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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

Syzygies of abelian varieties


Author: Giuseppe Pareschi
Journal: J. Amer. Math. Soc. 13 (2000), 651-664
MSC (2000): Primary 14K05; Secondary 14F05
Published electronically: April 10, 2000
MathSciNet review: 1758758
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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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Giuseppe Pareschi
Affiliation: Dipartimento di Matematica, Università di Roma, Tor Vergata V.le della Ricerca Scientifica, I-00133 Roma, Italy
Email: pareschi@mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00335-0
Keywords: Homogeneous ideal, Pontrjagin product, vector bundles
Received by editor(s): August 24, 1998
Received by editor(s) in revised form: March 8, 2000
Published electronically: April 10, 2000
Article copyright: © Copyright 2000 American Mathematical Society