Graphs of trigonal curves and rigid isotopies of singular real algebraic curves of bidegree $(4,3)$ on a hyperboloid

TL;DR

This paper completes the classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid by analyzing graphs of real trigonal curves on Hirzebruch surfaces, providing missing proofs for component uniqueness.

Contribution

It finalizes the rigid isotopic classification of these curves, including proofs of component uniqueness for 16 classes with singularities, using graph-based methods on Hirzebruch surfaces.

Findings

Completed classification of real algebraic curves of bidegree (4,3) on hyperboloids.

Proved uniqueness of connected components for 16 classes with singularities.

Utilized graphs of real trigonal curves on Hirzebruch surfaces as main technical tools.

Abstract

A rigid isotopy of real algebraic curves of a certain class is a path in the space of curves of this class. The paper's study completes the rigid isotopic classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, begun by the author in earlier works. There are given the missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces.