A note on the cone conjecture for K3 surfaces in positive characteristic
  We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group. 
Max Lieblich and Davesh Maulik
math.AG math.NT