Higher order embeddings of abelian varieties

Abstract

The strongest notion is k-jet ampleness; it implies k-very ampleness (cf. [BeSo2, Proposition 2.2]) which of course implies k-spannedness. For k = 0 or k = 1 all the three notions are equivalent and correspond to global generation resp. very ampleness. In this note we give criteria for k-jet ampleness of line bundles on abelian varieties. A naive way to obtain such a criterion is as follows: According to [BeSo2, Corollary 2.1] a tensor product of k very ample line bundles is always k-jet ample. Now on an abelian variety, by the generalization of Lefschetz’ classical theorem [LB,