On the 2 very ampleness of the adjoint bundle. (With an appendix by E. Ballico)
Let \(S\) be a smooth complex projective surface. A line bundle \(L\) on \(S\) is said to be \(k\)-very ample \((k\) a positive integer) if for any 0- dimensional subscheme \((Z,{\mathcal O}_ Z)\) of length \(k+1\) of \(S\) the restriction homomorphism \(\Gamma(L)\to\Gamma(L\otimes{\mathcal O}_ Z)\) is surjective. This definition was introduced by \textit{M. Beltrametti} and \textit{A. Sommese} [``Problems in the theory of surfaces and their classification'' in Computational algebraic geometry and commutative algebra, Proc. Conf., Cortona 1988, Symp. Math. 32, 33-48 (1993)] in connection with the notion of higher order embedding. In fact for \(k=1\) it simply means that \(L\) is very ample. In the paper under review the authors prove the following analogue of the very ampleness result for the adjoint bundle to a very ample line bundle obtained by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)]. Assume that \(L\) is 2-very ample with \(L^ 2\geq 13\); then the adjoint bundle \(K_ S\otimes L\) is 2-very ample up to contracting the (--1)-conics of \(S\), apart from a list of exceptions. Recently \textit{M. Beltrametti} and \textit{A. Sommese} [Math. Z. 212, 257-283 (1993)] obtained a more general result of this kind concerning \(k\)-very ample line bundles. In the appendix by \textit{E. Ballico} the above result is extended to higher dimensions.
scientific article; zbMATH DE number 60114
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