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Transactions of the American Mathematical Society

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Maximal degree subsheaves of torsion free sheaves on singular projective curves

Author: E. Ballico
Journal: Trans. Amer. Math. Soc. 353 (2001), 3617-3627
MSC (2000): Primary 14H20, 14H60
Published electronically: April 18, 2001
MathSciNet review: 1837251
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Fix integers $r,k,g$ with $r>k>0$ and $g\ge 2$. Let $X$ be an integral projective curve with $g:=p_a(X)$ and $E$ a rank $r$ torsion free sheaf on $X$which is a flat limit of a family of locally free sheaves on $X$. Here we prove the existence of a rank $k$ subsheaf $A$ of $E$ such that $r(\deg(A))\ge k(\deg (E))-k(r-k)g$. We show that for every $g\ge 9$ there is an integral projective curve $X,X$ not Gorenstein, and a rank 2 torsion free sheaf $E$ on $X$ with no rank 1 subsheaf $A$ with $2(\deg (A))\ge \deg(E)-g$. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.

References [Enhancements On Off] (What's this?)

  • 1. E. Ballico, On the number of components of the moduli schemes of stable torsion-free sheaves on integral curves, Proc. Amer. Math. Soc. 125 (1997), 2819-2824. MR 98d:14013
  • 2. -, Stable sheaves on reduced projective curves, Ann. Mat. Pura Appl. (4) 175 (1998), 375-393. MR 2001a:14029
  • 3. C. Banica, M. Putinar, and G. Schumaker, Variation deg globalen Ext in Deformationen kompakter Räume, Math. Ann. 250 (1980), 135-155. MR 82e:32015
  • 4. V. Barucci, D. E. Dobbs, and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Memoirs Amer. Math. Soc. 598, 1997. MR 97g:13039
  • 5. V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), 418-442. MR 98a:13033
  • 6. D. Eisenbud, J. Harris, J. Koh, and M. Stillman, Determinantal equations for curves of high degree, by D. Eisenbud, J. Koh, M. Stillman, Amer. J. Math. 110 (1988), 513-519. MR 89g:14023
  • 7. A. Grothendieck, Fondements de la géométrie algébriques (extraits du Séminaire Bourbaki), Secrétarial Math., Paris, 1962. MR 26:3566
  • 8. R. Hartshorne and A. Hirschowitz, Cohomology of a general instanton bundle, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 365-390. MR 84c:14011
  • 9. A. Hirschowitz, Problémes de Brill-Noether en rang supérieur, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 153-156. MR 89i:14010
  • 10. H. Lange, Universal families of extensions, J. Algebra 83 (1983), 101-112. MR 86e:14006
  • 11. H. Lange, Zur Klassifikation von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), 447-459. MR 85b:14019
  • 12. H. Lange and M. S. Narasimhan, Maximal subbundles of rank two vector bundles, Math. Ann. 266 (1983), 55-72. MR 85f:14013
  • 13. M. Maruyama, On the classification of ruled surfaces, Lecture in Mathematics Kyoto Univ. No. 3, Tokyo, 1970. MR 43:1990
  • 14. M. Maruyama, Elementary transformations of algebraic vector bundles. In: Algebraic Geometry - Proceedings La Rabida, pp. 241-266, Lecture Notes in Math. 961, Springer-Verlag, 1981. MR 85b:14020
  • 15. S. Mukai and F. Sakai, Maximal subbundles of vector bundles on a curve, Manuscripta Math. 52 (1985), 251-256. MR 86k:14013
  • 16. P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Inst. Lecture Notes, Springer-Verlag, 1978. MR 81k:14002
  • 17. C. J. Rego, Compactification of the space of vector bundles on a singular curve, Comm. Math. Helv. 57 (1982), 226-236. MR 84f:14015

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Additional Information

E. Ballico
Affiliation: Dipartimento di Matematicà, Università di Trento, 38050 Povo (TN) - Italy

Received by editor(s): September 25, 1998
Published electronically: April 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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