Moduli of genus six curves and K-stability

The K-moduli theory provides different compactifications of various moduli spaces, including moduli of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces M ¯ K (c) MK (c) of the quintic log Fano pairs. We classify the strata of genus six curves C C appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by Geometric Invariant Theory (GIT) and moduli of K3 surfaces via Hodge theory.