Special subvarieties of EPW sextics

Abstract

We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.