The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$
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| Format: | Preprint |
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2024
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| _version_ | 1866910054900301824 |
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| author | Finashin, Sergey Kharlamov, Viatcheslav |
| author_facet | Finashin, Sergey Kharlamov, Viatcheslav |
| contents | 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 $H_1(X_{\mathbb R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$ Finashin, Sergey Kharlamov, Viatcheslav Algebraic Geometry Geometric Topology 14P25, 14J27, 14J26 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 $H_1(X_{\mathbb R})$. |
| title | The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$ |
| topic | Algebraic Geometry Geometric Topology 14P25, 14J27, 14J26 |
| url | https://arxiv.org/abs/2409.01202 |