Geometric vertex decomposition and F-splitting

Geometric vertex decomposition was introduced by Knutson, Miller, and Yong while studying the Groebner geometry of matrix Schubert varieties. It extends the idea of vertex decomposition for simplicial complexes to the setting of algebraic geometry and polynomial ideals and provides an inductive strategy for reducing questions to smaller or simpler pieces. Frobenius splitting is a technique used to study singularities in positive characteristic. Knutson showed that Frobenius splittings descend through geometric vertex decomposition to the link and the deletion, the two pieces in which the original object is decomposed. In this talk, I define and discuss these ideas and their relations and I present a joint result with De Negri, Klein, Rajchgot, and Seccia, which establishes a partial converse of the result of Knutson.