Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair

Fernández de Bobadilla de Olarzábal, Javier José and Luengo Velasco, Ignacio and Melle Hernández, Alejandro and Némethi , A. (2007) Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair. In Real and complex singularities. Trends in Mathematics . Birkhauser Boston, Birkhauser Boston, 675 Massachusetts Ave, Cambridge, Ma 02139-2333 Usa, pp. 31-45. ISBN 3-7643-7775-5

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Abstract

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e., C is unicuspidal) with one Puiseux pair.

Item Type:	Book Section
Additional Information:	Conference: 8th Workshop on Real and Complex Singularities Location: Luminy, France Date: Jul. 19-23, 2004
Uncontrolled Keywords:	Cuspidal rational plane curves; Logarithmic Kodaira dimension
Subjects:	Sciences > Mathematics > Algebraic geometry
ID Code:	16589
References:	Fernández de Bobadilla J., Luengo I., Melle-Hernández A., Némethi A.: On rational cuspidal projective plane curves, preprint at arXiv:math.AG/0410611. Dimca, A.: Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992. Fujita, T.: On the topology of non-complete algeraic surfaces, J. Fac. Sci. Univ. Tokyo (Ser1A), 29 (1982), 503-566. Kashiwara, H.: Fonctions rationelles de type (0,1) sur le plan projectif complexe, Osaka J. Math., 24 (1987), 521-577. Namba, M.: Geometry of projective algebraic curves. Monographs and Textbooks in Pure and Applied Mathematics, 88 Marcel Dekker, Inc., New York, 1984. Orevkov, S. Yu.: On rational cuspidal curves, I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657-673. Matsuoka, T. and Sakai, F.: The degree of of rational cuspidal curves, Math. Ann., 285 (1989), 233-247. Tono, K.: On rational unicuspidal plane curves with logarithmic Kodaira dimension one, preprint. Tsunoda, Sh.: The complements of projective plane curves, RIMS-Kôkyûroku, 446 (1981), 48-56. Vajda, S.: Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press New York, 1989. Varchenko, A.N.: On the change of discrete characteristics of critical points of functions under deformations, Uspekhi Mat. Nauk, 38:5 (1985), 126-127. Varchenko, A. N. : Asymptotics of integrals and Hodge structures.Science rewievs:current problems in mathematics 1983. 22, 130-166; J. Sov. Math. 27 (1984). Wall, C.T.C.: Singular Points of Plane Curves, London Math. Soc. Student Texts 63, Cambridge University Press, 2004. Yoshihara, Y.: Rational curves with one cusp (in Japanese), Sugaku, 40 (1988), 269–271.
Deposited On:	02 Oct 2012 08:21
Last Modified:	02 Oct 2012 08:21

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