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.
"Gröbner basis, Mordell-Weil lattices and deformation of singularities, I." Proc. Japan Acad. Ser. A Math. Sci. 86 (2) 21 - 26, February 2010. https://doi.org/10.3792/pjaa.86.21