Inventiones mathematicae

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

Asymptotic syzygies of algebraic varieties

  • Lawrence Ein
  • Robert LazarsfeldEmail author


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.


Vector Bundle Algebraic Variety Smooth Projective Variety Ample Divisor Segre Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eisenbud, D.: The Geometry of Syzygies. Graduate Texts in Math., vol. 229. Springer, Berlin (2005) zbMATHGoogle Scholar
  11. 11.
    Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Restricting linear syzygies: algebra and geometry. Compos. Math. 141, 1460–1478 (2005) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984) zbMATHGoogle Scholar
  14. 14.
    Green, M.: Koszul cohomology and the geometry of projective varieties, II. J. Differ. Geom. 20, 279–289 (1984) zbMATHGoogle 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) zbMATHMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle 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) zbMATHMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pareschi, G.: Syzygies of abelian varieties. J. Am. Math. Soc. 13, 651–664 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rubei, E.: A result on resolutions of Veronese embeddings. Ann. Univ. Ferrara, Sez. 7 50, 151–165 (2004) zbMATHMathSciNetGoogle Scholar
  26. 26.
    Schreyer, F.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sidman, J., Van Tuyl, A., Wang, H.: Multigraded regularity: coarsenings and resolutions. J. Algebra 301, 703–727 (2006) zbMATHCrossRefMathSciNetGoogle 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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