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

We undertake a study of topological properties of the real Mordell-Weil group ${\mathrm{MW}}_{\mathbb{R}}$ 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 ${\mathrm{MW}}_{\mathbb{R}}$ in the mapping class group $\mathrm{Mod}(X_{\mathbb{R}})$. Combining these results we establish an explicit formula for the action of ${\mathrm{MW}}_{\mathbb{R}}$ in $H_1(X_{\mathbb{R}})$.