lectures by Christian Liedtke

Supersingular K3 surfaces are unirational

Abstract: We show that supersingular K3 surfaces are related by purely inseparable isogenies. As an application, we deduce that they are unirational, which confirms conjectures of 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 relates supersingular K3 surfaces of different Artin invariants by purely inseparable isogenies. If time permits, we will show how these "moving torsors" exhibit the moduli space of rigidified supersingular K3 crystals as an iterated projective bundle over a finite field.