Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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.


References [Enhancements On Off] (What's this?)

  • [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 $k$-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 $(1,4)$, 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 $D(X)$ and $D(\hat{X})$ 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.

American Mathematical Society