Abstracts of talks
Christophe Eyral
Blow-ADE singularities and μ*-constant deformations
Abstract. In this talk, we will introduce a class of complex surface singularities — the blow-ADE singularities — which are likely to be stable with respect to \(\mu^*\)-constant deformations. We will prove such a stability property in several special cases. Here, we emphasize that we are not just considering deformation families for small values of the deformation parameter but families connecting any two elements in the \(\mu^*\)-constant stratum. This is a joint work with Mutsuo Oka.
Evelia R. García Barroso
Behaviour of the topology of the generic polar curve with respect to the Zariski invariant for branches of genus one
Abstract.
We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023, since we filter the topological type fixed for the branch by the possible values of Zariski invariants.
This is a joint work with M.E. Hernandes and M.F. Hernández Iglesias.
Janusz Gwoździewicz
Newton diagram of the discriminant
Abstract. With every local analytic mapping \((f,g):(\mathbb C^2,0)\to (\mathbb C^2,0)\) one can associate two curves. These are: the jacobian curve which zero locus is the set of critical points, and the discriminant curve which zero locus is the set of critical values. During my talk I will concentrate on the discriminant curve. I will show that the Newton diagram of the discriminant depends only on the equisingularity type of a pair \((f,g)\). Then I will present a similar statement about the initial Newton polynomial of the discriminant.
- J. Gwoździewicz, Invariance of the Jacobian Newton diagram, Mathematical Research Letters {2012), 19, 377--382;
- B. Gryszka, J. Gwoździewicz, and A. Parusińki, Initial Newton polynomial of the discriminant, Bulletin of the London Mathematical Society (2022), 54.2 1584--1594.
Zbigniew Jelonek
Manifold with infinitely many fibrations over the sphere
Abstract. We show that the manifold \(X=S^2\times S^3\) has infinitely many structures of a fiber bundle over the base \(B = S^2\). In fact for every lens space \(L(p,1)\) there is a fiber bundle \(L(p,1)\to X\to B\).
Dmitry Kerner
Approximation results of Artin-Płoski type for left-right equivalence of map-germs
Abstract. Consider an implicit function equation \(F(x,y) = 0\). Artin approximation ensures: any formal solution is approximated by analytic/Nash solutions. A. Płoski has strengthened this: any formal solution is a (formal) specialization from an analytic family of solutions. (Thus the fomal solutions are not "intrinsic".)
These approximation results are quite useful when studying singularities of zero sets/schemes. In the study of map-germs (up to left-right equivalence) the relevant functional equations are not of implicit-function type. I extend the approximation results of Artin-Płoski to this setup.
An offshoot (of separate importance) is the class of maps of "weakly finite" singularity type. One allows non-isolated singularities that are not too pathological. To study this one uses the (higher) critical loci for maps with singular target, \(f: X \to Y\). This version of the critical locus (seemingly not well known) contains much more information about f than theclassical singular locus.
Tadeusz Krasiński
Scientific achievements of Arkadiusz Płoski
Abstract. The lecture is a memorial tribute to the late Professor Arkadiusz Płoski. I will present selected his main mathematical results.
Wojciech Kucharz
Approximation of maps between real algebraic varieties
Abstract. A nonsingular real algebraic variety \(Y\) is said to have the approximation property if for every nonsingular real algebraic variety \(X\) the following holds: if \(f: X\to Y\) is a \(\mathcal{C}^\infty\) map that is homotopic to a regular map, then \(f\) can be approximated in the (weak) \(\mathcal{C}^\infty\) topology by regular maps. In my talk, I will characterize the varieties \(Y\) with the approximation property. I will also characterize the varieties \(Y\) with the approximation property combined with a suitable interpolation condition. Some of these results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties. This talk is based on a joint paper with Juliusz Banecki.
Andrzej Lenarcik
Arkadiusz Płoski - the scientist and the men
Abstract. Professor Arkadiusz Płoski died February 8, 2024. During the lecture, some facts of his biography will be presented. Since Tadeusz Krasiński is preparing a separate presentation regarding the scientific achievements of Arkadiusz Płoski, during this lecture greater emphasis will be placed on the Professor’s family, travels, social and organizational activities and personal values which were important to him.
Andrzej Lenarcik
On the jacobian Newton polygon, branches and dandelions
Abstract. Let \(f\in\mathbb{C}\{X,Y\}\) be a convergent series and let \(f=0\) be a germ of an isolated plane curve singularity. We call this germ a dandelion if all the black vertices of the Eggers tree of \(f\) form one chain and all the branches go through these vertices. We consider the jacobian Newton polygon \(νJ(f)\) introduced by Bernard Teissier. We say that the jacobian Newton polygon determines the equisingularity class of \(f=0\) if for any germ \(g=0\) the equality \(νJ(f) = νJ(g)\) implies the equisingularity of the germs \(f = 0\) and \(g = 0\). Evelia García Barroso and Janusz Gwoździewicz proved that the jacobian Newton polygon determines the equisingularity class for branches. In this way they obtained a new criterion of irreducibility. We present an analogous result for dandelions.
Stanisław Spodzieja
On some effective version of Bertini's theorem
Abstract. The clasical Bertini theorem on generic intersection of an algebraic set with hyperplanes
says that (see Theorem 8.18 in the book by Robin Hartshorne, Algebraic Geometry,
Springer Science+Business Media, Inc. 1977):
Theorem (Bertini's Theorem). Let \(X\) be a nonsingular closed subvariety of \(\mathbb{P}^n_k\), where \(k\) is an algebraically closed field. Then there exists a hyperplane \(H\subset\mathbb{P}^n_k\), not containing \(X\), and
such that the scheme \(H\cap X\) is regular at every point. (In fact, if \(\dim X \ge 2\), then \(H \cap X\) is
connected, hence irreducible, and so \(H\cap X\) is a nonsingular variety.) Furthermore, the set
of hyperplanes with this property forms an open dense subset of the complete linear
system \(|H|\), considered as a projective space.
We will show that one can effectively indicate a finite family of hyperplanes \(H\), of which at
least one satisfies the assertion of the Bertini theorem.
This is a joint work with Tomasz Rodak and Adam Różycki.