On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Abstract

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math. 22 (1983) 1401–1476; Surfaces of type [Formula: see text] with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank [Formula: see text] that are the Néron–Severi lattice of projective [Formula: see text] surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such [Formula: see text] surfaces [Formula: see text], when these surfaces have moreover no elliptic fibrations. In that case, we show that such [Formula: see text] surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of [Formula: see text] surfaces (among the eight we study) are unirational.