Reconstructing a special double covering of \({\mathbb{P}}^ 3\) from a family of straight lines on it
Consider a double covering Y of \({\mathbb{P}}^ 3\) branched at a quartic A containing no straight lines. If A is a smooth family of straight lines on the covering Y is parametrized by a surface. \textit{A. S. Tikhomirov} proved that A and consequently Y are recovered from that surface [Math. USSR, Izv. 16, 373-397 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 415-442 (1980; Zbl 0434.14023)]. In this paper the same is shown for the desingularization X of Y when A has a single double point, after giving a suitable definition of straight lines on X.
desingularization of covering of projective 3-space
scientific article; zbMATH DE number 4189877