A note on the normal generation of ample line bundles on abelian varieties
Let k be an algebraically closed field, let A be an abelian variety defined over k and let L be an ample line bundle on A.- Our purpose is to give a condition for the normal generation of \(L^{\otimes 2}\). The result is as follows: Theorem. If char(k)\(\neq 2\) and L is a symmetric ample line bundle, then \(L^{\otimes 2}\) is normally generated if and only if the origin of A is not contained in \(Bs| L\otimes P_{\alpha}|\) for any \(\alpha \in \hat A_ 2=\{\alpha \in \hat A; 2\alpha =0\}\) where \(\hat A\) is the dual abelian variety of A, P is the Poincaré bundle on \(A\times \hat A\), \(P_{\alpha}=P_{A\times \{\alpha \}}\) for \(\alpha\in \hat A\) and \(Bs| L\otimes P_{\alpha}|\) is the set of all base points of \(L\otimes P_{\alpha}\).
scientific article; zbMATH DE number 4146117