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A generalization of Max Noether's theorem

Author: Renato Vidal Martins
Journal: Proc. Amer. Math. Soc. 140 (2012), 377-391
MSC (2010): Primary 14H20; Secondary 14H45, 14H51
Published electronically: May 31, 2011
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Abstract: Max Noether's theorem asserts that if $ \omega$ is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve, then the natural morphisms Sym$ ^nH^0(\omega)\to H^0(\omega^n)$ are surjective for all $ n\geq 1$. This is true for Gorenstein nonhyperelliptic curves as well. We prove that this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The results are independent and have different proofs. The first one is extrinsic, the second intrinsic.

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

  • 1. Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., ``Geometry of Algebraic Curves,'' Springer-Verlag, 1985. MR 770932 (86h:14019)
  • 2. Barucci, V., and Fröberg, R., One-Dimensional Almost Gorenstein Rings, J. Alg. 188 (1997), 418-442. MR 1435367 (98a:13033)
  • 3. Barucci, V., D'Anna, M., and Fröberg, R., Analytically Unramified One-Dimensional Semilocal Rings and Their Value Semigroups, J. Pure Appl. Alg. 147 (2000), 215-254. MR 1747441 (2001g:13054)
  • 4. Eisenbud, D., Koh, J., and Stillman, M. (appendix with Harris, J.), Determinantal Equations for Curves of High Degree, Amer. J. Math. 110 (1988), 513-539. MR 944326 (89g:14023)
  • 5. Kleiman, S. L., and Martins, R. V., The Canonical Model of a Singular Curve, Geom. Dedicata 139 (2009), 139-166. MR 2481842 (2010d:14038)
  • 6. Matsuoka, T., On the Degree of Singularity of One-Dimensional Analytically Irreducible Noetherian Local Rings, J. Math. Kyoto Univ. 11-3 (1971), 485-494. MR 0285526 (44:2744)
  • 7. Rosenlicht, M., Equivalence Relations on Algebraic Curves, Annals of Math. (2) 56 (1952), 169-191. MR 0048856 (14:80c)
  • 8. Rosa, R., Stöhr, K.-O., Trigonal Gorenstein Curves, J. Pure Appl. Alg. 174 (2002), 187-205. MR 1921820 (2003k:14037)
  • 9. Stöhr, K.-O., On the Poles of Regular Differentials of Singular Curves, Bull. Brazilian Math. Soc. 24 (1993), 105-135. MR 1224302 (94e:14035)
  • 10. Stöhr, K.-O., Hyperelliptic Gorenstein Curves, J. Pure Appl. Alg. 135 (1999), 93-105. MR 1667447 (99m:14049)

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Additional Information

Renato Vidal Martins
Affiliation: Departamento de Matemática, Instituto de Ciencias Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 30123-970 Belo Horizonte MG, Brazil

Keywords: Singular curve, non-Gorenstein curve, Max Noether theorem
Received by editor(s): September 14, 2009
Received by editor(s) in revised form: November 16, 2010
Published electronically: May 31, 2011
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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