The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree K²=1

2024/09/02 by Finashin, Sergey, Kharlamov, Viatcheslav
#14J26 #14J27 #14P25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2409.01202

Abstract

We undertake a study of topological properties of the real Mordell-Weil group 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 MW_{\mathbb R} in the mapping class group Mod(X_{\mathbb R}). Combining these results we establish an explicit formula for the action of MW_{\mathbb R} in H₁(X_{\mathbb R}).

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