On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface

TL;DR

This paper classifies real anti-bicanonical curves with one double point on the 4th real Hirzebruch surface, linking their isotopy types to degenerations of nonsingular curves through a detailed moduli space analysis.

Contribution

It provides a comprehensive enumeration of isotopy types and degenerations of these curves, revealing a novel correspondence similar to known classifications of sextic curves.

Findings

List of all candidate isotopy types for the curves.

Enumeration of possible degenerations of nonsingular curves.

Discovery of a correspondence between isotopy types and degenerations.

Abstract

We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,\theta)=((3,1,1), -id). We also list up all the candidates for the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF_4. We find an interesting correspondence between the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on RF_4 and the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF_4. This correspondence is very similar to the one provided by the rigid isotopic classification of real sextic curves on RP^2 with one real nondegenerate double point by I. Itenberg.