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The analytic classification of plane curves

Part of: Singularities Curves

Published online by Cambridge University Press:  19 March 2024

Marcelo Escudeiro Hernandes Open the ORCID record for Marcelo Escudeiro Hernandes [Opens in a new window]  and
Maria Elenice Rodrigues Hernandes Open the ORCID record for Maria Elenice Rodrigues Hernandes [Opens in a new window]
Marcelo Escudeiro Hernandes
Affiliation:
Universidade Estadual de Maringá, Av. Colombro 5790, Maringá-Pr 87020-900, Brazil mehernandes@uem.br
Maria Elenice Rodrigues Hernandes
Affiliation:
Universidade Estadual de Maringá, Av. Colombro 5790, Maringá-Pr 87020-900, Brazil merhernandes@uem.br
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Abstract

In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allows us to recover some particular cases found in the literature.

Keywords

plane curvesanalytic classificationanalytic invariants

MSC classification

Primary: 14H20: Singularities, local rings
Secondary: 14H10: Families, moduli (algebraic) 32S05: Local singularities 32S10: Invariants of analytic local rings
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 4 , April 2024 , pp. 915 - 944
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The authors were partially supported by grant 2019/07316-0, São Paulo Research Foundation (FAPESP) and the first author was partially supported by CNPq-Brazil Proc. 303638/2020-6.

