On deformation types of real elliptic surfaces. (English) Zbl 1184.14065
The paper under review addresses the question of quasi-simplicity for families of elliptic surfaces.
Recall that a deformation class of complex varieties is quasi-simple if a real variety within the complex class is determined up to equivariant deformation by the diffeomorphism type of the real structure. Considering equivariant deformation of real elliptic fibrations instead of equivariant deformation of real elliptic surfaces, we get a weaker form of the quasi-simplicity. Recall also that an almost generic elliptic fibration is a fibration with simplest singular fibers only.
The authors give an explicit description of the equivariant deformation classes of almost generic elliptic fibrations over a rational base, under the extra hypothesis that the total Betti number of the real part is maximal or sub-maximal.
They prove also the quasi-simplicity of almost generic maximal elliptic fibrations over a base of any genus and of almost generic sub-maximal elliptic fibrations over a rational base.
As in the original scheme of classification used by Kodaira in the complex case, the classification splits into two: the classification of real elliptic fibrations with a real section and the computation of the real Tate-Shafarevich group.
The real Weierstraßmodel of a real elliptic fibration with a real section is a double cover of a ruled surface whose branch locus is the union of a so-called trigonal curve and the exceptional section of the ruling. The functional invariant \(j\) of the elliptic fibration is also the moduli invariant of the trigonal curve. The classification is then reduced to the combinatorics of a real version of Grothendieck’s dessins d’enfants associated to \(j\). The key point is an extent of Orevkov’s decomposition of dessins into ribbons made of a few elementary pieces, [see S. Yu. Orevkov, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 12, No. 4, 517–531 (2003; Zbl 1078.14083)]. The authors produce the explicit list of the 8 non equivalent pieces.
Recall that a deformation class of complex varieties is quasi-simple if a real variety within the complex class is determined up to equivariant deformation by the diffeomorphism type of the real structure. Considering equivariant deformation of real elliptic fibrations instead of equivariant deformation of real elliptic surfaces, we get a weaker form of the quasi-simplicity. Recall also that an almost generic elliptic fibration is a fibration with simplest singular fibers only.
The authors give an explicit description of the equivariant deformation classes of almost generic elliptic fibrations over a rational base, under the extra hypothesis that the total Betti number of the real part is maximal or sub-maximal.
They prove also the quasi-simplicity of almost generic maximal elliptic fibrations over a base of any genus and of almost generic sub-maximal elliptic fibrations over a rational base.
As in the original scheme of classification used by Kodaira in the complex case, the classification splits into two: the classification of real elliptic fibrations with a real section and the computation of the real Tate-Shafarevich group.
The real Weierstraßmodel of a real elliptic fibration with a real section is a double cover of a ruled surface whose branch locus is the union of a so-called trigonal curve and the exceptional section of the ruling. The functional invariant \(j\) of the elliptic fibration is also the moduli invariant of the trigonal curve. The classification is then reduced to the combinatorics of a real version of Grothendieck’s dessins d’enfants associated to \(j\). The key point is an extent of Orevkov’s decomposition of dessins into ribbons made of a few elementary pieces, [see S. Yu. Orevkov, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 12, No. 4, 517–531 (2003; Zbl 1078.14083)]. The authors produce the explicit list of the 8 non equivalent pieces.
Reviewer: Frédéric Mangolte (Le Bourget du Lac)
