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

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.

**[BSz]**T. Bauer and T. Szemberg, Higher order embeddings of abelian varieties, Math. Z.**224**(1997), 449-455.**[BS]**M. Beltrametti and A. Sommese, On -jet ampleness, in*Complex analysis and geometry*, V. Ancona and A. Silva, eds., Plenum Press (1993), 355-376.**[BLvS]**C. Birkenhake, H. Lange and D. van Straten, Abelian surfaces of type , Math. Ann.**285**(1989), 625-646.**[Gr]**M. Green, Koszul cohomology and the geometry of projective varieties, I, J. Diff. Geom.**19**(1984), 125-171.**[GL]**M. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math.**83**(1985), 73-90.**[Kh]**A. Khaled, Equations definissant des varietes abeliennes, C.R. Acad. Sci. Paris, Ser. I Math**315**(1992) 571-576.**[Ke1]**G. Kempf, Linear systems on abelian varieties, Am. J. Math.**111**(1989), 65-94.**[Ke2]**G. Kempf, Projective coordinate rings of abelian varieties, in*Algebraic analysis, geometry and number theory*, J.I. Igusa ed., Johns Hopkins Press (1989), 225-236.**[Ke3]**G. Kempf,*Complex Abelian Varieties and Theta Functions*, Springer-Verlag, 1991.**[Ko]**S. Koizumi, Theta relations and projective normality of abelian varieties, Am. J. Math.**98**(1976), 865-889.**[LB]**H. Lange and C. Birkenhake,*Complex abelian varieties*, Springer-Verlag, 1992.**[La1]**R. Lazarsfeld, A sampling of vector bundles techniques in the study of linear series, in*Lectures on Riemann surfaces*(Cornalba, Gomez-Mont, Verjovsky, eds.), 500-559, World Scientific, 1989.**[La2]**R. Lazarsfeld, Lengths of periods and Seshadri constants of abelian varieties, Math. Res. Lett.**3**(1996), 439-447.**[Mu]**S. Mukai, Duality between and with its application to Picard sheaves, Nagoya Math. J.**81**(1981), 153-175.**[M1]**D. Mumford,*Abelian varieties*, Second edition, Oxford Univ. Press, 1974.**[M2]**D. Mumford, On the equations defining abelian varieties, Invent. Math.**1**(1966), 287-354.**[M3]**D. Mumford, Varieties defined by quadratic equations, in*Questions on algebraic varieties*, 31-100, Cremonese, Roma, 1970.**[Oh1]**A. Ohbuchi, Some remarks on simple line bundles on abelian varieties, Manuscripta Math.**57**(1987), 225-238.**[Oh2]**A. Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad.**64**(1988), 119-120.**[Pa]**G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc.**13**(2000), 651-664.**[PP]**G. Pareschi and M. Popa, Regularity on abelian varieties, I, J. Amer. Math. Soc.**16**(2003), 285-302.**[Se]**T. Sekiguchi, On the normal generation by a line bundle on an abelian variety, Proc. Japan Acad.**54**(1978), 185-188.

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.