On geometric invariants of singular plane curves

TL;DR

This paper introduces two geometric invariants for singular plane curves that measure inflections and vertices at the singularity, showing their finiteness, boundedness, and explicit values for generic cases, with computations for classical singularities.

Contribution

It defines and analyzes new geometric invariants for singular plane curves, establishing their finiteness, boundedness, and explicit values for generic and classical singularities.

Findings

Invariants are finite for generic representations of the curve.

Invariants are bounded and explicitly computable when the curve has no smooth components.

Relationships are established between invariants, Milnor number, and contact with osculating circle.

Abstract

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 show that for almost all representations of $f$, in the sense that their complement is of infinite codimension, these invariants are finite. Indeed when the curve has no smooth components they are always finite and bounded and we can be much more explicit about the values they can attain; the set of possible values is of course an analytic invariant of $f$. We illustrate our results by computing these invariants for Arnold's $\mathcal K$-simple singularities as well as singularities that have ${\mathcal A}$-simple parametrisations. We also obtain a relationship between these invariants, the Milnor number of $f$ and the contact of the curve germ with its osculating circle.