On the density of rational points on elliptic fibrations

Abstract

1. Introduction Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

Weighted Fano threefold hypersurfaces

“…Indeed, the surface S can be locally given near P by the equation (7,4,1) where P = (0, 0, 0). Let σ 1 be the weighted blow up of C 3 / Z 11 (7,4,1) at the singular point P with the weight 1 11 (10, 3, 1).…”

“…Indeed, the surface S can be locally given near P by the equation (7,4,1) where P = (0, 0, 0). Let σ 1 be the weighted blow up of C 3 / Z 11 (7,4,1) at the singular point P with the weight 1 11 (10, 3, 1). Then the blown up variety is covered by 3 affine charts, the first chart is isomorphic to C 3 /Z 10 (1, −3, −1), and in the first chart σ 1 is given by x = x 10/11 , y = x 3/11 y, z = x 1/11 z, where we denote the coordinates on C 3 /Z 10 (1, −3, −1) by the same letters x, y, z as the coordinates on C 3 / Z 11 (7,4,1).…”

“…Let σ 1 be the weighted blow up of C 3 / Z 11 (7,4,1) at the singular point P with the weight 1 11 (10, 3, 1). Then the blown up variety is covered by 3 affine charts, the first chart is isomorphic to C 3 /Z 10 (1, −3, −1), and in the first chart σ 1 is given by x = x 10/11 , y = x 3/11 y, z = x 1/11 z, where we denote the coordinates on C 3 /Z 10 (1, −3, −1) by the same letters x, y, z as the coordinates on C 3 / Z 11 (7,4,1). The full transform of S is given by the equation x 20/11 + x 9/11 y 3 + x 9/11 z 9 = 0, but the strict transformS of the surface S is given by the equation x + y 3 + z 9 = 0 ⊂ C 3 /Z 10 (1, −3, −1), and the exceptional divisor…”

“…, τ ℓ are actually elliptic and induced by a single elliptic fibration. This shows that it is worth our while to study birational transformations of the hypersurface 1 The weighted projective space Proj(F[x1, x2, · · · , xn]) defined over an arbitrary field F with wt(xi) = ai be denoted by P F (a1, a2, · · · , an). The weights ai are always assumed that a1 ≤ a2 ≤ .…”

“…The papers [1], [2], and [8] give us a stimulating result that rational points are potentially dense 3 on smooth Fano 3-folds possibly except double covers of P 3 ramified along smooth sextic surfaces. In the case N = 1, the potential density of rational points on the hypersurface X is proved in [8].…”