Let $X$ be the Fermat hypersurface of dimension $2m$
and of degree $q+1$ defined over an algebraically
closed field of characteristic $p>0$, where $q$ is
a power of $p$, and let $NL^m (X)$ be the free
abelian group of numerical equivalence classes of
linear subspaces of dimension $m$ contained in $X$.
By the intersection form, we regard $NL^m (X)$ as
a lattice. Investigating the configuration of these
linear subspaces, we show that the rank of $NL^m (X)$
is equal to the $2m$th Betti number of $X$, that the
intersection form multiplied by $(-1)^m$ is positive
definite on the primitive part of $NL^m (X)$, and that
the discriminant of $NL^m (X)$ is a power of~$p$.
Let ${\mathcal L}^m (X)$ be the primitive part of
$NL^m (X)$ equipped with the intersection form
multiplied by $(-1)^m$. In the case $p=q=2$,
the lattice ${\mathcal L}^m (X)$ is described in
terms of certain codes associated with the unitary
geometry over ${\mathbb F}_2$. Since
${\mathcal L}^1 (X)$ is isomorphic to the root
lattice of type $E_6$, the series of lattices
${\mathcal L}^m (X)$ can be considered as a
generalization of $E_6$.
The lattice ${\mathcal L}^2 (X)$ is isomorphic
to the laminated lattice of rank $22$.
This isomorphism explains Conway's identification
$\cdot 222\cong {\rm PSU}(6,2)$ geometrically.
The lattice ${\mathcal L}^3 (X)$ is of discriminant
$2^{16}\cdot 3$, minimal norm $8$, and kissing
number $109421928$. 2000 Mathematics Subject Classification:
14C25, 11H31, 51D25.