Very ampleness of \(K_ x\otimes {\mathcal L}^{\dim \,{\mathbf{X}}}\) for ample and spanned line bundles \({\mathcal L}\)
Let X be a normal projective variety of dimension n over the complex numbers. Let reg(X) (resp. sing(X)) denote the set of smooth (resp. non- smooth) points of X. Let K denote either the canonical sheaf or the Grauert-Riemenschneider canonical sheaf of X. Let L be an ample and spanned line bundle on X. Properties of \(K\otimes L^ t\) play an important role in the study of the holomorphic map determined by L, this paper is one in the sequence of papers studying the former. The case \(t=n-1\) was studied in detail by \textit{M. Andretta} and \textit{A. J. Sommese} [Forum Math. 1, No.2, 143-152 (1989; Zbl 0667.14022)]. For \(t\geq n\), \textit{A. J. Sommese} had shown spannedness of \(K\otimes L^ t\) on X where X is Gorenstein with finite non-rational singularities [Conf. complex geometry (Nancy 1985); see also Inst. Elie Cartan, Univ. Nancy I 10, 104-125 (1986; Zbl 0646.14007)]. The authors show that if \(t\geq n\), sections of \(K\otimes L^ t\) span it on reg(X) unless \(t=n\) and \((X,L)=({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(1))\). For \(t\geq n+1\), \(K\otimes L^ t\) is shown to be very ample on reg(X) unless \(t=n+1\), \((X,L)=({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(1))\). For \(t=n\), the very ampleness (of \(K\otimes L^ n)\) on reg(X) is studied under certain assumptions on X, L. - The results of the paper have many corollaries, we mention a few interesting ones: (1) Let \(f:X\to {\mathbb{P}}^ n\) be a branched cover. If Codim(Sing(X))\(\geq 3\) and deg(f)\(\geq 2\), then the ramification divisor of f is very ample on reg(X). With additional assumptions, a similar result is proved for \({\mathbb{P}}^ n\) replaced by a quadric hypersurface. (2) If X is smooth, \(K^ N\) is trivial for some \(N\geq 1\) and \(C_ 1(L)^ n\geq 5\), then \(L^ n\) is very ample.
scientific article; zbMATH DE number 4177245
scientific article