10 introduction

Table 1. Sporadic lines on cubic threefolds

a b Smallest point Point s.t. x = 0

19 18 (1 : 2 : 3 : -3 : -5) (0 : 7 : 1 : -7 : -18)

21 6 (1 : 2 : 3 : -3 : -3) (0 : 9 : 1 : -10 : -15)

22 5 (1 : -1 : 3 : 3 : -3) (0 : 27 : -4 : -60 : 49)

45 18 (1 : 1 : 3 : 3 : -3) (0 : 3 : -1 : 3 : -8)

73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96)

been found. These are the only non-obvious lines we know and the only ones

containing a point of height less than 5000.

We describe all the computations that were done as well some background on

the geometry of cubic and quartic threefolds. Observe that the lines on a cubic

threefold have a particular rich geometry. They form a smooth surface that is

of general type. Our observation that -rational lines are rare is therefore in

coincidence with Lang’s conjecture.

In Chapter VII we return to the more standard case of diagonal cubic surfaces.

The experiments are analogous to those described in Chapter VI for diagonal cubic

and quartic threefolds. The theory is, however, more complicated. The geometric

Picard rank is equal to 7 and, in the generic case, there is a Brauer–Manin ob-

struction to weak approximation excluding precisely two thirds of the adelic points.

The factors α(X) and β(X) appearing in the definition of Peyre’s constant are not

always the same and need to be considered.

We demonstrate experimentally the connection of Peyre’s constant with the

height m(X) of the smallest rational point. Under the Generalized Riemann Hy-

pothesis, we prove that there is no constant C such that m(X)

C

τ(X)

for every

diagonal cubic surface. We also prove that, for diagonal cubic surfaces, the recip-

rocal

1

τ(X)

behaves like a height function, i.e.,

1

τ(X)

admits a fundamental finite-

ness property.