Wall crossing for K-moduli spaces of plane curves
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866929330402099200 |
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| author | Ascher, Kenneth DeVleming, Kristin Liu, Yuchen |
| author_facet | Ascher, Kenneth DeVleming, Kristin Liu, Yuchen |
| contents | 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 $d \geq 4$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_04576 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Wall crossing for K-moduli spaces of plane curves Ascher, Kenneth DeVleming, Kristin Liu, Yuchen Algebraic Geometry Differential Geometry 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 $d \geq 4$ 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. |
| title | Wall crossing for K-moduli spaces of plane curves |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/1909.04576 |