On a sequence of singular ball quotient surfaces on the line $K^2=9χ-18$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915544767135744 |
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| author | Rito, Carlos Roulleau, Xavier |
| author_facet | Rito, Carlos Roulleau, Xavier |
| contents | Starting from computer experiments with the fundamental group of the Cartwright--Steger surface, we construct an infinite tower $(X_n)_{n\ge 1}$ of normal projective surfaces obtained by successive $\mathbb Z/3$-Galois covers $X_{n}\to X_{n-1}$. For $n>1$, their minimal resolutions $\widetilde{X}_n$ lie on the line $K^2 = 9χ- 18$ (equivalently $c_1^2 = 3c_2 - 72$), which is parallel to the Bogomolov--Miyaoka--Yau line $K^2 = 9χ$ of ball quotients. We compute the fundamental groups for the first cases, showing that $π_1(\widetilde{X}_n)=1$ for $n=1,\ldots,5$. Motivated by the geometry of the construction, we conjecture that all $\widetilde{X}_n$ are simply connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a sequence of singular ball quotient surfaces on the line $K^2=9χ-18$ Rito, Carlos Roulleau, Xavier Algebraic Geometry Primary 14J29, Secondary 14Q10, 20F05, 20F34 Starting from computer experiments with the fundamental group of the Cartwright--Steger surface, we construct an infinite tower $(X_n)_{n\ge 1}$ of normal projective surfaces obtained by successive $\mathbb Z/3$-Galois covers $X_{n}\to X_{n-1}$. For $n>1$, their minimal resolutions $\widetilde{X}_n$ lie on the line $K^2 = 9χ- 18$ (equivalently $c_1^2 = 3c_2 - 72$), which is parallel to the Bogomolov--Miyaoka--Yau line $K^2 = 9χ$ of ball quotients. We compute the fundamental groups for the first cases, showing that $π_1(\widetilde{X}_n)=1$ for $n=1,\ldots,5$. Motivated by the geometry of the construction, we conjecture that all $\widetilde{X}_n$ are simply connected. |
| title | On a sequence of singular ball quotient surfaces on the line $K^2=9χ-18$ |
| topic | Algebraic Geometry Primary 14J29, Secondary 14Q10, 20F05, 20F34 |
| url | https://arxiv.org/abs/2510.09588 |