Lines on quartic surfaces (Q2396238)

Let \(X\subset{\mathbb{P}}^3\) be an algebraic surface over \({\mathbb{C}}\). Let \(\text{{Fn}}(X)\) be the set of projective lines in \(X\). The paper under review discusses the the number of lines on various quartic surfaces. Theorem. Let \(X\subset{\mathbb{P}}^3\) be a nonsingular quartic surface, and assume that \(|\text{{Fn}}(X)|>52\). Then \(X\) is projectively equivalent to either {\parindent=0.7cm\begin{itemize}\item[--] Schur's quartic \(X_{64}\), or \item[--] one of the three quartics \(X'_{60}\), \(X''_{60}\), \( X'''_{60}\), or \item[--] the quartic \(Y_{56}\), or quartics \(X_{56}\), \(\bar{X}_{56}\), \(Q_{56}\), or \item[--] one of the two quartics \(X_{54}\), \(Q_{54}\). \end{itemize}} In particular, one has \(|\text{{Fn}}(X)|=64, 60, 56\) or \(54\), respectively. This rediscovers the result of \textit{B. Segre} [Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860)], and that of \textit{S. Rams} and \textit{M. Schütt} [Math. Ann. 362, No. 1--2, 679--698 (2015; Zbl 1319.14042)]. Corollary 1. Any nonsingular quartic in \({\mathbb{P}}^3\) contains at most \(64\) lines. Now if \(X\) is real, \(\text{{Fn}}_{\mathbb{R}}(X)\) denotes the set of real lines contained in \(X\). Corollary 2. Let \(X\subset{\mathbb{P}}^3\) be a nonsingular (over \({\mathbb{C}}\)) real quartic, and assume that \(|\text{{Fn}}_{\mathbb{R}}(X)|>52\). Then \(X\) is projectively equivalent (over \({\mathbb{R}})\) to the quartic \(Y_{56}\). In particular, one has \(|\text{{Fn}}(X)|=56\), and this is the maximal number of real lines that can be contained in a nonsingular real quartic. Facts. For any number \(n\in \{0,1,2,\dots, 51, 52,54, 56, 60, 64\}\), there exists a nonsingular quartic \(X\subset{\mathbb{P}}^3\) such that \(|\text{{Fn}}(X)|=n\). For any number \(m\in\{0,1,\dots, 46, 48, 5 2, 56\}\), there exists a nonsingular real quartic \(X\subset {\mathbb{P}}^3\) such that \(|\text{{Fn}}_{\mathbb{R}}(X)|=m\). The proof rests on the study of the Picard lattice of the surface.