Gr\"obner deformation and $F$-singularities

Mitra Koley; Matteo Varbaro

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arXiv · 2107.12116

Gr\"obner deformation and $F$-singularities

Mitra Koley·Matteo Varbaro

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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.

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Mitra Koley, Matteo Varbaro. 2021-07-26. Gr\"obner deformation and $F$-singularities. https://arxiv.org/abs/2107.12116

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