Frobenius splittings of toric and T-varieties

Hendrik Suess, University of Manchester. Part of the pure mathematics algebra seminar series

In positive characteristic some standard tools of algebraic geometry stop working, such as Kodaira vanishing or resolution of singularities, but as a compensation we also obtain a very powerful new tool: The Frobenius morphism. For example, if there exists a section of this morphism we get back the Kodaira vanishing theorem. Such sections are called Frobenius splittings. Following a paper of Sam Payne I am going to discuss Frobenius splittings on toric varieties and connect them to the combinatorics of lattice polytopes. Then I will present a generalisation of Payne's results to so-called T-varieties. This is joint work with Piotr Achinger and Nathan Ilten.