Topological types of real regular Jacobian elliptic surfaces

Abstract

Abstract. We present the topological classification of real parts of real regular elliptic surfaces with a real section.

Real rational surfaces

“…Besides the progresses in the theory of real rational surfaces, the classification of other real algebraic surfaces has considerably advanced during the last decade (see [Kha06] for a survey): topological types and deformation types of real Enriques surfaces [DIK00], deformation types of geometrically (2) rational surfaces [DK02], deformation types of real ruled surfaces [Wel03], topological types and deformation types of real bielliptic surfaces [CF03], topological types and deformation types of real elliptic surfaces [AM08,BM07,DIK08].…”

Fibre singulière d’un pinceau réel en courbes de genre 2

“…-428 -La classification locale des surfaces elliptiques réelles aété réalisée en 1984 par Robert Silhol [22]. Pour des avancées plus récentesà propos de la classification réelle globale, voir [2], [3], [4] et [7]. C'est dans la direction des travaux de Silhol que s'inscrit le présent article pour le cas du genre 2.…”

Real trigonal curves and real elliptic surfaces of type I

“…e.g. [2], [1], etc., using the computation of the real version of the Tate-Shafarevich group found in [2], one can extend these results, almost literally, to real elliptic surfaces.…”

“…In view of 3.2.4(1), over the interior of each region R of the pull-back Γ ′ ⊂ B there is a canonical way to label the three sheets of C by 1, 2, 3, according to the increasing of the opposite side of the triangle ∆ b over any point b ∈ R. This labelling is obviously preserved by the real structure c : B → B and hence descends to the regions of Γ. The passage through an edge of Γ results in the following transformation:-solid edge: the transposition (23); -bold edge: the transposition (12); -dotted edge: the change of the orientation of ∆.The transpositions above represent a change of the labelling rather than a nontrivial monodromy.…”

“…In view of 3.2.4(1), over the interior of each region R of the pull-back Γ ′ ⊂ B there is a canonical way to label the three sheets of C by 1, 2, 3, according to the increasing of the opposite side of the triangle ∆ b over any point b ∈ R. This labelling is obviously preserved by the real structure c : B → B and hence descends to the regions of Γ. The passage through an edge of Γ results in the following transformation:…”