Noncomplete linear systems on abelian varieties

Let $X$ be a smooth projective variety. Every embedding $X\hookrightarrow \mathbb{P}_N$ is the linear projection of an embedding defined by a complete linear system. In this paper the geometry of such not necessarily complete embeddings is investigated in the special case of abelian varieites. To be more precise, the properties $N_p$ of complete embeddings are extended to arbitrary embeddings, and criteria for these properties to be satisfied are elaborated. These results are applied to abelian varieties. The main result is: Let $(X,L)$ be a general polarized abelian variety of type $(d_1,\dots,d_g)$ and $p\ge1$, $n\ge 2p+2$ such that $nd_g\ge 6$ is even, and $c\le n^{g-1}$. The general subvector space $V\subseteq H^0(L^n)$ of codimension $c$ satisfies the property $N_p$.