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 $M$-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 $M$-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]bauer T. Bauer and T. Szemberg, Higher order embeddings of abelian varieties, Math. Z. **224** (1997), 449–455.
[BS]sommese M. Beltrametti and A. Sommese, On $k$-jet ampleness, in *Complex analysis and geometry*, V. Ancona and A. Silva, eds., Plenum Press (1993), 355–376.
[BLvS]blvs C. Birkenhake, H. Lange and D. van Straten, Abelian surfaces of type $(1,4)$, Math. Ann. **285** (1989), 625–646.
[Gr]green M. Green, Koszul cohomology and the geometry of projective varieties, I, J. Diff. Geom. **19** (1984), 125–171.
[GL]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]khaled A. Khaled, Equations definissant des varietes abeliennes, C.R. Acad. Sci. Paris, Ser. I Math **315** (1992) 571–576.
[Ke1]kempf1 G. Kempf, Linear systems on abelian varieties, Am. J. Math. **111** (1989), 65–94.
[Ke2]kempf2 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]kempf3 G. Kempf, *Complex Abelian Varieties and Theta Functions*, Springer-Verlag, 1991.
[Ko]koizumi S. Koizumi, Theta relations and projective normality of abelian varieties, Am. J. Math. **98** (1976), 865–889.
[LB]lange H. Lange and C. Birkenhake, *Complex abelian varieties*, Springer-Verlag, 1992.
[La1]lazarsfeld 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]lazarsfeld2 R. Lazarsfeld, Lengths of periods and Seshadri constants of abelian varieties, Math. Res. Lett. **3** (1996), 439–447.
[Mu]mukai S. Mukai, Duality between $D(X)$ and $D(\hat {X})$ with its application to Picard sheaves, Nagoya Math. J. **81** (1981), 153–175.
[M1]mumford D. Mumford, *Abelian varieties*, Second edition, Oxford Univ. Press, 1974.
[M2]mumford1 D. Mumford, On the equations defining abelian varieties, Invent. Math. **1** (1966), 287–354.
[M3]mumford2 D. Mumford, Varieties defined by quadratic equations, in *Questions on algebraic varieties*, 31–100, Cremonese, Roma, 1970.
[Oh1]ohbuchi1 A. Ohbuchi, Some remarks on simple line bundles on abelian varieties, Manuscripta Math. **57** (1987), 225–238.
[Oh2]ohbuchi2 A. Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad. **64** (1988), 119–120.
[Pa]pareschi G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. **13** (2000), 651–664.
[PP]us G. Pareschi and M. Popa, Regularity on abelian varieties, I, J. Amer. Math. Soc. **16** (2003), 285–302.
[Se]Sekiguchi T. Sekiguchi, On the normal generation by a line bundle on an abelian variety, Proc. Japan Acad. **54** (1978), 185–188.

[BSz]bauer T. Bauer and T. Szemberg, Higher order embeddings of abelian varieties, Math. Z. **224** (1997), 449–455.
[BS]sommese M. Beltrametti and A. Sommese, On $k$-jet ampleness, in *Complex analysis and geometry*, V. Ancona and A. Silva, eds., Plenum Press (1993), 355–376.
[BLvS]blvs C. Birkenhake, H. Lange and D. van Straten, Abelian surfaces of type $(1,4)$, Math. Ann. **285** (1989), 625–646.
[Gr]green M. Green, Koszul cohomology and the geometry of projective varieties, I, J. Diff. Geom. **19** (1984), 125–171.
[GL]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]khaled A. Khaled, Equations definissant des varietes abeliennes, C.R. Acad. Sci. Paris, Ser. I Math **315** (1992) 571–576.
[Ke1]kempf1 G. Kempf, Linear systems on abelian varieties, Am. J. Math. **111** (1989), 65–94.
[Ke2]kempf2 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]kempf3 G. Kempf, *Complex Abelian Varieties and Theta Functions*, Springer-Verlag, 1991.
[Ko]koizumi S. Koizumi, Theta relations and projective normality of abelian varieties, Am. J. Math. **98** (1976), 865–889.
[LB]lange H. Lange and C. Birkenhake, *Complex abelian varieties*, Springer-Verlag, 1992.
[La1]lazarsfeld 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]lazarsfeld2 R. Lazarsfeld, Lengths of periods and Seshadri constants of abelian varieties, Math. Res. Lett. **3** (1996), 439–447.
[Mu]mukai S. Mukai, Duality between $D(X)$ and $D(\hat {X})$ with its application to Picard sheaves, Nagoya Math. J. **81** (1981), 153–175.
[M1]mumford D. Mumford, *Abelian varieties*, Second edition, Oxford Univ. Press, 1974.
[M2]mumford1 D. Mumford, On the equations defining abelian varieties, Invent. Math. **1** (1966), 287–354.
[M3]mumford2 D. Mumford, Varieties defined by quadratic equations, in *Questions on algebraic varieties*, 31–100, Cremonese, Roma, 1970.
[Oh1]ohbuchi1 A. Ohbuchi, Some remarks on simple line bundles on abelian varieties, Manuscripta Math. **57** (1987), 225–238.
[Oh2]ohbuchi2 A. Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad. **64** (1988), 119–120.
[Pa]pareschi G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. **13** (2000), 651–664.
[PP]us G. Pareschi and M. Popa, Regularity on abelian varieties, I, J. Amer. Math. Soc. **16** (2003), 285–302.
[Se]Sekiguchi 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

MR Author ID:
653676

Email:
mpopa@math.harvard.edu

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.