Grobner 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 Grobner 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.