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Asymptotic syzygies of algebraic varieties

Inventiones mathematicae volume 190, pages 603–646 (2012)Cite this 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. Aprodu, M.: Green–Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math. Res. Not. 63, 3409–3416 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bombieri, E.: Canonical models of surfaces of general type. Publ. Math. IHÉS 42, 171–219 (1973)

    MathSciNet  Google Scholar 

  4. Bruns, W., Conca, A., Römer, T.: Koszul homology and syzygies of Veronese subalgebras. Math. Ann. 351, 761–779 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  7. Catanese, F.: Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications. Invent. Math. 63, 433–465 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  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. Ein, L., Lazarsfeld, R.: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math. 111, 51–67 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Eisenbud, D.: The Geometry of Syzygies. Graduate Texts in Math., vol. 229. Springer, Berlin (2005)

    MATH  Google Scholar 

  11. Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Restricting linear syzygies: algebra and geometry. Compos. Math. 141, 1460–1478 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Eisenbud, D., Schreyer, F.: Betti numbers of graded modules and cohomology of vector bundles. J. Am. Math. Soc. 22, 859–888 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)

    MATH  Google Scholar 

  14. Green, M.: Koszul cohomology and the geometry of projective varieties, II. J. Differ. Geom. 20, 279–289 (1984)

    MATH  Google Scholar 

  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. Green, M., Lazarsfeld, R.: Some results on the syzygies of finite sets and algebraic curves. Compos. Math. 67, 301–314 (1988)

    MATH  MathSciNet  Google Scholar 

  17. Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1985)

    Article  MathSciNet  Google Scholar 

  18. Hering, M., Schenck, H., Smith, G.: Syzygies, multigraded regularity and toric varieties. Compos. Math. 142, 1499–1506 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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. Maclagan, D., Smith, G.: Multigraded Castelnuovo–Mumford regularity. J. Reine Angew. Math. 571, 179–212 (2004)

    MATH  MathSciNet  Google Scholar 

  21. Mumford, D.: On the equations defining abelian varieties. Invent. Math. 1, 287–354 (1966)

    Article  MathSciNet  Google Scholar 

  22. Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties, Corso CIME 1969, Rome, pp. 30–100 (1970)

    Google Scholar 

  23. Ottaviani, G., Paoletti, R.: Syzygies of Veronese embeddings. Compos. Math. 125, 31–37 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pareschi, G.: Syzygies of abelian varieties. J. Am. Math. Soc. 13, 651–664 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rubei, E.: A result on resolutions of Veronese embeddings. Ann. Univ. Ferrara, Sez. 7 50, 151–165 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Schreyer, F.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sidman, J., Van Tuyl, A., Wang, H.: Multigraded regularity: coarsenings and resolutions. J. Algebra 301, 703–727 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Snowden, A.: Syzygies of Segre varieties and Δ functors. To appear

  29. Zhou, X.: Thesis in preparation

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Authors and Affiliations

  1. Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL, 60607, USA

    Lawrence Ein

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Robert Lazarsfeld

Authors
  1. Lawrence Ein
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  2. Robert Lazarsfeld
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Correspondence to Robert Lazarsfeld.

Additional information

Research of the first author partially supported by NSF grant DMS-1001336.

Research of the second author partially supported by NSF grant DMS-0652845.

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Ein, L., Lazarsfeld, R. Asymptotic syzygies of algebraic varieties. Invent. math. 190, 603–646 (2012). https://doi.org/10.1007/s00222-012-0384-5

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  • DOI: https://doi.org/10.1007/s00222-012-0384-5

Keywords

  • Vector Bundle
  • Algebraic Variety
  • Smooth Projective Variety
  • Ample Divisor
  • Segre Variety