The MWL-algorithms for constructing cubic surfaces with preassigned 27 lines
It is classically known that a smooth cubic surface in \(\mathbb P^3\) contains exactly 27 straight lines; this holds over \(\mathbb C\), but also over any other algebraically closed field. However, it is not so obvious how to find cubics with all 27 lines defined over a given non-algebraically closed field \(k\). The present paper develops a systematic answer in form of an algorithm based on the theory of elliptic surfaces (with section) and Mordell-Weil lattices. In particular, the algorithm builds on the author's idea of excellent families, both of multiplicative and additive type. It can also be applied to the problem of the 28 bitangents of a plane quartic curve.NEWLINENEWLINEThe paper contains instructive examples, also concerning degenerations, and beautiful figures making visible the 27 lines on a cubic surface (or singularities, if any), and the bitangents to a plane quartic curve.
scientific article; zbMATH DE number 6587609