Generic flatness of the cohomology of thickenings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915836608905216 |
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| author | Ballico, Edoardo Cid-Ruiz, Yairon Singh, Anurag K. |
| author_facet | Ballico, Edoardo Cid-Ruiz, Yairon Singh, Anurag K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_09201 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generic flatness of the cohomology of thickenings Ballico, Edoardo Cid-Ruiz, Yairon Singh, Anurag K. Algebraic Geometry Commutative Algebra Primary 13D45, Secondary 13A02, 13A30, 14F17 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. |
| title | Generic flatness of the cohomology of thickenings |
| topic | Algebraic Geometry Commutative Algebra Primary 13D45, Secondary 13A02, 13A30, 14F17 |
| url | https://arxiv.org/abs/2602.09201 |