Gr\"obner deformation and $F$-singularities

TL;DR

This paper explores the relationship between Gr"obner deformations and $F$-singularities in polynomial ideals, providing characteristic-free results and insights into how $F$-singularities behave under Gr"obner deformation.

Contribution

It establishes new connections between $F$-singularities and Gr"obner deformations, including a characteristic-free result on squarefree monomial initial ideals of prime ideals.

Findings

Strong interactions between $F$-split rings and squarefree monomial initial ideals.

Characterization of $F$-singularities that deform under Gr"obner basis changes.

A characteristic-free theorem relating initial ideals of prime ideals and squarefree monomials.

Abstract

For polynomial ideals in positive charachteristic, defining $F$-split rings and admitting a squarefree monomial initial ideal are different notions. In this note we show that, however, there are strong interactions in both directions. Moreover we provide an overview on which $F$-singularities are Gr\"obner deforming. Also, we prove the following characteristic-free statement: if $\mathfrak{p}$ is a height $h$ prime ideal such that $\textrm{in}(\mathfrak{p}^{(h)})$ contains at least one squarefree monomial, then $\textrm{in}(\mathfrak{p})$ is a squarefree monomial ideal.