Projective manifolds with hyperplane sections being five-sheeted covers of projective space
Let \(X\) be a smooth complex projective variety of dimension \(n+1\) and let \(L\) be a very ample line bundle on \(X\). Suppose that \(| L| \) contains a smooth member \(A\) endowed with a branched covering \(\pi:A \to \mathbb P^n\) of degree \(d\). Classical results on surfaces with hyperelliptic curves as hyperplane sections and their revision made in the 1980's called the attention to the problem of classifying pairs \((X,L)\) as above. The problem has been considered by many authors according to the following values of \((n,d)\). For \((1,2)\) by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593--603 (1987; Zbl 0655.14001)]; for \((1,3)\) by Serrano (unpublished) and by \textit{M. L. Fania} [Manuscr. Math. 68, 17--34 (1990; Zbl 0729.14028)]; for \((n,2), (n,3)\) by \textit{M. Palleschi, A. J. Sommese} and the reviewer [Nagoya Math. J. 137, 1--32 (1995; Zbl 0820.14005); in: Classification of algebraic varieties. Conf. 1992, L'Aquila, Italy. Contemp. Math. 162, 277--292 (1994; Zbl 0841.14003)]; for \((n,d)\) with \(n > d=4,5\) by the reviewer in [Papers in honor of Giovanni Melzi. Milano: Univ. Cattolica del Sacro Cuore. Sci. Mat. 11, 231--248 (1994; Zbl 0831.14023)]. In the paper under review, the author focuses on case \((n,5)\) with \(n \geq 6\) again, providing a complete classification result. In particular, one case, described only in terms of numerical invariants and pointed out as possible by the reviewer, is settled positively. In addition to the four pairs \((X,L)\) already known, the author proves that \(X\) is a smooth weighted hypersurface \(W \subset \mathbb P(5,4,1^{n+1})\) of degree \(20\) with \(L=\mathcal O_W(5)\), which in fact turns out to be very ample. Recently the author considered also the case \((n,4)\) with \(n \geq 5\) [Proc. Japan Acad., Ser. A Math. Sci. 82, 8--13 (2006; Zbl 1106.14002)].
scientific article; zbMATH DE number 5119783