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.
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- S. Yuzvinsky
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 94703
- Email: firstname.lastname@example.org
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 1641-1648
- MSC (2000): Primary 14H50; Secondary 32S22, 52C35
- DOI: https://doi.org/10.1090/S0002-9939-08-09753-0
- MathSciNet review: 2470822