Very ample linear systems on abelian varieties

Abstract

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d\ge g+2$, then ${\varphi }_L:X\to {\mathbf{P}}^{d-1}$ defines a birational morphism onto its image. (2) If $d>2^g$, then $L$ is very ample. We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.

arXiv comment29 pages, typeset with AMS-LaTeX 1.1