Special geometry of quartic curves

TL;DR

This paper classifies maximal quartic generalised projective special real curves, explores their boundary behaviour, and describes their asymptotic properties, contributing to the understanding of their geometric structure.

Contribution

It provides a classification of these curves up to equivalence and analyzes the boundary behaviour of associated homogeneous spaces.

Findings

Quartic generalised projective special real manifolds have non-regular boundary behaviour.

The asymptotic behaviour of each classified curve is explicitly described.

Homogeneous spaces of these manifolds exhibit boundary degeneracies.

Abstract

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic homogeneous real polynomial $h:\mathbb{R}^2\to\mathbb{R}$ and its level set $\{h=1\}$. Two such curves are called equivalent if they are related by a linear coordinate transformation. As an application of our results we prove that quartic generalised projective special real manifolds that are homogeneous spaces have non-regular boundary behaviour, meaning that the differential of each of these spaces' defining polynomials vanishes identically on a ray in the boundary of the cone spanned by the corresponding manifold. Lastly we describe the asymptotic behaviour of each curve.