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

Abstract: Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the N\'eron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case we show that such K3 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 $4$ cases the description is sufficient, in each of the $4$ other cases there is exactly another one possibility which we study. We obtain that at least 5 moduli spaces of K3 surfaces (among the 8 we study) are unirational.