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arXiv · 2509.08198

Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

Carlos Rito·Christian Gleissner·Noah Ruhland
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Abstract

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.

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Carlos Rito, Christian Gleissner, Noah Ruhland. 2026-07-10. Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland). https://arxiv.org/abs/2509.08198

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