Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911691745263616 |
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| author | Fang, Hanlong Nguyen, Bin Wu, Xian Zhang, Zheng |
| author_facet | Fang, Hanlong Nguyen, Bin Wu, Xian Zhang, Zheng |
| contents | We study a family of canonically polarized surfaces introduced by Persson, which arise as Galois $G=(\mathbb{Z}/2\mathbb{Z})^4$-covers of $\mathbf{P}^2$ branched along eight general lines. For this family, we construct the compactified moduli space and explicitly describe the stable degenerations in the sense of Kollár, Shepherd-Barron, and Alexeev (KSBA) via stable pairs of weighted hyperplane arrangements. By computing the $\mathbb{Q}$-Gorenstein obstructions and using the KSBA wall crossings, we show that the resulting compactified moduli stack is smooth. Furthermore, we establish a generic global Torelli type result: up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its étale double cover, together with the associated $\widetilde{G}=(\mathbb{Z}/2\mathbb{Z})^5$-action. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17223 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem Fang, Hanlong Nguyen, Bin Wu, Xian Zhang, Zheng Algebraic Geometry We study a family of canonically polarized surfaces introduced by Persson, which arise as Galois $G=(\mathbb{Z}/2\mathbb{Z})^4$-covers of $\mathbf{P}^2$ branched along eight general lines. For this family, we construct the compactified moduli space and explicitly describe the stable degenerations in the sense of Kollár, Shepherd-Barron, and Alexeev (KSBA) via stable pairs of weighted hyperplane arrangements. By computing the $\mathbb{Q}$-Gorenstein obstructions and using the KSBA wall crossings, we show that the resulting compactified moduli stack is smooth. Furthermore, we establish a generic global Torelli type result: up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its étale double cover, together with the associated $\widetilde{G}=(\mathbb{Z}/2\mathbb{Z})^5$-action. |
| title | Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.17223 |