariant theory (GIT), with examples.
Title: The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity
Abstract: Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a "worst'' destabilizing 1-parameter subgroup $\lambda$. What are the worst 1-parameter subgroups for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples. This is joint work with Joshua Jackson (Cambridge).
Format: beamer