On the non-degeneracy of Enriques surfaces

I will report on joint work with G. Mezzedimi and D. Veniani on Enriques surfaces with small non-degeneracy. The non-degeneracy of an Enriques surface X is the maximal length of an isotropic sequence of effective curves on X. Roughly speaking, the higher the non-degeneracy of an Enriques surface is, the more well-behaved are its projective models. For example, Enriques surfaces of maximal non-degeneracy 10 are isomorphic to surfaces of degree 10 in P^5. We prove that, in characteristic different from 2, every Enriques surface has non-degeneracy at least 4, which implies that all of them arise via Enriques’ classical construction as a minimal resolution of a sextic in P^3 which is non-normal along the edges of a tetrahedron and all Enriques surfaces are birational to normal quintics in P^3.