Real algebraic curves of bidegree (5,5) on the quadric ellipsoid

TL;DR

This paper completes the topological classification of real algebraic curves of bidegree (5,5) on the quadric ellipsoid, confirming known restrictions and constructing new examples using advanced methods.

Contribution

It provides a complete classification for these curves and introduces a novel approach by degenerating the ellipsoid to the cone and applying classical construction techniques.

Findings

Restrictions form a complete system for bidegree (5,5)

Construction of curves achieved via degeneration and classical methods

Topological classification finalized for these curves

Abstract

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our strategy is first to reduce to the construction of curves in the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, we combine different classical construction methods on toric surfaces, such as Dessin d'enfants and and Viro's patchworking method.