## Osaka Journal of Mathematics

### Deformations of singularities of plane curves: Topological approach

Maciej Borodzik

#### Abstract

In this paper we use a knot invariant, namely the Tristram--Levine signature, to study deformations of singular points of plane curves. We bound, in some cases, the difference between the $M$-number of the singularity of the central fiber and the sum of $M$-numbers of the generic fiber.

#### Article information

Source
Osaka J. Math., Volume 51, Number 3 (2014), 573-585.

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.ojm/1414090791

Mathematical Reviews number (MathSciNet)
MR3272605

Zentralblatt MATH identifier
1342.14058

#### Citation

Borodzik, Maciej. Deformations of singularities of plane curves: Topological approach. Osaka J. Math. 51 (2014), no. 3, 573--585. https://projecteuclid.org/euclid.ojm/1414090791

#### References

• V.I. Arnol'd, A.N. Varchenko and S.M. Guseĭ n-Zade: Singularities of Differentiable Mappings, II, Nauka, Moscow, 1984, (Russian).
• M. Borodzik: Morse theory for plane algebraic curves, J. Topol. 5 (2012), 341–365.
• 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.
• M. Borodzik: A $\rho$-invariant of iterated torus knots, arxiv:0906.3660v3, an updated version of [3251-03?] with two sign mistakes corrected.
• M. Borodzik and H. Żo\lądek: Complex Algebraic Plane Curves via Poincaré–Hopf Formula, III, Codimension Bounds, J. Math. Kyoto Univ. 48 (2008), 529–570.
• 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.
• 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.
• \begingroup G.-M. Greuel, C. Lossen and E. Shustin: Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. \endgroup
• A. Hirano: Construction of plane curves with cusps, Saitama Math. J. 10 (1992), 21–24.
• R. Kirby and P. Melvin: Dedekind sums, $\mu$-invariants and the signature cocycle, Math. Ann. 299 (1994), 231–267.
• S.Yu. Orevkov: On rational cuspidal curves. I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657–673.
• S.Yu. Orevkov, M.G. Zaidenberg: On the number of singular points of plane curves..