On the spannedness of adjunction of vector bundles
Let \(X\) be a complex projective manifold of dimension \(n\) and \(E\) be an ample vector bundle on \(X\). The nefness and spannedness of the adjoint bundle \(K_X+c_1(E)\) has been investigated in several papers. Here the author asks for conditions ensuring that \(K_X+c_1(E)\) is spanned. \textit{M. C. Beltrametti, M. Schneider} and \textit{A. J. Sommese} [Math. Z. 214, No. 4, 593-600 (1993; Zbl 0804.14006)] had conjectured that, if \(E\) is of rank \(r\geq n\) and spanned, then the adjoint bundle is spanned unless \((X,E)\simeq ({\mathbb{P}}^n, \bigoplus^n{\mathcal O}_{{\mathbb{P}}^n}(1))\). The main result of this paper is an affirmative answer to this conjecture, under the additional condition that \(L\) is very ample, where \(L\) denotes the tautological line bundle on \(\text{Proj}_X(E)\), associated to \(E\).
scientific article; zbMATH DE number 1028280
scientific article