On the hyperbolicity locus of a real curve

TL;DR

This paper explores the hyperbolicity locus of a real algebraic curve in projective 3-space, providing an example where the locus is disconnected without linking numbers distinguishing its components.

Contribution

It presents the first example of a smooth irreducible curve with a disconnected hyperbolicity locus not characterized by linking numbers.

Findings

Hyperbolicity locus can be disconnected.

Linking numbers do not always distinguish hyperbolicity components.

Provides a counterexample to previous assumptions.

Abstract

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected but the connected components are not distinguished by the linking numbers with the connected components of the curve.