Noncomplete linear systems on abelian varieties

Abstract

Let X be a smooth projective variety. Every embeddingX \hookrightarrow\mathbbP_Nis the linear projection of an embedding defined by acomplete linear system. In this paper the geometry of such not necessarilycomplete embeddings is investigated in the special case of abelianvarieties. To be more precise,the properties Np of complete embeddingsare extended to arbitrary embeddings,and criteria for these propertiesto be satisfied are elaborated. These results are applied to abelianvarieties. The main result is: Let (X,L) be a general polarized abelianvariety of type (d1,...,dg) and p≥1,n≥2p + 2 such that ndg≥6is even,and c≤ng-1. The general subvector spaceV \subseteq H^0(L^n)ofcodimension c satisfies the property Np.