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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.13526 |
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| _version_ | 1866915556049813504 |
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| author | Arai, Hayato |
| author_facet | Arai, Hayato |
| contents | We study autoequivalences and stability conditions on the derived category of coherent sheaves on a singular surface $X$ which arises as an open subvariety of a type III Kulikov degeneration of K3 surfaces. The surface $X$ consists of four irreducible components, one of which is $\mathbb{P}^2$, and the others are non-compact rational surfaces. Using a comparison with the total space of the degeneration, we show that the connected component $\mathrm{Stab}^\dagger(D^b_{\mathbb{P}^2}(X))$ of the space of stability conditions on the supported derived category $D^b_{\mathbb{P}^2}(X)$ containing geometric stability conditions is simply connected, and describe its wall-and-chamber structure via half-spherical twists. As consequences, we determine the subgroup of the autoequivalence group $\mathrm{Aut}(D^b(X))$ that preserves this component; it is isomorphic to $\mathbb{Z} \times Γ_1(3) \times \mathrm{Aut}(X)$, where $Γ_1(3) \subset \mathrm{SL}(2,\mathbb{Z})$ is the congruence subgroup of level~3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13526 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Autoequivalences and stability conditions on a degenerate K3 surface Arai, Hayato Algebraic Geometry We study autoequivalences and stability conditions on the derived category of coherent sheaves on a singular surface $X$ which arises as an open subvariety of a type III Kulikov degeneration of K3 surfaces. The surface $X$ consists of four irreducible components, one of which is $\mathbb{P}^2$, and the others are non-compact rational surfaces. Using a comparison with the total space of the degeneration, we show that the connected component $\mathrm{Stab}^\dagger(D^b_{\mathbb{P}^2}(X))$ of the space of stability conditions on the supported derived category $D^b_{\mathbb{P}^2}(X)$ containing geometric stability conditions is simply connected, and describe its wall-and-chamber structure via half-spherical twists. As consequences, we determine the subgroup of the autoequivalence group $\mathrm{Aut}(D^b(X))$ that preserves this component; it is isomorphic to $\mathbb{Z} \times Γ_1(3) \times \mathrm{Aut}(X)$, where $Γ_1(3) \subset \mathrm{SL}(2,\mathbb{Z})$ is the congruence subgroup of level~3. |
| title | Autoequivalences and stability conditions on a degenerate K3 surface |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.13526 |