Abstract:
This is the paper as published. The topology of a complex plane curve singularity with real branches is deduced from any real deformation having delta crossings. An example of the computation of the global geometric monodromy of a polynomial mapping is added.

Abstract:
Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.

Abstract:
For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.

Abstract:
We introduce a new method for the construction of smoothings of a real plane branch $(C, 0)$ by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of $(C,0)$. We characterize the $M$-smoothings obtained by this method by the local data. In particular, we analyze the class of multi-Harnack smoothings, those smoothings arising in a sequence $M$-smoothings of the strict transforms of (C,0) which are in maximal position with respect to the coordinate lines. We prove that there is a unique the topological type of multi-Harnack smoothings, which is determined by the complex equisingularity type of the branch. This result is a local version of a recent Theorem of Mikhalkin.

Abstract:
We classify simple parametrisations of complex curve singularities. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal deformation of the parametrisation. This leads to a smaller list than that obtained by looking at the neighbours in a fixed space of multi-germs. Our simple parametrisations are the same as the fully simple singularities of Zhitomirskii, who classified real plane and space curve singularities. The list of simple parametrisations of plane curves is the A-D-E list. Also for space curves the list coincides with the lists of simple curves of Giusti and Fr\"uhbis-Kr\"uger, in the sense of deformations of the curve. For higher embedding dimension no classification of simple curves is available, but we conjecture that even there the list is exactly that of curves with simple parametrisations.

Abstract:
We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. This improves a result by M. Avendano. Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant.

Abstract:
For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced by complex conjugation and another involution. This topological property holds for all isolated complex plane curve singularities. Using real morsifications, we compute the action of complex conjugation and of the other involution on the Milnor fiber of real plane curve singularities. These involutions have nice descriptions in terms of divides for the singularity.

Abstract:
We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

Abstract:
Let C be a smooth curve in P^3 . Trisecant lines to the curve C are, in general, sweeping out a (reduced) surface in P^3 . In this note we attempt to describe some of its singularities, and in particular we show that if the curve C has only a finite number of quadrisecant lines, then the singular locus contains the quadrisecant lines to C, and the points through which pass more trisecants than through a generic point of the trisecant surface. Several explicit examples are discussed in the last section.

Abstract:
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.