On quartics with lines of the second kind (Q468767)

Consider a quartic surface \(X\) in \(\mathbb{P}^3\) which contains a line \(L\). Projecting from \(L\) onto another line, not intersecting \(L\) induces an elliptic fibration on \(X\). A line of the second kind is a line contained in the closure of the locus of the flexes of the fibers of this fibration.NEWLINENEWLINELines of the second kind play an important role in the study of the lines on quartic surfaces. \textit{B. Segre} [Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860)] used lines of the second kind to prove that a smooth complex quartic surface has at most 64 lines. However, in the proof of his result, he falsely assumed that there is a unique unique family \((X,L)\) of lines of the second kind on smooth quartic surfaces. It turns out that there are actually three different families of lines of the second kind. The authors noticed this in a previous paper [``64 lines on smooth quartic surfaces'', \url{arXiv:1212.3511}]. In that paper they showed that Segre's bound was nevertheless correct in characteristic 0.NEWLINENEWLINEIn the paper under review, the author study various claims on lines of the second kind, made by Segre. They discuss whether and how these claims can be corrected.