Lengths of Local Cohomology of Thickenings

TL;DR

This paper investigates the growth of local cohomology lengths of thickenings in specific algebraic settings, establishing the existence and rationality of an invariant in a new non-monomial case.

Contribution

It proves the existence and rationality of the invariant for rings defined by size two minors of a 2 by m matrix, extending prior monomial ideal results.

Findings

Invariant exists and is rational for the given class

First non-monomial calculation of this invariant

Extends understanding of local cohomology in algebraic geometry

Abstract

Let $R$ be a standard graded polynomial ring that is finitely generated over a field of characteristic $0$, let $\mathfrak{m}$ be the homogeneous maximal ideal of $R$, and let $I$ be a homogeneous prime ideal of $R$. Dao and Monta\~{n}o defined an invariant that, in the case that $\operatorname{Proj}(R/I)$ is lci and for cohomological index less than $\dim(R/I)$, measures the asymptotic growth of lengths of local cohomology modules of thickenings. They showed its existence and rationality for certain classes of monomial ideals $I$. The following affirms that the invariant exists and is rational for rings $R = \mathbb{C}[X]$ where $X$ is a $2 \times m$ matrix and $I$ is the ideal generated by size two minors and is to our knowledge, the first non-monomial calculation of this invariant.