We undertake a study of topological properties of the real Mordell-Weil group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {MW}_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>MW</mo> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Y_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Y</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation> and an explicit presentation of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\operatorname {MW}_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>MW</mo> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation> in the mapping class group <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\operatorname {Mod}(X_\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Mod</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi mathvariant="double-struck">R</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Combining these results we establish an explicit formula for the action of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\operatorname {MW}_\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>MW</mo> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H_1(X_\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi mathvariant="double-struck">R</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

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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2025年12月19日

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S. Finashin,
V. Kharlamov

摘要

We undertake a study of topological properties of the real Mordell-Weil group $\operatorname {MW}_\mathbb {R}$ ${MW}_{R}$ 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 $Y_\mathbb {R}$ $Y_{R}$ and an explicit presentation of $\operatorname {MW}_\mathbb {R}$ ${MW}_{R}$ in the mapping class group $\operatorname {Mod}(X_\mathbb {R})$ $Mod (X_{R})$ . Combining these results we establish an explicit formula for the action of $\operatorname {MW}_\mathbb {R}$ ${MW}_{R}$ in $H_1(X_\mathbb {R})$ $H_{1} (X_{R})$ .

The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree \({K^{2}}=1\)

摘要