The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$

TL;DR

This paper explores the topological structure of the real Mordell-Weil group of rational elliptic surfaces and the configuration of real lines on these surfaces and related del Pezzo surfaces, providing explicit descriptions and formulas.

Contribution

It offers a detailed description of the isotopy types of real lines on del Pezzo surfaces and an explicit presentation of the real Mordell-Weil group within the mapping class group.

Findings

Explicit description of isotopy types of real lines on del Pezzo surfaces.

Presentation of the real Mordell-Weil group in the mapping class group.

Formula for the action of the Mordell-Weil group on the first homology.

Abstract

We undertake a study of topological properties of the real Mordell-Weil group $\operatorname{MW}_{\mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the "subordinate" del Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy types of real lines on $Y_{\mathbb R}$ and an explicit presentation of $\operatorname{MW}_{\mathbb R}$ in the mapping class group $\operatorname{Mod}(X_{\mathbb R})$. Combining these results we establish an explicit formula for the action of $\operatorname{MW}_{\mathbb R}$ in $H_1(X_{\mathbb R})$.