Syzygies of abelian varieties

Pareschi, Giuseppe¹

¹Dipartimento di Matematica, Università di Roma, Tor Vergata V.le della Ricerca Scientifica, I-00133 Roma, Italy

Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 651-664

Résumé

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.

Détail

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

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