Lines in supersingular quartics (Q2165756)

It is a classical result that a smooth quartic surface over \(\mathbb{C}\) contains at most \(64\) lines, with the first proof going back to \textit{B. Segre} [Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860)]. Recently \textit{S. Rams} and \textit{M. Schütt} [Math. Ann. 362, No. 1--2, 679--698 (2015; Zbl 1319.14042)] fixed a gap in Segre's proof, and estabilished the bound of \(64\) lines for quartics over algebraically closed field of characteristic \(\ne 2,3\). It was then showed by \textit{A. Degtyarev} et al. [Math. Ann. 368, No. 1--2, 753--809 (2017; Zbl 1368.14052)] that there are only \(10\) quartics containing more than \(52\) lines. However the situation is different in characteristic \(2\) and \(3\): here the maximal number of lines on a smooth quartic is \(60\) and \(112\), respectively. The goal of the present paper is to obtain a classification of quartics with \textit{many} lines in characteristic \(2\) and \(3\). More precisely, the author shows that: \begin{itemize} \item In characteristic \(2\), the maximal number of lines on a supersingular quartic is \(40\), and there are at most \(5\) types of configuration of lines on them; moreover there are no other supersingular quartics with more than \(32\) lines. \item In characteristic \(3\), the maximal number of lines on a supersingular quartic is \(112\), attained only by the Fermat quartic. Otherwise the quartic contains at most \(58\) lines, with this second best record achieved by at most \(3\) configurations; moreover these are the only supersingular quartics containing more than \(52\) lines. \end{itemize} Concerning non-supersingular quartics, the author shows that in characteristic \(2\) and \(3\) they cannot contain more than \(60\) lines. The proofs of the main results of the paper rely on a lattice-theoretical study that leads to the different possible configurations of many lines on supersingular quartics.