Projective spaces in Fermat varieties

In 1981, T. Shioda showed that the straight lines contained in a
Fermat surface of degree prime to 6 generate the Picard group of
the surface *over the rationals*, and he conjectured that the same
lines also generate the Picard group *over the integers*. This
conjecture was settled in the affirmative in 2013, when it was also
extended to certain---but not all---more general Delsarte surfaces.
(The dependence of the discrepancy between the two groups on the
surface is the subject of an ongoing investigation.)

Another generalization is the higher dimensional version of the
conjecture, i.e., whether the projective spaces of middle dimension
contained in a Fermat variety generate---over the integers---the
group of algebraic classes in the middle homology of the variety. I
will discuss an algebraic reduction of this problem (similar to the
one that worked well in the case of surfaces), the numerical
evidence, and the open questions that arise. This part is a joint
project with Ichiro Shimada (Hiroshima University).