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Chebyshev curves, free resolutions and rational curve arrangements

Published online by Cambridge University Press:  28 February 2012

ALEXANDRU DIMCA  and
GABRIEL STICLARU
ALEXANDRU DIMCA
Affiliation:
Institut Universitaire de France et Laboratoire J.A. Dieudonné, UMR du CNRS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. e-mail: dimca@unice.fr
GABRIEL STICLARU
Affiliation:
Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania. e-mail: gabrielsticlaru@yahoo.com
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Abstract

First we construct a free resolution for the Milnor (or Jacobian) algebra M(f) of a complex projective Chebyshev plane curve d : f = 0 of degree d. In particular, this resolution implies that the dimensions of the graded components M(f)k are constant for k ≥ 2d − 3.

Then we show that the Milnor algebra of a nodal plane curve C has such a behaviour if and only if all the irreducible components of C are rational.

For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 385 - 397
Copyright
Copyright © Cambridge Philosophical Society 2012

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