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

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.