Complutense University Library

On rational cuspidal plane curves, open surfaces and local singularities

Fernández de Bobadilla de Olarzábal, Javier José and Luengo Velasco, Ignacio and Melle Hernández, Alejandro and Némethi , A. (2007) On rational cuspidal plane curves, open surfaces and local singularities. In Singularity theory. World Scientific Publishing Co. , Singapore, pp. 411-442. ISBN 978-981-270-410-8

[img] PDF
Restricted to Repository staff only until 31 December 2020.

273kB

Official URL: http://www.worldscientific.com/doi/abs/10.1142/9789812707499_0015

View download statistics for this eprint

==>>> Export to other formats

Abstract

Let C be an irreducible projective plane curve in the complex projective space P(2). The classification of such curves, up to the action of the automorphism group PGL(3, C) on P(2), is a very difficult open problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when C is a rational curve. The aim of this article is to present some of the old conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors.

Item Type:Book Section
Additional Information:Conference: Marseille Singularity School and Conference Location: CIRM, Luminy, France Date: Jan 24-Feb 25, 2005
Uncontrolled Keywords:Cuspidal rational plane curves, logarithmic Kodaira dimension, Nagata-Coolidge problem, Flenner-Zaidenberg rigidity conjecture, surface singularities, Q-homology spheres, Seiberg-Witten invariant, graded roots, Heegaard Floer homology, Ozsváth-Szabó invariants.
Subjects:Sciences > Mathematics > Geometry
ID Code:16567
References:

Aluffi, P. and Faber, C.: Plane curves with small linear orbits II, International Journal of Mathematics , 11 (2000), 591–608.

Artal Bartolo, E., Luengo, I. and Melle-Hernández, A.: Superisolated Surface Singularities, to appear in Proceedings of the Conference ”Singularities and Computer Algebra” on Occasion of Gert-Martin Greuel’s 60th Birthday, LMS Lecture Notes.

Coolidge, J.L.: A treatise of algebraic plane curves, Oxford Univ. Press. Oxford, (1928).

Diaz, S. and Harris, J.: Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468.

Eisenbud, D. and Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110, Princeton University Press, (1985).

Fenske, T.: Rational 1- and 2-cuspidal plane curves, Beiträge Algebra Geom. 40 (1999), no. 2, 309–329.

Fenske, T.: Rational cuspidal plane curves of type (d, d − 4) with χ(V hDi) ≤ 0. Manuscripta Math. 98 (1999), no. 4, 511–527.

Flenner, H. and Zaidenberg, M.: Q-acyclic surfaces and their deformations, Contemporary Math. 162 (1994), 143–208.

Flenner, H. and Zaidenberg, M.: On a class of rational cuspidal plane curves. Manuscripta Math. 89 (1996), no. 4, 439–459.

Flenner, H. and Zaidenberg, M.: Rational cuspidal plane curves of type (d, d−3). Math. Nachr.210 (2000), 93–110.

Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A. and Némethi, A.: On rational cuspidal projective plane curves, Proc. London Math. Soc. (3) 92 (2006), no. 1, 99–138

Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A. and Némethi, A.: Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair, to appear in Real and Complex Singularities (Luminy, 2004).

Greuel, G-M. and Lossen, C.: Equianalytic and equisingular families of curves on surfaces, Manuscripta Math. 91 (1996) 323–342.

Gurjar R.V., Kaliman S., Kumar N.M., Miyanishi M., Russell R., Sakai F., Wright D., and Zaidenberg M.: Open problems on open algebraic varieties. arXiv:math.AG/9506006.

Gusein-Zade, S.M., Delgado, F. and Campillo, A.: On the monodromy of a plane curve singularity and the Poincaré series of the ring of functions on the curve, Functional Analysis and its Applications, 33(1) (1999), 56-67.

Harris, J.; Morrison, I.: Moduli of curves, Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.

Iitaka, S.: On logarithmic Kodaira dimension of algebraic varities, Complex Analysis and Algebraic Geometry, Cambridge Univ. Press, Cambridge, (1977), 175–190.

Iitaka,S.:On irreducible plane curves, Saitama Math. J.,1 (1983) 47–63.

Iitaka, S.: Birational geometry of plane curves, Tokyo J. Math. 22 (1999) 289–321.

de Jong, Th.: Equisingular Deformations of Plane Curve and of Sandwiched Singularities, arXiv:math.AG/0011097.

Kashiwara, H.: Fonctions rationelles de type (0,1) sur le plan projectif complexe, Osaka J. Math. 24 (1987) 521–577.

Kumar, N.M. and Murthy, M.P.: Curves with negative self intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1983) 767–777.

Lescop, C.: Global Surgery Formula for the Casson-Walker Invariant, Annals of Math. Studies, vol. 140, Princeton University Press, 1996.

Looijenga, E.: Isolated singular points of complete intersections. London Math. Soc. Lecture Notes Series 77, (1983).

Luengo, I.: The μ-constant stratum is not smooth, Invent. Math., 90 (1) (1987), 139–152.

Luengo, I., Melle-Hernández, A. and Némethi, A.: Links and analytic invariants of superisolated singularities, Journal of Algebraic Geometry, 14 (2005), 543-565.

Matsuoka, T. and Sakai, F.: The degree of rational cuspidal curves, Math. Ann., 285 (1989), 233–247.

Mattei, J.-F.: Modules de feuilletages holomorphes singuliers. I. quisingularit, Invent. Math. 103 (1991), no. 2, 297–325.

Miyanishi, M. and Sugie, T.: On a projective plane curve whose complement has logarithmic Kodaira dimension −∞, Osaka J. Math. , 18 (1981), 1–11.

