Clifford Indices for Vector Bundles on Curves

  • Herbert Lange
  • Peter E. Newstead
Part of the Trends in Mathematics book series (TM)


For smooth projective curves of genus g ≥ 4, the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.

Mathematics Subject Classification (2000)

Primary: 14H60 Secondary: 14F05 32L10 


Semistable vector bundle Clifford index gonality Brill-Noether theory 


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  1. [1]
    D. Arcara: A lower bound for the dimension of the base locus of the generalized theta divisor. C. R. Math. Acad. Sci. Paris 340 (2005), no. 2, 131–134.zbMATHMathSciNetGoogle Scholar
  2. [2]
    E. Ballico: Spanned vector bundles on algebraic curves and linear series. Rend. Istit. Mat. Univ. Trieste 27 (1995), no. 1–2, 137–156 (1996).zbMATHMathSciNetGoogle Scholar
  3. [3]
    E. Ballico: Brill-Noether theory for vector bundles on projective curves. Math. Proc. Camb. Phil. Soc. 128 (1998), 483–499.CrossRefGoogle Scholar
  4. [4]
    P. Belkale: The strange duality conjecture for generic curves. J. Amer. Math. Soc. 21 (2008), no. 1, 235–258.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Belkale: Strange duality and the Hitchin/WZW connection. J. Diff. Geom. 82 (2009), 445–465.zbMATHMathSciNetGoogle Scholar
  6. [6]
    U.N. Bhosle, L. Brambila-Paz, P.E. Newstead: On coherent systems of type (n, d, n+1) on Petri curves. Manuscr. Math. 126 (2008), 409–441.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    L. Brambila-Paz: Non-emptiness of moduli spaces of coherent systems. Int. J. Math. 19 (2008), 777–799.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    L. Brambila-Paz, I. Grzegorczyk, P.E. Newstead: Geography of Brill-Noether loci for small slopes. J. Alg. Geom. 6 (1997), 645–669.zbMATHMathSciNetGoogle Scholar
  9. [9]
    L. Brambila-Paz, V. Mercat, P.E. Newstead, F. Ongay: Nonemptiness of Brill-Noether loci. Int. J. Math. 11 (2000), 737–760.zbMATHMathSciNetGoogle Scholar
  10. [10]
    L. Brambila-Paz, A. Ortega: Brill-Noether bundles and coherent systems on special curves. Moduli spaces and vector bundles, London Math. Soc. Lecture Notes Ser., 359, Cambridge Univ. Press, Cambridge, 2009, pp. 456–472.Google Scholar
  11. [11]
    D.C. Butler: Normal generation of vector bundles over a curve. J. Diff. Geom. 39 (1994), 1–34.zbMATHGoogle Scholar
  12. [12]
    D.C. Butler: Birational maps of moduli of Brill-Noether pairs. arXiv:alg-geom/9705009v1.Google Scholar
  13. [13]
    S. Casalaina-Martin, T. Gwena, M. Teixidor i Bigas: Some examples of vector bundles in the base locus of the generalized theta divisor. arXiv:0707.2326v1.Google Scholar
  14. [14]
    C. Ciliberto: Alcuni applicazioni di un classico procedimento di Castelnuovo. Sem. di Geom., Dipart. di Matem., Univ. di Bologna, (1982–83), 17–43.Google Scholar
  15. [15]
    J. Cilleruelo, I. Sols: The Severi bound on sections of rank two semistable bundles on a Riemann surface. Ann. Math. 154 (2001), 739–758.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Coppens: The gonality of general smooth curves with a prescribed plane nodal model. Math. Ann. 289 (1991), 89–93.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Coppens, C. Keem, G. Martens: Primitive linear series on curves. Manuscr. Math. 77 (1992), 237–264.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    M. Coppens, G. Martens: Linear series on 4-gonal curves. Math. Nachr. 213 (2000), 35–55.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Coppens, G. Martens: Linear series on a general k-gonal curve. Abh.Math. Sem. Univ. Hamburg 69 (1999), 347–371.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    L. Ein, R. Lazarsfeld: Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves. London Math. Soc. Lecture Notes Ser., 179, Cambridge Univ. Press, Cambridge, 1992, pp. 149–156.Google Scholar
  21. [21]
    D. Eisenbud: Linear sections of determinantal varieties. Amer. J. Math. 110 (1988), 541–575.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    D. Eisenbud, H. Lange, G. Martens, F.-O. Schreyer: The Clifford dimension of a projective curve. Comp. Math. 72 (1989), 173–204.zbMATHMathSciNetGoogle Scholar
