Syzygies of abelian varieties

Pareschi, Giuseppe

doi:10.1090/S0894-0347-00-00335-0

Syzygies of abelian varieties
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by Giuseppe Pareschi

J. Amer. Math. Soc. 13 (2000), 651-664

DOI: https://doi.org/10.1090/S0894-0347-00-00335-0

Published electronically: April 10, 2000

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Abstract:

We prove a conjecture of R. Lazarsfeld on the syzygies (of the homogeneous ideal) of abelian varieties embedded in projective space by multiples of an ample line bundle. Specifically, we prove that if $A$ is an ample line on an abelian variety, then $A^{\otimes n}$ satisfies the property $N_{p}$ as soon as $n\ge p+ 3$. The proof uses a criterion for the global generation of vector bundles on abelian varieties (generalizing the classical one for line bundles) and a criterion for the surjectivity of multiplication maps of global sections of two vector bundles in terms of the vanishing of the cohomology of certain twists of their Pontrjagin product.

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