Wall crossing for K‐moduli spaces of plane curves

Abstract

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.