Higher order embeddings of abelian varieties
A line bundle \(L\) on a smooth projective variety \(X\) is called \(k\)-jet ample if the restriction map \(H^0 (L)\to H^0 (L\otimes {\mathcal O}_X/m^{k_1}_{y_1} \otimes \cdots \otimes m^{k_r}_{y_r})\) is surjective for any choice of distinct points \(y_1, \dots, y_r\) in \(X\) and positive integers \(k_1, \dots, k_r\) with \(\sum k_i= k+1\). In particular \(L\) is 1-jet ample if and only if \(L\) is very ample. The main result of the paper is the following theorem: Let \(L_1, \dots, L_{k+ 2}\) be ample line bundles on an abelian variety \(A\). Then \(L_1 \otimes \cdots \otimes L_{k+2}\) is \(k\)-jet ample. This is an improvement of the authors' generalization of Lefschetz theorem, saying that a tensor product of any three ample line bundles on an abelian variety is very ample [see \textit{Th. Bauer} and \textit{T. Szemberg}, Math. Z. 223, No. 1, 79-85 (1996; Zbl 0863.14009)]. The result is sharp in the sense that not every tensor product of \(k+1\) ample line bundles on an abelian variety is \(k\)-jet ample.
scientific article; zbMATH DE number 1028153