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arXiv · 2602.09201

Generic flatness of the cohomology of thickenings

Edoardo Ballico·Yairon Cid-Ruiz·Anurag K. Singh
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Abstract

We prove a generic flatness result for the cohomology of thickenings of a projective scheme that is smooth over a Noetherian domain containing a field of characteristic zero. Our study is motivated, in part, by a classical question in algebraic geometry: Given a set of $m$ distinct points in projective space over a field, and $t$ a positive integer, determine the least degree of a hypersurface that passes through each point with multiplicity at least $t$. Related to this, it remains unresolved whether there exists a dense open set of $m$-tuples of points for which this least degree is constant for each $t\ge 1$. Investigating this connection in the case of nine points in projective plane, we construct a local cohomology module that is not generically free; moreover, we show that it has infinitely many associated prime ideals.

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Edoardo Ballico, Yairon Cid-Ruiz, Anurag K. Singh. 2026-02-09. Generic flatness of the cohomology of thickenings. https://arxiv.org/abs/2602.09201

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