Gröbner deformations and F-singularities

For polynomial ideals in positive characteristic, 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 𝔭 is a height ℎ prime ideal such that in(𝔭(ℎ)) contains at least one squarefree monomial, then in(𝔭) is a squarefree monomial ideal.