Wall crossing for K-moduli spaces of plane curves

Abstract

We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-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 $c$ varies. The main application in this paper is the case of plane curves of degree $d\ge 4$ as boundary divisors of ${\mathbb{P}}^2$. 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.

V2: 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

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• K-stability of Fano varieties: an algebro-geometric approach
• On K-stability of some del Pezzo surfaces of Fano index 2
• Applications of the moduli continuity method to log K-stable pairs
• Equivariant K-stability under finite group action
• K-stability of cubic fourfolds
• K-moduli of curves on a quadric surface and K3 surfaces
• Polystable log Calabi-Yau varieties and Gravitational instantons
• On wall crossing for K-stability
• On the volume of some Fano K-moduli spaces
• On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line
• Log K-stability of GIT-stable divisors on Fano varieties