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

TL;DR

This paper investigates real anti-bicanonical curves with a single double point on the 4th real Hirzebruch surface, providing a criterion to determine the degeneracy of the double point, thereby advancing understanding of real K3 surfaces.

Contribution

It establishes a criterion for the degeneracy of real double points on anti-bicanonical curves on the 4th Hirzebruch surface, completing a previous partial proof.

Findings

Criterion for degeneracy of real double points

Completion of proof for inverse direction of a key proposition

Enhanced understanding of real anti-bicanonical curves on Hirzebruch surfaces

Abstract

A real 2-elementary K3 surfaces of type ((3,1,1),- id) yields a real anti-bicanonical curve s \cup A^\prime_1 (disjoint union) on the 4-th real Hirzebruch surface F_4 where s is the exceptional section of F_4 and the real curve A^\prime_1 has one real double point. We give a criterion (see Proposition 2.4) which determines whether the real double point is degenerate or not. One direction of the assertion of this proposition has already been proved in Lemma 4.6 of the preceding paper [9] (2015) by the author. In this paper we prove the inverse direction.