  23. [23]
    M.L. Green: Koszul cohomology and the geometry of projective varieties. J. Differential Geom. 19 (1984), no. 1, 125–171.zbMATHMathSciNetGoogle Scholar
  24. [24]
    M.L. Green, R. Lazarsfeld: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83 (1985), no. 1, 73–90.CrossRefMathSciNetGoogle Scholar
  25. [25]
    T. Gwena, M. Teixidor i Bigas: Maps between moduli spaces of vector bundles and the base locus of the theta divisor. Proc. Amer. Math. Soc. 137 (2009), no. 3, 853–861.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    S. Kim: On the Clifford sequence of a general k-gonal curve. Indag. Mathem. 8 (1997), 209–216.zbMATHCrossRefGoogle Scholar
  27. [27]
    H. Lange, M.S. Narasimhan: Maximal subbundles of rank two vector bundles on curves. Math. Ann. 266 (1983), 55–72.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    H. Lange, P.E. Newstead: On Clifford’s theorem for rank-3 bundles. Rev. Mat. Iberoamericana 22 (2006), 287–304.zbMATHMathSciNetGoogle Scholar
  29. [29]
    Y. Li: Spectral curves, theta divisors and Picard bundles. Int. J. Math. 2 (1991), 525–550.zbMATHCrossRefGoogle Scholar
  30. [30]
    A. Marian, D. Oprea: The level-rank duality for non-abelian theta functions. Invent. Math. 168 (2007), no. 2, 225–247.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    G. Martens, F.-O. Schreyer: Line bundles and syzygies of trigonal curves. Abh. Math. Sem. Univ. Hamburg 56 (1986), 169–189.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    V. Mercat: Le problème de Brill-Noether pour des fibrés stables de petite pente. J. reine angew. Math. 506 (1999), 1–41.zbMATHMathSciNetGoogle Scholar
  33. [33]
    V. Mercat: Fibrés stables de pente 2. Bull. London Math. Soc. 33 (2001), 535–542.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    V. Mercat: Clifford’s theorem and higher rank vector bundles. Int. J. Math. 13 (2002), 785–796.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    K. Paranjape, S. Ramanan: On the canonical ring of a curve. Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (1987), 503–516.Google Scholar
  36. [36]
    M. Popa: On the base locus of the generalized theta divisor. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 6, 507–512.zbMATHGoogle Scholar
  37. [37]
    M. Popa, M. Roth: Stable maps and Quot schemes. Invent. Math. 152 (2003), no. 3, 625–663.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    R. Re: Multiplication of sections and Clifford bounds for stable vector bundles on curves. Comm. in Alg. 26 (1998), 1931–1944.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    O. Schneider: Sur la dimension de l’ensemble des points base du fibré déterminant sur SU C >(r). Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 481–490.zbMATHGoogle Scholar
  40. [40]
    X.-J. Tan: Clifford’s theorems for vector bundles. Acta Math. Sin. 31 (1988), 710–720.zbMATHGoogle Scholar
  41. [41]
    M. Teixidor i Bigas: Brill-Noether theory for stable vector bundles. Duke Math. J. 62 (1991), 385–400.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    M. Teixidor i Bigas: Rank two vector bundles with canonical determinant. Math. Nachr. 265 (2004), 100–106.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    M. Teixidor i Bigas: Petri map for rank two vector bundles with canonical determinant. Compos. Math. 144 (2008), 705–720.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    M. Teixidor i Bigas: Syzygies using vector bundles. Trans. Amer. Math. Soc. 359 (2007), no. 2, 897–908.zbMATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    C. Voisin: Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri. Acta Math. 168 (1992), 249–272.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Herbert Lange
    • 1
  • Peter E. Newstead
    • 2
  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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