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arXiv · 2504.07591

On the Cox rings of some hypersurfaces

Andrew Pollock·Atsushi Ito·Balazs Szendroi
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Abstract

We introduce a cohomological method to compute Cox rings of hypersurfaces in the ambient space P^1 x P^n, which is more direct than existing methods. We prove that smooth hypersurfaces defined by regular sequences of coefficients are Mori dream spaces, generalizing a result of Ottem. We also compute Cox rings of certain specialized examples. In particular, we compute Cox rings in the well-studied family of Calabi--Yau threefolds of bidegree (2,4) in P^1 x P^3, determining explicitly how the Cox ring can jump discontinuously in a smooth family.

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Andrew Pollock, Atsushi Ito, Balazs Szendroi. 2025-04-10. On the Cox rings of some hypersurfaces. https://arxiv.org/abs/2504.07591

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