Geometric invariant theory of syzygies, with applications to moduli spaces

TL;DR

This paper develops a geometric invariant theory framework for syzygies of projective schemes, applying it to moduli spaces of K3 surfaces and genus six curves, leading to new insights and constructions.

Contribution

It introduces the concept of syzygy points and explores their GIT stability, providing new results for moduli spaces of K3 surfaces and genus six curves.

Findings

Progress in GIT stability of syzygy points for K3 surfaces and genus six curves

Effectivity results for divisor classes on moduli spaces of K3 surfaces

A new construction in the Hassett-Keel program for genus six curves

Abstract

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and of genus six canonical curves. Applications of our results include effectivity statements for divisor classes on the moduli space of odd genus K3 surfaces, and a new construction in the Hassett-Keel program for the moduli space of genus six curves.