Double covers of smooth hyperquadrics as ample and very ample divisors
The object of this paper is the classification of pairs \((X,L)\), where \(X\) is a smooth projective variety of dimension \(n\geq 4\) and \(L\) is a very ample (or ample) divisor, such that one of the smooth elements of the linear system \(|L|\) is a double cover of a smooth quadric. This problem may be regarded as a generalization of the classical question of which smooth projective surfaces admit a hyperelliptic curve as a hyperplane section, solved by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)] and by \textit{F. Serrano} [J. Reine Angew. Math. 381, 90-109 (1987; Zbl 0618.14001)]. The author solves the problem completely, and moreover he generalizes along the same lines an earlier result of his, namely the classification of pairs \((X,L)\), with \(L\) ample, such that one of the smooth elements of \(|L|\) is a double cover of \(\mathbb{P}^n\). The proof rests on Fujita's classification of varieties according to \(\Delta\)-genus.
scientific article; zbMATH DE number 706665