Ideal Containments under Flat Extensions

TL;DR

This paper investigates the behavior of symbolic and ordinary powers of ideals under flat extensions, extending known results from matroids to point configurations in projective space, and provides new counterexamples to a conjectured containment.

Contribution

It generalizes a recent result on matroid ideals to saturated homogeneous ideals of point configurations and derives new counterexamples to symbolic power containments.

Findings

Proves the containment result for point configuration ideals under flat extensions.

Generates new counterexamples to the conjectured symbolic power containment.

Extends the understanding of ideal containments in algebraic geometry.

Abstract

Let $\varphi : S = k[y_0,..., y_n] \to R = k[y_0,...,y_n]$ be given by $y_i \to f_i$ where $f_0,...,f_n$ is an $R$-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, $I_\Delta \subseteq S$, of matroids, $\Delta$, that $I_\Delta^{(m)} \subseteq I^r$ if and only if $\varphi_*(I_\Delta)^{(m)} \subseteq \varphi_*(I_\Delta)^r$ where $\varphi_*(I_\Delta)$ is the ideal generated in $R$ by $\varphi(I_\Delta)$. We prove this result for saturated homogeneous ideals $I$ of configurations of points in $\mathbb{P}^n$ and use it to obtain many new counterexamples to $I^{(rn - n + 1)} \subseteq I^r$ from previously known counterexamples.