A formula relating inflections, bitangencies and the Milnor number of a plane curve
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- by Fabio Scalco Dias, Raúl Oset Sinha and Maria Aparecida Soares Ruas PDF
- Proc. Amer. Math. Soc. 142 (2014), 2353-2368 Request permission
Abstract:
In this article we obtain a formula relating inflections, bitangencies and the Milnor number of a plane curve germ. Moreover, we present an extension of the formula obtained by the first author and Luis Fernando Mello for a class of plane curves with singularities.References
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR 777682, DOI 10.1007/978-1-4612-5154-5
- J. W. Bruce and T. J. Gaffney, Simple singularities of mappings $\textbf {C},0\rightarrow \textbf {C}^{2},0$, J. London Math. Soc. (2) 26 (1982), no. 3, 465–474. MR 684560, DOI 10.1112/jlms/s2-26.3.465
- David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
- Fabio Scalco Dias and Luis Fernando Mello, Geometry of plane curves, Bull. Sci. Math. 135 (2011), no. 4, 333–344. MR 2799811, DOI 10.1016/j.bulsci.2011.03.007
- F. S. Dias and J. J. Nuño-Ballesteros, Plane curve diagrams and geometrical applications, Q. J. Math. 59 (2008), no. 3, 287–310. MR 2444062, DOI 10.1093/qmath/ham039
- Freddy Dumortier, Jaume Llibre, and Joan C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006. MR 2256001
- Fr. Fabricius-Bjerre, On the double tangents of plane closed curves, Math. Scand. 11 (1962), 113–116. MR 161231, DOI 10.7146/math.scand.a-10656
- Fr. Fabricius-Bjerre, A relation between the numbers of singular points and singular lines of a plane closed curve, Math. Scand. 40 (1977), no. 1, 20–24. MR 444673, DOI 10.7146/math.scand.a-11672
- Emmanuel Ferrand, On the Bennequin invariant and the geometry of wave fronts, Geom. Dedicata 65 (1997), no. 2, 219–245. MR 1451976, DOI 10.1023/A:1004936711196
- Benjamin Halpern, Global theorems for closed plane curves, Bull. Amer. Math. Soc. 76 (1970), 96–100. MR 262936, DOI 10.1090/S0002-9904-1970-12380-1
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
- John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- David Mond, Looking at bent wires—${\scr A}_e$-codimension and the vanishing topology of parametrized curve singularities, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 2, 213–222. MR 1307076, DOI 10.1017/S0305004100073060
- Raúl Oset Sinha and Farid Tari, Projections of space curves and duality, Q. J. Math. 64 (2013), no. 1, 281–302. MR 3032100, DOI 10.1093/qmath/har035
- C. T. C. Wall, Singular points of plane curves, London Mathematical Society Student Texts, vol. 63, Cambridge University Press, Cambridge, 2004. MR 2107253, DOI 10.1017/CBO9780511617560
- Joel L. Weiner, A spherical Fabricius-Bjerre formula with applications to closed space curves, Math. Scand. 61 (1987), no. 2, 286–291. MR 947479, DOI 10.7146/math.scand.a-12205
Additional Information
- Fabio Scalco Dias
- Affiliation: Instituto de Ciências Exatas, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil
- Email: scalco@unifei.edu.br
- Raúl Oset Sinha
- Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, CEP 13.560–970, São Carlos-SP, Brazil
- Email: raul.oset@uv.es
- Maria Aparecida Soares Ruas
- Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, CEP 13.560–970, São Carlos-SP, Brazil
- MR Author ID: 239264
- ORCID: 0000-0001-8890-524X
- Email: maasruas@icmc.usp.br
- Received by editor(s): November 11, 2011
- Received by editor(s) in revised form: August 5, 2012
- Published electronically: March 31, 2014
- Additional Notes: The first author was supported by FAPESP grant No. 2011/01946-0.
The second author was partially supported by FAPESP grant No. 2010/01501-5 and DGCYT and FEDER grant No. MTM2009-08933
The third author was supported by FAPESP, grant No. 08/54222-6 and CNPq, grant No. 303774/2008-8. - Communicated by: Lev Borisov
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2353-2368
- MSC (2010): Primary 14H20; Secondary 53A55, 58K60
- DOI: https://doi.org/10.1090/S0002-9939-2014-11980-0
- MathSciNet review: 3195759