Inventiones mathematicae

, Volume 190, Issue 3, pp 603–646 | Cite as

Asymptotic syzygies of algebraic varieties

Article

Abstract

We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo–Mumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.

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References

  1. 1.
    Aprodu, M.: Green–Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math. Res. Not. 63, 3409–3416 (2004) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aprodu, M., Voisin, C.: Green–Lazarsfeld’s conjecture for generic curves of large gonality. C. R. Math. Acad. Sci. Paris 336, 335–339 (2003) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bombieri, E.: Canonical models of surfaces of general type. Publ. Math. IHÉS 42, 171–219 (1973) MathSciNetGoogle Scholar
  4. 4.
    Bruns, W., Conca, A., Römer, T.: Koszul homology and syzygies of Veronese subalgebras. Math. Ann. 351, 761–779 (2011) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bruns, W., Conca, A., Römer, T.: Koszul cycles. In: Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Proceedings of the Abel Symposium 2009, pp. 17–32 (2011) CrossRefGoogle Scholar
  6. 6.
    Castelnuovo, G.: Sui multipli di uni serie lineare di gruppi di punti apparetmenente as una curva algebrica. Rend. Circ. Mat. Palermo 7, 99–119 (1893) CrossRefGoogle Scholar
  7. 7.
    Catanese, F.: Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications. Invent. Math. 63, 433–465 (1981) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Catanese, F.: Commutative algebra methods and equations of regular surfaces. In: Algebraic Geometry, Bucharest 1982. Lecture Notes in Math., vol. 1056, pp. 30–50. Springer, Berlin (1983) Google Scholar
  9. 9.
    Ein, L., Lazarsfeld, R.: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math. 111, 51–67 (1993) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eisenbud, D.: The Geometry of Syzygies. Graduate Texts in Math., vol. 229. Springer, Berlin (2005) MATHGoogle Scholar
  11. 11.
    Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Restricting linear syzygies: algebra and geometry. Compos. Math. 141, 1460–1478 (2005) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Eisenbud, D., Schreyer, F.: Betti numbers of graded modules and cohomology of vector bundles. J. Am. Math. Soc. 22, 859–888 (2009) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984) MATHGoogle Scholar
  14. 14.
    Green, M.: Koszul cohomology and the geometry of projective varieties, II. J. Differ. Geom. 20, 279–289 (1984) MATHGoogle Scholar
  15. 15.
    Green, M., Lazarsfeld, R.: A simple proof of Petri’s theorem on canonical curves. In: Geometry Today (Rome 1984). Progr. Math., vol. 60. Birkhäuser, Basel (1985) Google Scholar
  16. 16.
    Green, M., Lazarsfeld, R.: Some results on the syzygies of finite sets and algebraic curves. Compos. Math. 67, 301–314 (1988) MATHMathSciNetGoogle Scholar
  17. 17.
    Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1985) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Hering, M., Schenck, H., Smith, G.: Syzygies, multigraded regularity and toric varieties. Compos. Math. 142, 1499–1506 (2006) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. In: Lectures on Riemann Surfaces, pp. 500–559. World Scientific, Singapore (1989) Google Scholar
  20. 20.
    Maclagan, D., Smith, G.: Multigraded Castelnuovo–Mumford regularity. J. Reine Angew. Math. 571, 179–212 (2004) MATHMathSciNetGoogle Scholar
  21. 21.
    Mumford, D.: On the equations defining abelian varieties. Invent. Math. 1, 287–354 (1966) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties, Corso CIME 1969, Rome, pp. 30–100 (1970) Google Scholar
  23. 23.
    Ottaviani, G., Paoletti, R.: Syzygies of Veronese embeddings. Compos. Math. 125, 31–37 (2001) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pareschi, G.: Syzygies of abelian varieties. J. Am. Math. Soc. 13, 651–664 (2000) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rubei, E.: A result on resolutions of Veronese embeddings. Ann. Univ. Ferrara, Sez. 7 50, 151–165 (2004) MATHMathSciNetGoogle Scholar
  26. 26.
    Schreyer, F.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sidman, J., Van Tuyl, A., Wang, H.: Multigraded regularity: coarsenings and resolutions. J. Algebra 301, 703–727 (2006) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Snowden, A.: Syzygies of Segre varieties and Δ functors. To appear Google Scholar
  29. 29.
    Zhou, X.: Thesis in preparation Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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