We call a section of an elliptic surface to be everywhere integral if it is disjoint from the zero-section. The set of everywhere integral sections of an elliptic surface is a finite set under a mild condition. We pose the basic problem about this set when the base curve is P1. In the case of a rational elliptic surface, we obtain a complete answer, described in terms of the root lattice E8 and its roots. Our results are related to some problems in Gröbner basis, Mordell-Weil lattices and deformation of singularities, which have served as the motivation and idea of proof as well.
Publié le : 2010-02-15
Classification:
Gröbner basis,
integral section,
Mordell-Weil lattice,
deformation of singularities,
14J26,
14J27,
11G05
@article{1265033217,
author = {Shioda, Tetsuji},
title = {Gr\"obner basis, Mordell-Weil lattices and deformation of singularities, I},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {86},
number = {1},
year = {2010},
pages = { 21-26},
language = {en},
url = {http://dml.mathdoc.fr/item/1265033217}
}
Shioda, Tetsuji. Gröbner basis, Mordell-Weil lattices and deformation of singularities, I. Proc. Japan Acad. Ser. A Math. Sci., Tome 86 (2010) no. 1, pp. 21-26. http://gdmltest.u-ga.fr/item/1265033217/