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Very ample linear systems on abelian varieties. (English) Zbl 0806.14031
Let $$(X,L)$$ be a complex abelian variety of dimension $$g$$ with a polarization of type $$(1, \dots, 1,d)$$. For $$(X,L)$$ generic we prove the following:
1. If $$d \geq g + 2$$, then $$\varphi_{| L |} : X \to \mathbb{P}^{d- 1}$$ 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.

##### MSC:
 14K05 Algebraic theory of abelian varieties 14C20 Divisors, linear systems, invertible sheaves
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