On automorphisms of Enriques surfaces

Abstract

An Enriques surface over an algebraically closed field k of characteristic 4=2 is a nonsingular projective surface F with Hi(F, (gv)= H2(F, Or)=0, 2Kv=0. The unramified double cover of F defined by the torsion class K v is a K3-surface F, a nonsingular projective surface with HI(F,(gr)=0, Kr=0. The study of Enriques surfaces is equivalent to the study of K3-surfaces with a fixed-point-free involution z. In particular, the automor_phism group Aut(F) of F is isomorphic to the group Aut(ff, z)/(z), where Aut(F,z) is the centralizer of z in the automorphism group Aut(F) of ft. In the case k = ~ , the field of complex numbers, the study of Aut(F) is based on the Global Torelli Theorem for K3-surfaces proven by I. Piatetski-Shapiro and I. Shafarevich in [19]. It follows from this theorem that up to a finite group the group Aut(ff) is isomorphic to the quotient group O(Pic(F))/W, where O(Pic(F)) is the orthogonal group of the Picard lattice of ff and W its normal subgroup generated by the reflections into the classes of nonsingular rational curves. For a generic Enriques surface F this theorem allows to compute Aut(F) (see [3] and also [17], where this result is not stated explicitly). For an arbitrary F the relation between F and ff does not help, since it is very difficult to compute the action of z in Pie(if). However, by other means, we can prove the following analog of Piatetski-Shapiro and Shafarevich's result: