A formula relating inflections, bitangencies and the Milnor number of a plane curve

Authors:
Fabio Scalco Dias, Raúl Oset Sinha and Maria Aparecida Soares Ruas

Journal:
Proc. Amer. Math. Soc. **142** (2014), 2353-2368

MSC (2010):
Primary 14H20; Secondary 53A55, 58K60

Published electronically:
March 31, 2014

MathSciNet review:
3195759

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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****[2]**J. W. Bruce and T. J. Gaffney,*Simple singularities of mappings 𝐶,0→𝐶²,0*, J. London Math. Soc. (2)**26**(1982), no. 3, 465–474. MR**684560**, 10.1112/jlms/s2-26.3.465**[3]**David Cox, John Little, and Donal O’Shea,*Using algebraic geometry*, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR**1639811****[4]**Fabio Scalco Dias and Luis Fernando Mello,*Geometry of plane curves*, Bull. Sci. Math.**135**(2011), no. 4, 333–344. MR**2799811**, 10.1016/j.bulsci.2011.03.007**[5]**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**, 10.1093/qmath/ham039**[6]**Freddy Dumortier, Jaume Llibre, and Joan C. Artés,*Qualitative theory of planar differential systems*, Universitext, Springer-Verlag, Berlin, 2006. MR**2256001****[7]**Fr. Fabricius-Bjerre,*On the double tangents of plane closed curves*, Math. Scand**11**(1962), 113–116. MR**0161231****[8]**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**0444673****[9]**Emmanuel Ferrand,*On the Bennequin invariant and the geometry of wave fronts*, Geom. Dedicata**65**(1997), no. 2, 219–245. MR**1451976**, 10.1023/A:1004936711196**[10]**Benjamin Halpern,*Global theorems for closed plane curves*, Bull. Amer. Math. Soc.**76**(1970), 96–100. MR**0262936**, 10.1090/S0002-9904-1970-12380-1**[11]**Morris W. Hirsch,*Differential topology*, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR**0448362****[12]**John W. Milnor,*Topology from the differentiable viewpoint*, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965. MR**0226651****[13]**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****[14]**David Mond,*Looking at bent wires—𝒜ₑ-codimension and the vanishing topology of parametrized curve singularities*, Math. Proc. Cambridge Philos. Soc.**117**(1995), no. 2, 213–222. MR**1307076**, 10.1017/S0305004100073060- [15]
Raúl
Oset Sinha and Farid
Tari,
*Projections of space curves and duality*, Q. J. Math.**64**(2013), no. 1, 281–302. MR**3032100**, 10.1093/qmath/har035 **[16]**C. T. C. Wall,*Singular points of plane curves*, London Mathematical Society Student Texts, vol. 63, Cambridge University Press, Cambridge, 2004. MR**2107253****[17]**Joel L. Weiner,*A spherical Fabricius-Bjerre formula with applications to closed space curves*, Math. Scand.**61**(1987), no. 2, 286–291. MR**947479**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
14H20,
53A55,
58K60

Retrieve articles in all journals with MSC (2010): 14H20, 53A55, 58K60

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

Email:
maasruas@icmc.usp.br

DOI:
http://dx.doi.org/10.1090/S0002-9939-2014-11980-0

Keywords:
Milnor number,
plane curves,
double point,
inflection,
bitangency.

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

Article copyright:
© Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.