Algebraic Geometry Seminar

How much of the geometry of a positive characteristic variety is encoded in its Frobenius? I'll give exact answers to this vague question in the toric case. For this, I'll introduce a Frobenius-theoretic cone sitting inside the pseudo-effective cone whose interaction with the other nef/ample/big cones determines the type of extremal contractions a variety can undergo. This leads to a geometric characterization of when the kernel of the Frobenius trace is; respectively, big, ample, and nef. This talk is based on my joint work with Emrezavci (EPFL), arXiv:2506.02994.

Reider's Theorem is a remarkably precise criterion for very ampleness of line bundles of the form K_S + D on an algebraic surface. By using Bridgeland stability conditions and various theorems about the moduli spaces of stable objects, we can extract information about the birational geometry of the blow-up of the projective space |K_S + D|^vee along the surface, analogous to results of Thaddeus in the curve case. This is part of a program that I am working on with my graduate students in Utah and Macri's laboratory in Paris.