Wall crossing for K-moduli spaces of plane curves · arXivDesk
1909.04576Sep 10, 2019V2: 94 pages. Many typos fixed, and proofs of Theorems 5.2, 5.14, and 5.15 have been clarified (Lemma 5.16 is new). Postscript added (Remark 1.9) to discuss progress in K-moduli since the first version of this paper. To appear in Proceedings of the London Math Society
We construct proper good moduli spaces parametrizing K-polystable Q-Gorenstein smoothable log Fano pairs (X,cD), where X is a Fano variety and D is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as c
Nearby in the stack
varies. The main application in this paper is the case of plane curves of degree
d≥4
as boundary divisors of
P2
. In this case, we show that when the coefficient
c
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.