On isotropic points and contact invariants of curves

Abstract

Let $(C_f,0)\subset ({\mathbb{C}}^n,0)$ be a germ of an irreducible holomorphic curve, given implicity by $f(x)=0$ if $n=2$ and by $f(x)=g(x)=0$ if $n=3$, and parametrised by $\gamma :(\mathbb{C},0)\to ({\mathbb{C}}^n,0)$. We define the contact invariants ${\mathrm{Iso}}_f$ and ${\mathrm{Iso}}_{\gamma }$ for $f$ and $\gamma$, respectively, which count the number of isotropic points that appear after a generic deformation of the curve. We first consider the case where the curve is regular and establish a relation between those invariants and the number of vertices, inflection points and the zero-torsion points (for $n=3)$ that emerge from the deformation of the curve. Furthermore, we consider a germ of a singular curve with an isolated singularity and relate the number of isotropic points with the Milnor number of $f$