6 марта, "Unirationality, supersingularity, and formal Brauer groups"
Аннотация: An n-dimensional variety is called unirational if there exists a rational and dominant map from n-dimensional projective space onto it. Unirational varieties are thus (in some sense) close to projective space itself, and the classical Lüroth problem asks whether unirational varieties are in fact rational, that is, birational to projective space. This is true for curves (by Lüroth himself), as well as for surfaces over the complex numbers (Castelnuovo). Over the complex numbers it false in dimension 3 (counter-examples by Fano, Clemens-Griffiths, Artin-Mumford), as well as for surfaces in positive characteristic (Zariski). In this talk I will introduce the formal Brauer group, and various notions of supersingularity (via cycles, formal Brauer groups, and in terms of F-crystals). I will explain that unirational varieties are supersingular. The interesting (and open) question is whether supersingular varieties are unirational.
7 марта, "Supersingular K3 surfaces are unirational"
Аннотация: I show that supersingular K3 surfaces in positive characteristic are related by purely inseparable isogenies. As an application, I deduce that supersingular K3 surfaces are unirational, which confirms conjectures Artin, Rudakov, Shafarevich, and Shioda. The main ingredient in the proof is to use the formal Brauer group of a Jacobian elliptically fibered supersingular K3 surface to construct a family of "moving torsors" under this fibration that eventually related supersingular K3 surfaces of different Artin invariants by purely inseparable isogenies. If time permits, I will also explain how these moving torsors exhibit Ogus' moduli space of supersingular K3 crystals as an iterated projective bundle over a finite field.
11 марта, "Rational curves on K3 surfaces"
Аннотация: Although a projective K3 surface over the complex numbers cannot be unirational, Bogomolov conjectured that it contains nevertheless infinitely many rational curves. In this talk, I will prove Bogomolov's conjecture for K3 surfaces of odd Picard rank - this includes the generic case of Picard rank 1. The idea is to reduce to positive characteristic p, and to use the Tate conjecture to find infinitely many rational curves (albeit for infinitely many distinct primes). Then we use Kontsevich's moduli space of stable maps and a rigidification argument to lift cycles of rational curves to characteristic zero, which eventually establishes infinitely many rational curves on the original K3 surface. This work is joint with Jun Li and extends an approach due to Bogomolov, Hassett, and Tschinkel.
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