$F$-singularities under generic linkage

TL;DR

This paper investigates how $F$-singularities behave under generic linkage in polynomial rings over perfect fields of positive characteristic, providing criteria for $F$-rationality and comparing $F$-pure thresholds.

Contribution

It describes the parameter test submodule of linked ideals in terms of the test ideal, and establishes criteria for $F$-rational singularities in this context.

Findings

Parameter test submodule expressed via test ideal for certain linked ideals

Criterion for $F$-rationality of linked ideals

Comparison of $F$-pure thresholds before and after linkage

Abstract

Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over a prefect field of positive characteristic. Let $I$ be an unmixed ideal in $R$ and let $J$ be a generic link of $I$ in $S=R[u_{ij}]_{c \times r}$. We describe the parameter test submodule of $S/J$ in terms of the test ideal of the pair $(R, I)$ when a reduction of $I$ is a complete intersection or almost complete intersection. As an application, we deduce a criterion for when $S/J$ has $F$-rational singularities in these cases. We also compare the $F$-pure threshold of $(R, I)$ and $(S, J)$.