### Abstract

We apply the theory of M-regularity developed by the authors [Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), 285-302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called M-regularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-664], This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.

Original language | English (US) |
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Pages (from-to) | 167-193 |

Number of pages | 27 |

Journal | Journal of Algebraic Geometry |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Journal of Algebraic Geometry*,

*13*(1), 167-193. https://doi.org/10.1090/S1056-3911-03-00345-X