Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A new bound on the number of special fibers in a pencil of curves


Author: S. Yuzvinsky
Journal: Proc. Amer. Math. Soc. 137 (2009), 1641-1648
MSC (2000): Primary 14H50; Secondary 32S22, 52C35
Published electronically: November 19, 2008
MathSciNet review: 2470822
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Abstract | References | Similar Articles | Additional Information

Abstract: In a paper by J. V. Pereira and the author it was proved that any pencil of plane curves of degree $ d>1$ with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such fibers had ever been found. Recently Janis Stipins has proved that if a pencil has a base of $ d^2$ points, then it cannot have five completely reducible fibers. In this paper we generalize Stipins' result to arbitrary pencils. We also include into consideration more general special fibers that are the unions of lines and non-reduced curves. These fibers are important for characteristic varieties of hyperplane complements.


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

  • 1. A. Bodin, Reducibility of rational functions in several variables, math.AG/0510434.
  • 2. Julian Lowell Coolidge, A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959. MR 0120551
  • 3. A. Dimca, Pencils of plane curves and characteristic varieties, math.AG/0606442.
  • 4. Michael Falk and Sergey Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), no. 4, 1069–1088. MR 2339840, 10.1112/S0010437X07002722
  • 5. William Fulton, Algebraic curves. An introduction to algebraic geometry, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss; Mathematics Lecture Notes Series. MR 0313252
  • 6. J. Hadamard, Sur les conditions de décomposition des formes, Bull. Soc. Math. France 27 (1899), 34–47 (French). MR 1504330
  • 7. G. Halphen, Oeuvres de G.-H. Halphen, t. III, Gauthier-Villars, 1921, 1-260.
  • 8. Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337–361. MR 1761630, 10.1023/A:1001826010964
  • 9. J. V. Pereira and S. Yuzvinsky, Completely reducible hypersurfaces in a pencil, Advances in Math. 219 (2008), 672-688.
  • 10. J. Stipins, On finite $ k$-nets in the complex projective plane, Ph.D. thesis, The University of Michigan, 2007.
  • 11. Angelo Vistoli, The number of reducible hypersurfaces in a pencil, Invent. Math. 112 (1993), no. 2, 247–262. MR 1213102, 10.1007/BF01232434
  • 12. Sergey Yuzvinsky, Realization of finite abelian groups by nets in ℙ², Compos. Math. 140 (2004), no. 6, 1614–1624. MR 2098405, 10.1112/S0010437X04000600

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Additional Information

S. Yuzvinsky
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 94703
Email: yuz@uoregon.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09753-0
Received by editor(s): January 10, 2008
Received by editor(s) in revised form: July 27, 2008
Published electronically: November 19, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society