Gröbner deformation and $F$-singularities

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 Gröbner deforming. Also, we prove the following characteristic-free statement: if $\mathfrak{p}$ is a height $h$ prime ideal such that $\text{in}({\mathfrak{p}}^{(h)})$ contains at least one squarefree monomial, then $\text{in}(\mathfrak{p})$ is a squarefree monomial ideal.

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