A new bound on the number of special fibers in a pencil of curves
By Sergey Yuzvinsky. 2008
63 citations recorded by the source corpus.
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.
Subfields: Computer Graphics and Computer-Aided Design, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/2cf198807ec9e4949b2bdc548fb9990abcb3cf12
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In Computer Graphics and Computer-Aided Design
In Geometry and Topology
Citations in the corpus
- Hyperplane arrangements and Milnor fibrations (2013, result): “As shown by Falk and Yuzvinsky in [25], the irreducible components of V1(A) passing through the origin can be described in terms of multinets on the intersection lattice of A.”
- The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy (2014, background): “The basic structure of the (complex) resonance varieties of arrangements is explained in the following theorem, which summarizes the works of Cohen–Suciu [9] and Libgober–Yuzvinsky [38].”
- Classical Algebraic Geometry: A Modern View (2012, background): “Stipins [548] (see also [607]) that N(2,m) ≤ 4 for all m and it is conjectured that N(2,m) = 3 for m 6= 3.”