Regularity on abelian varieties II: Basic results on linear series and defining equations

Authors:
Giuseppe Pareschi and Mihnea Popa

Journal:
J. Algebraic Geom. **13** (2004), 167-193

DOI:
https://doi.org/10.1090/S1056-3911-03-00345-X

Published electronically:
August 21, 2003

MathSciNet review:
2008719

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Abstract | References | Additional Information

Abstract: We apply the theory of -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 -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.

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Additional Information

**Giuseppe Pareschi**

Affiliation:
Dipartamento di Matematica, Università di Roma, Tor Vergata, V.le della Ricerca Scientifica, I-00133 Roma, Italy

Email:
pareschi@mat.uniroma2.it

**Mihnea Popa**

Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138

Email:
mpopa@math.harvard.edu

DOI:
https://doi.org/10.1090/S1056-3911-03-00345-X

Received by editor(s):
October 21, 2001

Published electronically:
August 21, 2003

Additional Notes:
The second author was partially supported by a Clay Mathematics Institute Liftoff Fellowship during the preparation of this paper.