Abstract
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on the intersection theory combined with liftings and degenerations. The procedure computes the homology class in question whenever a given singularity type is defined. Our method does not require knowledge of all the possible deformations of a given singularity, as it was in previously known procedures.
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[AR] M. Alberich-Carraminana and J. Roe,Enriques diagrams and adjacency of planar curve singularities, Canadian Journal of Mathematics57 (2005), 3–16. Preprint arXiv:math.AG/0108223.
[AGV] V. I. Arnol'd, S. M. Guseîn-Zade and A. N. Varchenko,Singularities of Differentiable Maps, Vol. I.The Classification of Critical Points, Caustics and Wave Fronts, Monographs in Mathematics, 82, Birkhäuser Boston, Boston, MA, 1985.
[Al] P. Aluffi,Characteristic classes of discriminants and enumerative geometry, Communications in Algebra26 (1998), 3165–3193.
[CH1] L. Caporaso and J. Harris,Enumerating rational curves: the rational fibration method, Composition Mathematica113 (1998), 209–236.
[CH2] L. Caporaso and J. Harris,Counting plane curves of any genus, Inventiones Mathematicae131 (1998), 345–392.
[Fr,Itz] P. di Francesco and C. Itzykson,Quantum intersection rings, inThe Moduli Space of Curves (R. Dijkgraaf, C. Faber and G. van der Geer, eds.), Progress in Mathematics, Vol. 129, Birkhäuser, Boston, 1995, pp. 81–148.
[Ful] W. Fulton,Intersection Theory, Second edition, A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998.
[G] G.-M. Greuel,Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscripta Mathematica56 (1986), 159–166.
[GLS1] G.-M. Greuel, C. Lossen and E. Shustin,Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
[GLS2] G.-M. Greuel, C. Lossen and E. Shustin,The variety of plane curves with ordinary singularities is not irreducible, International Mathematics Research Notices11 (2001), 543–550.
[Kaz1] M. Kazarian,Classifying spaces of singularities and Thom polynomals, inNew Developments in Singularity Theory (Cambridge 2000), NATO Science Series II Mathematical Physics and Chemistry, 21, Kluwer Academic, Dordrecht, 2001, pp. 117–134.
[Kaz2] M. Kazarian,Thom polynomials for Lagrange, Legendre, and critical point function singularities, Proceedings of the London Mathematical Society (3)86 (2003), 707–734.
[Kaz3] M. Kazarian,Multisingularities, cobordisms, and enumerative geometry, Russian Mathematical Surveys58 (2003), 665–724.
[Kaz4] M. Kazarian,Characteristic Classes in Singularity Theory, Doctoral Dissertation (habilitation thesis), Steklov Institute of Mathematics, 2003.
[Kl1] S. L. Kleiman,The enumerative theory of singularities, inReal and Complex Singularities (Proceedings of the Ninth Nordic Summer School/NAVF Symposium in Mathematics, Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 297–396.
[Kl2] S. L. Kleiman,Intersection theory and enumerative geometry: a decade in review, With the collaboration of Anders Thorup on 3. Proceedings of Symposium in Pure Mathematics, 46, Part 2, Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), American Mathematical Society, Providence, RI, 1987, pp. 321–370.
[KlPi1] S. Kleiman and R. Piene,Enumerating singular curves on surfaces, Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemporary Mathematics, Vol. 241, American Mathematical Society, Providence, RI, 1999, pp. 209–238.
[KlPi2] S. Kleiman and R. Piene,Node polynomials for families: methods and applications, Mathematische Nachrichten271 (2004), 69–90.
[Li] A.-K. Liu,The Algebraic Proof of the University Theorem, Preprint arXiv:math.AG/0402045.
[Lu] I. Luengo,The μ-constant stratum is not smooth, Inventiones Mathematicae90 (1987), 139–152.
[PH] F. Pham,Remarque sur l'équisingularité universelle, Prépublication, Université de Nice Faculté des Sciences, 1970.
[R] Z. Ran,Enumerative geometry of singular plane curves, Inventiones Mathematicae97 (1989), 447–465.
[V1] I. Vainsencher,Counting divisors with prescribed singularities, Transactions of the American Mathematical Society267 (1981), 399–422.
[V2] I. Vainsencher,Hypersurfaces with up to six double points, Special issue in honor of Steven L. Kleiman, Communications in Algebra31 (2003), 4107–4129.
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Kerner, D. Enumeration of singular algebraic curves. Isr. J. Math. 155, 1–56 (2006). https://doi.org/10.1007/BF02773947
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DOI: https://doi.org/10.1007/BF02773947