Frobenius Splitting and Gröbner Degeneration

We define and study Frobenius splittings for schemes and compatible splittings for closed subschemes. We introduce consequences of the existence of a splitting (e.g. Positroid varieties are shown to be reduced using Frobenius splitting). We introduce an important class of near-splittings, which naturally comes with a compatibly split closed subscheme. Finally, we introduce Allen Knutson's criterion about Gröbner degeneration and use it to show splittings for a few examples.