References

Ayuso, P. F. and Ribón, J., The action of a plane singular holomorphic flow on a non-invariant branch, Canad. J. Math. 72 (2020), 835–866.10.4153/S0008414X19000221CrossRefGoogle Scholar
Bayer, V. S., Guzmán, E. M. N., Hefez, A. and Hernandes, M. E., Tjurina number of a local complete intersection, Preprint (2020), arXiv:2001.11123.Google Scholar
Brieskorn, E. and Knörrer, H., Plane algebraic curves (Birkhäuser, 1986).10.1007/978-3-0348-5097-1CrossRefGoogle Scholar
Bruce, J. W. and Gaffney, T. J., Simple singularities of mappings $\mathbb {C}, 0 \longrightarrow \mathbb {C}^2,0$, J. Lond. Math. Soc. (2) 26 (1982), 465–474.10.1112/jlms/s2-26.3.465CrossRefGoogle Scholar
Bruce, J. W., Kirk, N. P. and du Plessis, A. A., Complete transversals and the classification of singularities, Nonlinearity 10 (1997), 253–275.10.1088/0951-7715/10/1/017CrossRefGoogle Scholar
Câmara, L. M. and Scárdua, B., A comprehensive approach to the moduli space of quasi-homogeneous singularities, in Singularities and foliations. Geometry, topology and applications, Springer Proceedings in Mathematics & Statistics, vol. 222, eds Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M. and Snoussi, J. (Springer, 2018), 459–487.10.1007/978-3-319-73639-6_15CrossRefGoogle Scholar
Carvalho, E. and Hernandes, M. E., The value semiring of an algebroid curve, Comm. Algebra 48 (2020), 3275–3284.Google Scholar
Carvalho, E. and Hernandes, M. E., Standard bases for fractional ideals of the local ring of an algebroid curve, J. Algebra 551 (2020), 342–361.10.1016/j.jalgebra.2020.01.018CrossRefGoogle Scholar
Delorme, C., Sur les modules des singularités des courbes planes, Bull. Soc. Math. France 106 (1978), 417–446.10.24033/bsmf.1880CrossRefGoogle Scholar
Ebey, S., The classification of singular points of algebraic curves, Trans. Amer. Math. Soc. 118 (1965), 454–471.10.1090/S0002-9947-1965-0176983-8CrossRefGoogle Scholar
Garcia, A., Semigroups associated to singular points of plane curves, J. Reine Angew. Math. 336 (1982), 165–184.Google Scholar
Genzmer, Y., Dimension of the moduli space of a germ of curve in $\mathbb {C}^2$, Int. Math. Res. Not. IMRN 5 (2022), 3805–3859.10.1093/imrn/rnaa237CrossRefGoogle Scholar
Genzmer, Y. and Paul, E., Normal forms of foliations and curves defined by a function with a generic tangent cone, Mosc. Math. J. 11 (2011), 41–72.10.17323/1609-4514-2011-11-1-41-72CrossRefGoogle Scholar
Genzmer, Y. and Paul, E., Moduli spaces for topologically quasi-homogeneous functions, J. Singul. 14 (2016), 3–33.Google Scholar
Guzmán, E. M. N. and Hefez, A., On the colength of fractional ideals, J. Singul. 21 (2020), 119–131.Google Scholar
Gorenstein, D., An arithmetic theory of adjoint plane curves, Trans. Amer. Math. Soc. 72 (1952), 414–436.10.1090/S0002-9947-1952-0049591-8CrossRefGoogle Scholar
Granger, J.-M., Sur un espace de modules de germe de courbe plane, Bull. Sci. Math. 103 (1979), 3–16.Google Scholar
Greuel, G.-M. and Pfister, G., Moduli for singularities, in Singularities, London Mathematical Society Lecture Note Series, vol. 201 (Cambridge University Press, 1994), 119–146.10.1017/CBO9780511752520.008CrossRefGoogle Scholar
Hefez, A. and Hernandes, M. E., The analytic classification of plane branches, Bull. Lond. Math. Soc. 43 (2011), 289–298.10.1112/blms/bdq113CrossRefGoogle Scholar
Hefez, A. and Hernandes, M. E., The analytic classification of irreducible plane curve singularities, in Handbook of geometry and topology of singularities II, eds Cisneros-Molina, J. L., Lê, D. T. and Seade, J. (Springer, 2021), 1–65.Google Scholar
Hefez, A., Hernandes, M. E. and Rodrigues Hernandes, M. E., The analytic classification of plane curves with two branches, Math. Z. 279 (2015), 509–520.10.1007/s00209-014-1379-2CrossRefGoogle Scholar
Kang, C., Analytic types of plane curve singularities defined by weighted homogeneous polynomials, Trans. Amer. Math. Soc. 352 (2000), 3995–4006.10.1090/S0002-9947-00-02478-8CrossRefGoogle Scholar
Kolgushkin, P. A. and Sadykov, R. R., Simple singularities of multigerms of curves, Rev. Mat. Complut. 2 (2001), 311–344.Google Scholar
Laudal, O. A. and Pfister, G., Local moduli and singularities, Lecture Notes in Mathematics, vol. 1310 (Springer, 1988).10.1007/BFb0078937CrossRefGoogle Scholar
Mata, F. D., The semigroup of values of a curve singularity with several branches, Manuscripta Math. 59 (1987), 347–374.10.1007/BF01174799CrossRefGoogle Scholar
Pol, D., On the values of logarithmic residues along curves, Ann. Inst. Fourier (Grenoble) 68 (2018), 725–766.10.5802/aif.3176CrossRefGoogle Scholar
Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 27 (1980), 265–291.Google Scholar
Waldi, R., Wertehalbgruppe und Singularitäten einer ebenen algebraischen Kurve, Dissertation, Regensburg (1972).Google Scholar
Wall, C. T. C., Finite determinacy of smooth map-germs, Bull. Lond. Math. Soc. 13 (1981), 481–539.10.1112/blms/13.6.481CrossRefGoogle Scholar
Washburn, S., Book review: Le problème des modules pour les branches planes, by Oscar Zariski with an appendix by Bernard Teissier, Hermann, Paris, 1986, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 209–214.10.1090/S0273-0979-1988-15651-0CrossRefGoogle Scholar
Zariski, O., Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Natl. Acad. Sci. USA 56 (1966), 781–786.10.1073/pnas.56.3.781CrossRefGoogle ScholarPubMed
Zariski, O., General theory of saturation and of saturated local rings II: Saturated local rings of dimension 1, Amer. J. Math. 93 (1971), 872–964.10.2307/2373741CrossRefGoogle Scholar
Zariski, O., Le problème des modules pour les branches planes (Hermann, Paris, 1986) (French); B. Lichtin, The moduli problem for plane branches, University Lecture Series, vol. 39 (American Mathematical Society, Providence, RI, 2006) (English).Google Scholar