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 $γ:(\mathbb{C},0) \rightarrow (\mathbb{C}^{n},0)$. We define the contact invariants $\operatorname{Iso}_{f}$ and $\operatorname{Iso}_γ$ for $f$ and $γ$, 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 $f$