Generic flatness of the cohomology of thickenings

TL;DR

This paper establishes a generic flatness property for the cohomology of thickenings of smooth projective schemes over a Noetherian domain with characteristic zero, linking algebraic geometry and cohomological behavior.

Contribution

It proves a new generic flatness result for cohomology of thickenings and explores its implications for classical point configuration problems in projective space.

Findings

Constructed a local cohomology module that is not generically free.

Showed the existence of infinitely many associated prime ideals in a specific cohomology module.

Connected cohomological properties to classical problems in algebraic geometry.

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.