Pathologies of the Hilbert scheme of points of a supersingular Enriques surface

Abstract

We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2, Hilbn(X), for n≥2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number h2,0>1, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for Göttsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showing that there are strictly more classes of simply connected varieties with trivial canonical class in characteristic 2 than over C as given by Beauville-Bogolomov decomposition theorem.