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.

An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. The first instance of such a formula is Kodaira�s canonical bundle formula for surfaces which admit a fibration with elliptic fibres. It describes the relation between the canonical bundles in terms of the singularities of the fibres and their j-invariants. In higher dimension, we do not have an equivalent of the j-invariant, but we can still define a moduli part. Over fields of characteristic 0, positivity properties of the moduli part have been studied using variations of Hodge structures. Recently, the problem has been addressed with the minimal model program for foliations, which is known to fail in positive characteristic. In this talk I will explain an approach to the canonical bundle formula which adapts these methods in positive characteristic.