## Noncomplete linear systems on abelian varieties

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- by Christina Birkenhake PDF
- Trans. Amer. Math. Soc.
**348**(1996), 1885-1908 Request permission

## Abstract:

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 $p\ge 1$, $n\ge 2p+2$ 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 $N_p$.*

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## Additional Information

**Christina Birkenhake**- Affiliation: Mathematisches Institut, Universität Erlangen Bismarckstrasse 1$\frac 12$, D-91054 Erlangen, Germany
- Email: Birkenhake@mi.uni-erlangen.de
- Received by editor(s): June 9, 1995
- Additional Notes: Supported by EC Contract No. CHRXCT 940557
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**348**(1996), 1885-1908 - MSC (1991): Primary 14C20, 14K05
- DOI: https://doi.org/10.1090/S0002-9947-96-01570-X
- MathSciNet review: 1340170