F-signatures and Frobenius alpha invariants of del Pezzo surfaces

Ponente(s): Devlin Mallory N/a

The F-signature is a well-studied but extremely subtle invariant of singular rings of positive characteristic. In particular, computing the F-signature is extremely difficult, even for very "simple" rings. One natural class of rings with interesting F-signature consists of the anticanonical rings of Fano varieties (this includes, for example, all hypersurfaces and complete intersections with nonzero F-signature). The first nontrivial case is that of dimension-2 Fano varieties. In this talk, we will describe the F-signatures of anticanonical rings of 2-dimensional Fano varieties, and show that (in contrast to previous calculations) a very general such ring has "maximal F-signature". Equivalently, we will show that such varieties have "maximal Frobenius alpha invariant", a recently developed invariant that provides a computational tool for demonstrating maximal F-signature.