2026/02/28 by DRAGOS GHIOCA, Dragos Ghioca, NEGIN SHADGAR +1
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.1017/s0004972726101592
Abstract Let C be a smooth projective curve defined over <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> \mathbb Q double struck upper Q overbar , let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> </mml:mrow> <mml:mo stretchy="false">⟶</mml:mo> <mml:mi>C</mml:mi> </mml:math> π :\mathcal E\longrightarrow C pi colon script upper E long right arrow upper C be an elliptic surface and let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> σ P1,σ P2,σ Q sigma Subscript upper P 1 Baseline comma sigma Subscript upper P 2 Baseline comma sigma Subscript upper Q Baseline be sections of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>π</mml:mi> </mml:math> π pi (corresponding to points <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> </mml:math> P1,P2, Q upper P 1 comma upper P 2 comma upper Q of the generic fibre E of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> </mml:mrow> </mml:math> \mathcal E script upper E ). We obtain a precise characterisation, expressed solely in terms of the dynamical relations between the points <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo>