Tetsuji Shioda

Generators of the Néron-Severi Group of a Fermat Surface

The Neron-Severi group of a (nonsingular projective) variety is, by definition, the group of divisors modulo algebraic equivalence, which is known to be a finitely generated abelian group (cf. [2]). Its rank is called the Picard number of the variety. Thus the Neron-Severi group is defined in purely algebro-geometric terms, but it is a rather delicate invariant of arithmetic nature. Perhaps, because of this reason, it usually requires some nontrivial work before one can determine the Picard number of a given variety, let alone the full structure of its Neron-Severi group. This is the case even for algebraic surfaces over the field of complex numbers, where it can be regarded as the subgroup of the cohomology group H 2(X, ℤ) characterized by the Lefschetz criterion.

Geometry of Fermat Varieties

In studying geometry of Fermat varieties, there are two basic facts. One is readily visible (and hence a well-known) fact that a large finite group of automorphisms acts on a Fermat variety. The other is the existence of the so-called “inductive structure” of Fermat varieties of a fixed degree. By combining these, we can deal with various geometric questions concerning Fermat varieties and their products (or varieties closely related to them) such as the Hodge Conjecture [1] or the Tate Conjecture [11].

On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines. We give the isomorphism explicitly.

An infinite family of elliptic curves over Q with large rank via Néron's method.

In his famous article [N1], N6ron has given a method for constructing an infinite family of elliptic curves over Q with rank at least 11. By studying this from the viewpoint of Mordell-Weil lattices IS 1, S 2, S 3], we find that N6rons method gives a complete algorithm for such a construction. In particular, we can write down an explicit numerical example. N6rons idea is a very ingeneous, beautiful combination of some deep results in algebraic geometry and number theory. The former is based on the theory of del Pezzo surfaces (cf. [M, DP]) and the latter is the specialization argument originated by N6ron, which has been strengthened by Silverman and Tate (cf. [Sil , T]). Another ingredient implicit in IN1] seems to be the idea leading to the Kodaira-N6ron model (an elliptic surface) of an elliptic curve over a function field, which was to be developed later in [N2]. Our contribution, if any, will be:

Gröbner basis, Mordell-Weil lattices and deformation of singularities, II

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 always a finite set. We pose the basic problem about this set when the base curve is P. 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.

The abc-theorem, Davenport’s inequality and elliptic surfaces

We consider the application of the abc-theorem and Davenports inequality to elliptic surfaces over the projective line P 1 , with special attention to the case of equality in the abc-theorem. Some existence theorem and the finiteness results will be given for certain type of elliptic surfaces.

On Q-split Bézout intersection

Abstract This paper deals with the following question: Can one find a linear pencil of plane curves of a given degree m , defined over the rational number field Q , with m 2 distinct Q -rational base points, such that every curve belonging to the pencil is irreducible? The answer for m = 3 is well known, which gives an elliptic curve defined over Q ( t ) with Mordell-Weil rank r = 8. For general m , an affirmative answer will give an algebraic curve over Q ( t ) with rank r = m 2 − 1 (cf) [7]. The case m = 4 is solved in the affirmative in this paper. The question is open for m >4.