Session 39. Contributed talks

Remarks on the sequence of jumps of Milnor numbers

Justyna Walewska, Faculty of Mathematics and Computer Science, University of Łódź, Poland
The talk is based on the joint work with Maria Michalska.
Consider a non-degenerated isolated singularity \[ f_0=\sum\limits_{m\alpha+l\beta \geq lm } a_{\alpha\beta}x^{\alpha}y^{\beta} \] such that \(a_{l0}a_{0m}\not= 0\) and \(l, m > 2\). Consider an arbitrary holomorphic deformation \((f_s)_{s\in S}\) of \(f_0\), where \(s\) is a single parameter defined in a neighborhood \(S\) of \(0 \in \mathbb{C}\). By the semi-continuity (in the Zariski topology) of Milnor numbers in families of singularities \(\mu(f_s)\) is constant for sufficiently small \(s \not = 0\) and \(\mu(f_s) \leq \mu(f_0)\). Denote this constant value by \(\mu((f_s))\) and call it generic Milnor number of the deformation \((f_s)\). Let \[ \mathcal{M}(f_0) = (\mu_0(f_0),\mu_1(f_0),\ldots,\mu_k(f_0)) \] be the strictly decreasing sequence of generic Milnor numbers of all possible deformations of \(f_0\). In particular \[ \mu_0(f_0) = \mu(f_0) > \mu_1(f_0) > \ldots > \mu_k(f_0) = 0. \] We find first few terms of the sequence \(\mathcal{M}(f_0)\) in the case of non-degenerate deformations.
References
  1. S. Brzostowski, T. Krasiński, The jump of the Milnor number in the \(X_9\) singularity class , Cent. Eur. J. Math. 12, 2014, 429-435.
  2. S. Brzostowski, T. Krasiński, J. Walewska, Milnor numbers in deformations of homogeneous singularities , arXiv:1404.7704v1, 2014.
  3. M. Michalska, J. Walewska, Remarks on the sequence of jumps of Milnor numbers , Proceedings of the XXXV Conference and Workshop of Analytic and Algebraic Geometry, 2014, 29-34 (in Polish).
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