Polar Invariants of Plane Curve Singularities: Intersection Theoretical Approach

This paper addresses a very classical topic that goes back at least to Pl\"ucker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data ...

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the GSV-index in terms of polar curves. We also interpret the proofs concerning the Poincar\'e problem with polar invariants.

In this paper, we study polar quotients and \L ojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the \L ojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the \L ojasiewicz exponents in the gradient and classical inequalities of pol...

We present an intersection-theoretical approach to the invariants of plane curve singularities $\mu$, $\delta$, $r$ related by the Milnor formula $2\delta=\mu+r-1$. Using Newton transformations we give formulae for $\mu$, $\delta$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities.

Given a germ of a smooth plane curve $(\{f(x,y)=0\},0)\subset (\mathbb K^2,0), \mathbb K=\mathbb R, \mathbb C$, with an isolated singularity, we define two invariants $I_f$ and $V_f\in \mathbb N\cup\{\infty\}$ which count the number of inflections and vertices (suitably interpreted in the complex case) concentrated at the singular point; the first is an affine invariant and the second is invariant under similarities of $\mathbb R^2$, and their analogue for $\mathbb C^2$. We s...

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.

In this survey we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some demanding open problems that can indicate new directions in research, for instance by introducing weak Ziegler pairs of curve arrangements.

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which have intersections with the original curves and surfaces at the singular points. Three algorithms are presented for three cases of plane curves, space curves and surfaces.