On 2-spannedness for the adjunction mapping
Let X be a smooth, complete variety of dimension greater than or equal to 1. An (effective) 0-cycle \(Z\subset X\) is called curvilinear, if \(Z\subset C\subset X\), where C is a smooth projective curve. A line bundle L on X is k-spanned if for all curvilinear cycles \(Z\subset C\), with \(\ell ({\mathcal O}_ Z)=k+1\), the natural map \(H^ 0(L)\to H^ 0(L\otimes {\mathcal O}_ Z)\) is surjective. Notice that 0-spanned is the same as saying that L is very ample. The authors prove the following: Let X be a smooth complete variety of dimension greater than or equal to 3 and L a line bundle on X. If L is 3-spanned then \(K_ X\otimes L^{n- 1}\) is 2-spanned. The case \(n=2\), which is not covered by this theorem, is studied in depth in the following preprints: 1. \textit{M. Beltrametti} et al., ``On Reider's method and higher order embeddings'' and 2. \textit{M. Beltrametti}, and \textit{A. J. Sommese}, ``On k-spannedness for projective surfaces'', both Max Planck Institut Math. preprints, No. 7 and 14 (1988).
scientific article; zbMATH DE number 4079586
scientific article