The \(k\)-very ampleness on an elliptic quasi bundle
Let \(S\) be a smooth projective surface over \(\mathbb{C}\). A line bundle \(L\) on \(S\) is \(k\)-very ample \((k \geq 0)\) if for any zero-dimensional subscheme \((Z,{\mathcal O}_ Z)\) of \(S\) with \(k+1=\) length \(({\mathcal O}_ Z)\) the restriction map \(\Gamma (L) \to \Gamma ({\mathcal O}_ Z(L))\) is a surjection. In this paper \(S\) is a relatively minimal elliptic fibration over a rational curve with \(b_ 2=2\) and Kodaira dimension \(k(S) \geq 0\). First of all, the authors describe the line bundles on \(S\) spanned by their global sections, i.e. 0-very ample line bundles. Then they give both necessary and sufficient conditions for a line bundle to be \(k\)-very ample on \(S\). These results are being made more precise in the case when \(S\) is a hyperelliptic surface. For each of the seven types of such surfaces, as classified by Bagnera and De Franchis [see \textit{F. Serrano}, Math. Z. 203, No. 3, 527-533 (1990; Zbl 0722.14023)], the authors give a complete characterization of \(k\)-very ampleness of a line bundle on \(S\).
scientific article; zbMATH DE number 492594