Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$

Benjamin Church

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arXiv · 2508.14876

Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$

Benjamin Church

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Abstract

We construct a surface over $\overline{\mathbb{F}}_p$ with $\pi_1^{\'{e}t}(X) = 1$ that is supersingular -- in the sense that $H^2_{\'{e}t}(X, \mathbb{Q}_{\ell}(1))$ 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.

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Benjamin Church. 2025-08-20. Obstructions to unirationality for product-quotient surfaces over $\overline{\mathbb{F}}_p$. https://arxiv.org/abs/2508.14876

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