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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.14876 |
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| _version_ | 1866911112484618240 |
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| author | Church, Benjamin |
| author_facet | Church, Benjamin |
| contents | We construct a surface over $\overline{\mathbb{F}}_p$ with $π_1^{ét}(X) = 1$ that is supersingular -- in the sense that $H^2_{ét}(X, \mathbb{Q}_{\ell}(1))$ is spanned by algebraic cycles -- but is not unirational. This provides a counterexample to a 1977 conjecture of Shioda. To achieve this, we produce new obstructions to unirationality for product-quotient surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$ Church, Benjamin Algebraic Geometry Primary: 14E08. Secondary: 14G17 and 14J29 We construct a surface over $\overline{\mathbb{F}}_p$ with $π_1^{ét}(X) = 1$ that is supersingular -- in the sense that $H^2_{ét}(X, \mathbb{Q}_{\ell}(1))$ is spanned by algebraic cycles -- but is not unirational. This provides a counterexample to a 1977 conjecture of Shioda. To achieve this, we produce new obstructions to unirationality for product-quotient surfaces. |
| title | Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$ |
| topic | Algebraic Geometry Primary: 14E08. Secondary: 14G17 and 14J29 |
| url | https://arxiv.org/abs/2508.14876 |