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

TL;DR

This work finalizes the rigid isotopy classification of real nonsingular curves of bidegree $$(4,3)$$ on a hyperboloid by translating the problem into the study of real trigonal curves on the Hirzebruch surface $\Sigma_3$ and employing a robust graph-theoretic framework (graphs, blocks, and skeletons). It fills a gap by proving the uniqueness of connected components for 16 singular classes and provides a constructive description of maximally inflected trigonal curves, together with an extension to nodal-cuspidal cases via Nagata transformations. The methodology yields a concrete, combinatorial route to rigid isotopy through deformation- and weak-equivalence classes of graphs, bridging complex schemes and real combinatorics. The results offer a scalable toolkit for similar classifications on ruled surfaces and potentially for other bidegrees.

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.

Figures (17)

• Figure 1: Elementary moves of graphs
• Figure 2: Straightening/Creating a Zigzag
• Figure 3: Removing/creating adjacent jump and zigzag
• Figure 4: Removing/creating a pair of adjacent zigzags
• Figure 5: Passage of a nodal point through a cusp

Theorems & Definitions (28)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Definition 3
• Remark 1
• Definition 4