Seminar

Zoom Link

The F-signature is an invariant of positive characteristic local rings that captures asymptotic properties of the Frobenius map. It was first studied by Smith and Van den Bergh, and formally introduced by Huneke and Leuschke to provide new numerical characterizations of regularity and F-regularity of a local ring.

This talk concerns the behavior of the F-signature under reduction to characteristic p >> 0 of a fixed complex singularity. Motivated by applications to the sizes of local fundamental groups, Carvajal-Rojas, Schwede and Tucker conjectured that the F-signatures remain uniformly bounded above zero when we reduce a sufficiently nice (KLT) complex singularity to characteristic p >>0. We will present joint works with Yuchen Liu, and with Anna Brosowsky, Izzet Coskun and Kevin Tucker, in which we prove this conjecture in many new cases including for low degree smooth hypersurfaces and (cones over) most log del Pezzo surfaces. The techniques involved in the proof include Groebner degenerations, toric degenerations, and some recent results from the K-stability theory of Fano varieties. During the talk, we will introduce the relevant notions and give a brief sketch of the proofs.

Positivity of the Limit F-signature