Lines on \(K3\) quartic surfaces in characteristic \(3\)
I the last decade or so, the classical problem of estimating the number of lines (conics, etc.)\ on a polarized \(K3\)-surface, most notably quartic \(X\subset\mathbb{P}^3\), has become quite popular. The author addresses the number of lines on a quartic \(X\) with simple singular points defined over an algebraically closed field of characteristic \(3\). Recall that, in the smooth case, the number is \(112\) (and then \(X\) is the Fermat quartic) or at most \(60\); if \(X\) is known to be supersingular, the number is \(112\), \(58\), or at most \(52\). In the presence of singularities, the author proves, by mostly geometric arguments, the bound of at most \(67\) lines (at most \(58\) under an extra geometric assumption on the configuration of lines); on the other hand, he constructs an example with \(8\) singular points and \(48\) lines. The sharp upper bound in the presence of singularities remains open. Besides, the author finds explicit defining equations of the three \(1\)-parameter families of (necessarily supersingular, with the Artin invariant \(\sigma=2\)) smooth quartics with \(58\) lines; these families were recently discovered by abstract purely lattice theoretical means based on the global Torelli theorem.
scientific article; zbMATH DE number 7489552
scientific article