Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T12:35:43.045Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
GABRIEL STICLARU
Affiliation:
Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arnold, V. I., Gusein–Zadeh, S. M. and Varchenko, A. N.Singularities of Differentiable Maps. vols 1/2, Monographs in Math., 82/83 (Birkhäuser, Basel, 1985/1988).CrossRefGoogle Scholar
[2]Artal, E., Carmona, J., Cogolludo, J. I., Escario, M., Fernndez de Bobadilla, J., Luengo, I. and Melle, A.Topological and arithmetical properties of rational plane curves. Mathematical contributions in honor of Professor Enrique Outerelo Dominguez (Spanish), 223–242 (Homen. Univ. Complut., Editorial Complutense, Madrid, 2004).Google Scholar
[3]Bauer, I., Catanese, F. and Grunewald, F.Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory. Mediterr. J. Math. 3 (2006), no. 2, 121–146.CrossRefGoogle Scholar
[4]Chmutov, S. V.Examples of projective surfaces with many singularities. J. Algebraic Geom. 1 (1992), 191–196.Google Scholar
[5]Choudary, A. D. R. and Dimca, A.Koszul complexes and hypersurface singularities. Proc. Amer. Math. Soc. 121 (1994), 1009–1016.CrossRefGoogle Scholar
[6]Cogolludo–Agustn, J. I.Topological invariants of the complement to arrangements of rational plane curves. Mem. Amer. Math. Soc. 159 (2002), no. 756.Google Scholar
[7]Dimca, A.Singularities and Topology of Hypersurfaces (Universitext, Springer-Verlag, 1992).CrossRefGoogle Scholar
[8]Dimca, A. and Saito, M.A generalization of Griffiths' theorem on rational integrals. Duke Math. J. 135 (2006), 303–326.CrossRefGoogle Scholar
[9]Dimca, A., Saito, M. and Wotzlaw, L.A generalization of Griffiths' theorem on rational integrals II. Michigan Math. J. 58 (2009), 603–625.CrossRefGoogle Scholar
[10]Dimca, A. and Sticlaru, G. Koszul complexes and pole order filtrations, arXiv:1108.3976.Google Scholar
[11]Dimca, A. and Sticlaru, G. On the syzygies and Alexander polynomials of nodal hypersurfaces, arXiv:1111.1533.Google Scholar
[12]Greuel, G.-M. and Pfister, G.A Singular Introduction to Commutative Algebra (with contributions by Bachmann, O., Lossen, C., and Schönemann, H.). (Springer-Verlag, 2002), (second edition 2007).CrossRefGoogle Scholar
[13]Griffiths, Ph.On the period of certain rational integrals I, II. Ann. Math. 90 (1969), 460–541.CrossRefGoogle Scholar
[14]Griffiths, Ph. and Harris, J.Principles of Algebraic Geometry (Wiley, New York, 1978).Google Scholar
[15]Hartshorne, R.Algebraic Geometry (GTM 52, Springer 1977).CrossRefGoogle Scholar
[16]Koseleff, P.-V., Pecker, D. and Rouillier, F. Computing Chebyshev knots diagrams, arXiv:1001.5192v2.Google Scholar
[17]Peters, C. and Steenbrink, J.Mixed Hodge Structures, Ergeb. der Math. und ihrer Grenz. 3. Folge 52 (Springer, 2008).Google Scholar
[18]Schenck, H. and Tohaneanu, S. O.Freeness of Conic–Line Arrangements in $\mathbb P^2$. Comment. Math. Helv. 84 (2009), 235–258.CrossRefGoogle Scholar
[19]Xu, Y.Lagrange interpolation on Chebyshev points of two variables. J. Approx. Theory 87 (1996), 220–238.CrossRefGoogle Scholar
[20]CoCoA: a system for doing Computations in Commutative Algebra Available at http://cocoa.dima.unige.itGoogle Scholar
[21]Decker, W., Greuel, G.-M., Pfister, G. and Schönemann, H.Singular 3-1-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2011).Google Scholar