Junyan Zhao, Moduli of Curves of Genus 6 and K-stability

A general curve C of genus 6 can be embedded into the unique quintic del Pezzo surface X₅ as a divisor of class -2K_X5. This embedding is unique up to the action of the symmetric group S₅. Taking a double cover of X₅ branched along C yields a K3 surface Y. Thus the K-moduli spaces of the pair (X₅, C) can be studied via wall-crossing and by relating them to the Hassett-Keel program for C and the HKL program for Y. On the other hand, X₅ can be embedded in P¹ × P² as a divisor of class O(1,2), under which -2K_X is linearly equivalent to O_X(2,2). One can study the VGIT-moduli spaces in this setting. In this talk, I will compare these four types of compactified moduli spaces and their different birational models given by wall-crossing.