Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909780146126848 |
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| author | Rito, Carlos Gleissner, Christian Ruhland, Noah |
| author_facet | Rito, Carlos Gleissner, Christian Ruhland, Noah |
| contents | We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two $\mathsf A_1$ and two $\mathsf A_3$ singularities. We show that this surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland) Rito, Carlos Gleissner, Christian Ruhland, Noah Algebraic Geometry 14J29 We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two $\mathsf A_1$ and two $\mathsf A_3$ singularities. We show that this surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves. |
| title | Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland) |
| topic | Algebraic Geometry 14J29 |
| url | https://arxiv.org/abs/2509.08198 |