Nagata, M.: On rational surfaces I. Irreducible curves of arithmetic genus 0 and 1, Memoirs of College of Science, Univ. of Kyoto, Series A, Vol. XXXII (3) (1960), 351-370.

Némethi, A.: On the spectrum of curve singularities, Proceedings of the Singularity Conference, Oberwolfach, July 1996; Progress in Mathematics, Vol. 162, 93-102, Birkhäuser (1998).

Némethi, A.: On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geometry and Topology 9 (2005), 991-1042.

Némethi,A.:On the Heegaard Floer homology of S3−d(K) and unicuspidal rational plane curves,Geometry and topology of manifolds,Editors:H.U.Boden,I.Hambleton,A.J.Nicas and B.D. Park,Fields Inst.Commun.,47,Amer.Math. Soc.,Providence, RI, 2005, 219–234.

Némethi, A.: On the Heegaard Floer homology of S3 −p/q(K), submitted (math.GT/0410570).

Némethi, A.: Line bundles associated with normal surface singularities, submitted (math.AG/0310084). Rational cuspidal curves, open surfaces and singularities 21

Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities, Geometry and Topology, Volume 6 (2002), 269-328.

Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities II (singularities with good C-action), J. London Math. Soc. (2) 69 (2004), no. 3, 593–607.

Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities splicings and cyclic covers, Selecta Math. New Series 11 Nr. 3-4, (2005), 399–451.

Neumann, W. and Wahl, J.: Casson invariant of links of singularities, Comment. Math. Helv. 65, 58-78, (1991).

Nicolaescu, L.I.: Seiberg-Witten invariants of rational homology 3-spheres, Comm. in Contemp. Math., 6 (2004), 833-866.

Orevkov, S.Yu.: On rational cuspidal curves, I. Sharp estimate for degree via multiplicity, Math. Ann. 324 (2002), 657-673.

Orevkov, S.Yu. and Zaidenberg, M.G.: On the number of singular points of plane curves, In: Algebraic Geometry. Proc.Conf.,Saintama Univ.,March 15–17,1995,alg-geom/9507005.

Ozsváth, P.S. and Szabó, Z.: Holomorphic disks and topological invariants for closed threemanifolds. Ann. of Math. (2) 159 (2004), no. 3, 1027–1158.

Ozsváth, P.S. and Szabó, Z.: On the Floer homology of plumbed three-manifolds, Geom. Topol.7 (2003), 185-224.

Sakai, F.: Kodaira dimensions of complements of divisors, Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 239–257.

Shustin, E. I.: On manifolds of singular algebraic curves, Selecta Math. Soviet. 10 (1991), no.1, 27–37.

Stevens, J.: Universal abelian covers of superisolated singularities, math.AG/0601669.

Teissier, B.: Résolution simultanée—I, II, Séminaire sur les Singularités des Surfaces. Edited by Michel Demazure, Henry Charles Pinkham and Bernard Teissier. Lecture Notes in Mathematics, 777. Springer, Berlin, (1980). 71–146

Teissier, B. and Zariski, O.: Le problème des modules pour les branches planes, (Hermann, Paris, 1986, Appendice).

Tono, K.: Defining equations of certain rational cuspidal plane curves, Manuscripta Math. 103 (2000) 47–62.

Tono, K.: Defining equations of certain rational cuspidal plane curves, doctoral thesis, Saitama University, 2000.

Tono, K.: On rational unicuspidal plane curves with ¯κ = 1, RIMS-Kˆokyˆuroku 1233 (2001)82–89.

Tono, K.: On the number of cusps of cuspidal plane curves, Math. Nachr. 278 (2005) 216–221.

Tsunoda, Sh.: The complements of projective plane curves, RIMS-Kôkyûroku 446 (1981) 48–56.

Tsunoda, Sh.: The Structure of Open Algebraic Surfaces and Its Application to Plane Curves, Proc. Japan Acad. Ser. A 57 (1981) 230–232.

Turaev, V.G.: Torsion invariants of Spinc-structures on 3-manifolds, Math. Res. Letters, 4 (1997), 679-695.

Varchenko, A.N.: On the change of discrete characteristics of critical points of functions under deformations, Uspekhi Mat. Nauk, 38:5 (1983), 126-127.

Varchenko, A.N.: Asymtotics of integrals and Hodge structures. Science rewievs: current problems in mathematics 1983, 22, 130-166; J. Sov. Math. (1984),Vol. 27.

Wakabayashi, I.: On the Logarithmic Kodaira Dimension of the Complement of a Curve in P2, Proc. Japan Acad., 54, Ser. A, (1978), 167162.

Wall, C. T.C.:Notes on the classification of singularities, Proc. London Math. Soc. 48(3) (1984), no. 3, 461–513.

Wall, C. T. C.: Singular points of plane curves, London Mathematical Society Student Texts, 63 Cambridge University Press, Cambridge, 2004.

Wahl, J. M.: Equisingular deformations of plane algebroid curves, Trans. Amer. Math. Soc. 193 (1974), 143–170.

Yoshihara, H: A problem concerning rational plane curves. (Japanese) Sûgaku 31 (1979), no. 3, 256–261.

Zaidenberg, M. G. and Lin, V. Yu: An irreducible simply connected algebraic curve in C2 is equivalent to a quasihomogeneous curve, Soviet Math. Dokl. 28 (1983) 200–204.

Zaidenberg, M. G.; Orevkov, S. Yu. On rigid rational cuspidal plane curves, Russian Math. Surveys 51 (1996), no. 1, 179–180

Zaidenberg, M. G.: Selected problems, arkiv:AG/0501457.

Deposited On:01 Oct 2012 09:55
Last Modified:01 Oct 2012 09:55

Repository Staff Only: item control page