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Deformations of singularities of plane curves: topological approach. (English) Zbl 1342.14058

The author studies deformations of singular points of plane curves, i.e. a smooth families of plane algebraic curves on two dimensional complex Euclidean space with distinguished members with a singular point. Using a knot invariant, i.e. the Tristram-Levine signature, he gets a main result which is an estimation of the difference between the \(M\)-number of the singularity of the central fiber and the sum of \(M\)-numbers of the generic fiber of the family.

MSC:

14H20 Singularities of curves, local rings
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:

[1] V.I. Arnol’d, A.N. Varchenko and S.M. Guseĭ n-Zade: Singularities of Differentiable Mappings, II, Nauka, Moscow, 1984, (Russian).
[2] M. Borodzik: Morse theory for plane algebraic curves , J. Topol. 5 (2012), 341-365. · Zbl 1251.57004
[3] M. Borodzik: Abelian \(\rho\)-invariants of iterated torus knots ; in Low-Dimensional and Symplectic Topology, Proc. Sympos. Pure Math. 82 , Amer. Math. Soc., Providence, RI, 2011, 29-38. · Zbl 1268.57003
[4] M. Borodzik: A \(\rho\)-invariant of iterated torus knots , arXiv: arXiv: 0906.3660v3 · Zbl 1268.57003
[5] M. Borodzik and H. Żołądek: Complex Algebraic Plane Curves via Poincaré-Hopf Formula, III, Codimension Bounds, J. Math. Kyoto Univ. 48 (2008), 529-570. · Zbl 1174.14028
[6] C. Christopher and S. Lynch: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces , Nonlinearity 12 (1999), 1099-1112. · Zbl 1074.34522
[7] D. Eisenbud and W. Neumann: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies 110 , Princeton Univ. Press, Princeton, NJ, 1985. · Zbl 0628.57002
[8] \begingroup G.-M. Greuel, C. Lossen and E. Shustin: Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. \endgroup
[9] A. Hirano: Construction of plane curves with cusps , Saitama Math. J. 10 (1992), 21-24. · Zbl 0796.14019
[10] R. Kirby and P. Melvin: Dedekind sums, \(\mu\)-invariants and the signature cocycle , Math. Ann. 299 (1994), 231-267. · Zbl 0809.11027
[11] S.Yu. Orevkov: On rational cuspidal curves . I. Sharp estimate for degree via multiplicities , Math. Ann. 324 (2002), 657-673. · Zbl 1014.14010
[12] S.Yu. Orevkov, M.G. Zaidenberg: On the number of singular points of plane curves ..
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