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On coherent systems of type (n, d, n  +  1) on Petri curves

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Abstract

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding “flip” has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill–Noether theory and the stability of kernels of evaluation maps, thus proving Butler’s conjecture in some cases in which it was not previously known.

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References

  1. Arbarello E., Cornalba M., Griffiths P.A., Harris J.: Geometry of Algebraic Curves, vol. I. Springer, New York (1985)

    Google Scholar 

  2. Ballico E.: Coherent systems with many sections on projective curves. Int. J. Math. 17, 263–267 (2006)

    Article  MATH  Google Scholar 

  3. Beauville A.: Some stable vector bundles with reducible theta divisor. Manuscripta Math. 110, 343–349 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bradlow S.B., Garcia-Prada O.: An application of coherent systems to a Brill–Noether problem. J. Reine Angew Math. 551, 123–143 (2002)

    MATH  MathSciNet  Google Scholar 

  5. Bradlow S.B., Garcia-Prada O., Muñoz V., Newstead P.E.: Coherent systems and Brill–Noether theory. Int. J. Math. 14, 683–733 (2003)

    Article  MATH  Google Scholar 

  6. Bradlow S.B., Garcia-Prada O., Mercat V., Muñoz V., Newstead P.E.: On the geometry of moduli spaces of coherent systems on algebraic curves. Int. J. Math. 18, 411–453 (2007)

    Article  MATH  Google Scholar 

  7. Bradlow, S.B., Garcia-Prada, O., Mercat, V., Muñoz, V., Newstead, P.E.: Moduli spaces of coherent systems of small slope on algebraic curves, preprint, arXiv:0707.0983

  8. Brambila-Paz, L.: Non-emptiness of moduli spaces of coherent systems. Int. J. Math., to appear, arXiv:math/0412285

  9. Brambila-Paz, L., Ortega, A.: Tensor product of coherent systems, preprint, arXiv: 0711.3944

  10. Brambila-Paz L., Grzegorczyk I., Newstead P.E.: Geography of Brill–Noether loci for small slopes. J. Alg. Geom. 6, 645–669 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Butler D.C.: Normal generation of vector bundles over a curve. J. Diff. Geom. 39, 1–34 (1994)

    MATH  MathSciNet  Google Scholar 

  12. Butler, D.C.: Birational maps of moduli of Brill–Noether pairs, preprint, arXiv:alg-geom/9705009

  13. Ein, L., Lazarsfeld, R.: Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves. In: Ellingsrud, G., Peskine, C., Sacchiero, G., Stromme, S.A. (eds.) Complex Projective Geometry (Trieste 1989/Bergen 1989). LMS Lecture Notes Series, vol. 179, pp. 149–156. CUP, Cambridge (1992)

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

    Google Scholar 

  15. He M.: Espaces de modules de systèmes cohérents. Int. J. Math. 9, 545–598 (1998)

    Article  MATH  Google Scholar 

  16. King A.D., Newstead P.E.: Moduli of Brill–Noether pairs on algebraic curves. Int. J. Math. 6, 733–748 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lange H., Newstead P.E.: Coherent systems of genus 0. Int. J. Math. 15, 409–424 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lange H., Newstead P.E.: Coherent systems on elliptic curves. Int. J. Math. 16, 787–805 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Le Potier, J.: Faisceaux semi-stables et systèmes cohérents. In: Hitchin, N.J., Newstead, P.E., Oxbury, W.M. (eds.) Vector Bundles in Algebraic Geometry, Durham 1993. LMS Lecture Notes Series, vol. 208, pp. 179–239. Cambridge University Press, Cambridge~(1995)

  20. Mercat V.: Le problème de Brill–Noether pour les fibrés stables de petite pente. J. Reine Angew Math. 506, 1–41 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Mercat V.: Le problème de Brill–Noether et la théorème de Teixidor. Manuscripta Math. 98, 75–85 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mercat V.: Fibrés stables de pente 2. Bull. Lond. Math. Soc. 33, 535–542 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mistretta, E.: Stability of line bundles transforms on curves with respect to low codimensional subspaces, preprint, arXiv:math/0703465

  24. Narasimhan M.S., Ramanan S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. 101, 391–417 (1975)

    Article  MathSciNet  Google Scholar 

  25. Paranjape, K., Ramanan, S.: On the canonical ring of a curve. In: Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, pp. 503–516 (1987)

  26. Raghavendra N., Vishwanath P.A.: Moduli of pairs and generalized theta divisors. Tôhoku Math. J. 46, 321–340 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  27. Re R.: Multiplication of sections and Clifford bounds for stable vector bundles on curves. Comm. Algebra 26, 1931–1944 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sundaram N.: Special divisors and vector bundles. Tôhoku Math. J. 39, 175–213 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  29. Teixidor i Bigas M.: Brill–Noether theory for stable vector bundles. Duke Math. J. 62, 385–400 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Teixidor i Bigas, M.: Existence of coherent systems. Int. J. Math., to appear, arXiv:math/ 0606348

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Authors and Affiliations

  1. TIFR, Homi Bhabha Road, Mumbai, 400005, India

    U. N. Bhosle

  2. CIMAT, Apdo. Postal 402, C.P. 36240, Guanajuato, Gto, Mexico

    L. Brambila-Paz

  3. Department of Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool, L69 7ZL, UK

    P. E. Newstead

Authors
  1. U. N. Bhosle
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  2. L. Brambila-Paz
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  3. P. E. Newstead
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Corresponding author

Correspondence to P. E. Newstead.

Additional information

The authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). The first two authors were supported by EPSRC grant GR/T22988/01 for a visit to the University of Liverpool. The second author acknowledges the support of CONACYT grant 48263-F. The third author thanks CIMAT, Guanajuato, México and California State University Channel Islands, where a part of this paper was completed, and acknowledges support from the Academia Mexicana de Ciencias, under its exchange agreement with the Royal Society of London.

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Bhosle, U.N., Brambila-Paz, L. & Newstead, P.E. On coherent systems of type (n, d, n  +  1) on Petri curves. manuscripta math. 126, 409–441 (2008). https://doi.org/10.1007/s00229-008-0190-y

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  • DOI: https://doi.org/10.1007/s00229-008-0190-y

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