可厚化方案余弦学的通用平坦性
Generic flatness of the cohomology of thickenings
AI总结：
本文研究了射影方案厚化方案的余弦学通用平坦性，通过构造非普遍自由的局部余弦学模块，证明其具有无限多个关联素理想，并探讨了相关问题的未解决状态。
摘要：
我们证明了对于一个在Noetherian域上光滑的射影方案的厚化方案的余弦学的通用平坦性结果。我们的研究部分受到代数几何中一个经典问题的启发：给定一个在射影空间上的m个不同的点，以及一个正整数t，确定通过每个点且乘数至少为t的最少数次 hypersurface 的次数。与此相关的是，是否还存在一个稠密开集的m个点的元组，使得对于每个t≥1，这个最少数次是常数的问题仍未解决。在九个点的射影平面情况下研究这一联系，我们构造了一个不是普遍自由的局部余弦学模块；此外，我们证明该模块有无限多个关联素理想。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
