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arXiv · 2011.03613

On Picard Groups of Perfectoid Covers of Toric Varieties

Gabriel Dorfsman-Hopkins·Anwesh Ray·Peter Wear
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Abstract

Let $X$ be a proper smooth toric variety over a perfectoid field of prime residue characteristic $p$. We study the perfectoid space $\mathcal{X}^{perf}$ which covers $X$ constructed by Scholze, showing that $\text{Pic}(\mathcal{X}^{perf})$ is canonically isomorphic to $\text{Pic}(X)[p^{-1}]$. We also compute the cohomology of line bundles on $\mathcal{X}^{perf}$ and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for "projectivoid space".

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Gabriel Dorfsman-Hopkins, Anwesh Ray, Peter Wear. 2020-11-06. On Picard Groups of Perfectoid Covers of Toric Varieties. https://doi.org/10.1007/s40879-023-00596-2

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