On \(k\)-very ampleness of tensor products of ample line bundles on abelian surfaces
Let \(X\) be an abelian variety over the complex numbers. It is well-known that if \(L\) is an ample line bundle on \(X\) then \(2L\) is spanned and \(3L\) is very ample. \textit{Th. Bauer} and \textit{T. Szemberg} [Math. Z. 224, No. 3, 449-455 (1997; Zbl 0897.14011)] extended these classical results by showing that if \(L_1,\dots,L_{k+2}\) are ample line bundles on \(X\) then \(L:=L_1+ \cdots+ L_{k+2}\) is \(k\)-very ample, i.e., the restriction map \(H^0(L)\to H^0(L\mid Z)\) is surjective for any 0-dimensional subscheme \(Z\) of \(X\) of length \(k+1\). They also gave a sufficient condition of \(k\)-very ampleness of \(L_1+\cdots +L_{k+1}\). The case of polarized abelian varieties \((X,L)\) with \(2L\) not very ample was previously investigated by \textit{A. Ohbuchi} [Manuscr. Math. 57, 225-238 (1987; Zbl 0595.14013)]. In this paper, the author studies abelian surfaces \(X\) and ample line bundles \(L_1,\dots,L_t\) on \(X\), \(t=k\), \(k+1\), such that \(L:=L_1+ \cdots + L_t\) is not \(k\)-very ample. He also studies polarized abelian surfaces \((X,L)\) such that \((k-t)L\) is not \(k\)-very ample, with \(t\geq 1\). In particular, he obtains a classification of \((X,L)\) such that \((k+2-t)L\) is not \(k\)-very ample for \(1\leq t\leq 4\). One of the ingredients used in the proofs is a result of \textit{H. Terakawa} [``The \(k\)-very ampleness and \(k\)-spannedness on polarized abelian surfaces'', Math. Nachr. 195, 237-250 (1998)] characterizing \(k\)-very ampleness on abelian surfaces.
scientific article; zbMATH DE number 1319603