Elliptic fibrations and involutions on \(K3\) surfaces (Q6611641)

Among algebraic surfaces, \(K3\)'s are the only ones who may and sometimes do host more than one non-trivial elliptic fibration with a section. So the first question immediately rises: how many? In 1985, Sterk proved that, up to automorphisms of the surface, a \(K3\) surface can only have a finite number of elliptic fibrations [\textit{H. Sterk}, Math. Z. 189, 507--513 (1985; Zbl 0545.14032)]. Given this result, the second natural question arises: is it possible to classify such elliptic fibrations?\N\NAlice Garbagnati and Cecília Salgado have been very active in this field and the paper under review is a survey on the results they achieved in four previous papers, partially in the frameworks of larger cooperations.\N\NIf a \(K3\) surface admits an elliptic fibration then it also admits a non-symplectic involution with non-empty fixed locus, induced by the multiplication by \(-1\) on the elliptic fibers; in many cases this involution is not the only one with the above properties. The quotient of a \(K3\) surface by such an involution is a rational surface. In the series of works under review, the main feature is to translate the study of the elliptic fibrations of a K3 surface \(X\) with a non-symplectic involution \(\iota\) with non-empty fixed locus to the study of the elliptic fibrations of the rational surface \(X/\iota\).\N\NUsing this approach, they introduce a classification of the elliptic fibrations of a \(K3\) surface based on the action of the double cover involution on them [\textit{A. Garbagnati} and \textit{C. Salgado}, J. Pure Appl. Algebra 223, No. 1, 277--300 (2019; Zbl 1443.14041)] and apply this classification to\N\begin{itemize}\N\item general K3 surfaces with a non-symplectic involution that fixes a curve of positive genus [\textit{A. Garbagnati} and \textit{C. Salgado}, Rev. Mat. Iberoam. 36, No. 4, 1167--1206 (2020; Zbl 1457.14081)] (the genus zero case had already been studied in works by Clingher-Malmendier, Kloosterman, and Oguiso);\N\item K3 surfaces that cover the modular elliptic surface of level \(5\) [\textit{F. Balestrieri} et al., Assoc. Women Math. Ser. 11, 159--197 (2018; Zbl 1407.14028)].\N\end{itemize}\NUsing the above classification, they determine the field of definition of the elliptic fibrations and Mordell-Weil group of K3 surfaces that are quadratic base changes of certain extremal rational surfaces [\textit{V. Cantoral-Farfán} et al., Assoc. Women Math. Ser. 24, 171--205 (2021; Zbl 1498.14097)].\N\NFor the entire collection see [Zbl 1542.11001].