Automorphisms of surfaces in a class of Wehler \(K3\) surfaces with Picard number \(4\) (Q309457)

The paper under review concerns the so-called Wehler \(K3\) surfaces, i.e. smooth surfaces given by a trihomogeneous \((2,2,2)\)-form in \(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\). Generically, such a surface has Picard number three, generated by the fibers of the three obvious projections, each of which generally is an elliptic curve.NEWLINENEWLINEThe author studies more specifically those Wehler \(K3\) surfaces of Picard number 4 where two of the fibrations attain a section (which features as a component of a reducible fiber of the third fibration). These \(K3\) surfaces are endowed with a set of four canonical involutions: three of them induced from the quadratic form structure over each \(\mathbb P^1\), one coming from either of the elliptic fibrations (with section!) which thus admit a hyperelliptic involution.NEWLINENEWLINEBased on the Torelli theorem and lattice theory, the author proves that these four involutions generate the automorphism group of the \(K3\) surface up to finite index. It is also stated that the techniques apply to other \(K3\) surfaces as well, especially those with small Picard number.