The following lemma is a (presumably well-known) result that Raymond Cheng and I happened upon while writing our paper Unbounded negativity on rational surfaces in positive characteristic (arXiv, DOI). Well, Raymond probably knew what he was doing, but to me it was a pleasant surprise.
Lemma. Let be a power of a prime
, and let
. Then
satisfy a linear relation over
if and only if
Proof. If for
, then
for all
since
. As
is a ring homomorphism, we find
so the determinant is zero. Conversely, the union of -rational hyperplanes
is a hypersurface
of degree
(where
denotes the dual projective space parametrising hyperplanes in
). Since the determinant above is a polynomial of the same degree
that vanishes on all
-rational hyperplanes, we conclude that it is the polynomial cutting out
, so any
for which the determinant vanishes lies on one of the hyperplanes.
Of course when the determinant is zero, one immediately gets a vector in the kernel. There may well be an immediate argument why this vector is proportional to an element of
, but the above cleverly circumvents this problem.
For concreteness, we can work out what this determinant is in small cases:
: a point
only satisfies a linear relation over
if it is zero.
: the polynomial
cuts out the
-rational points of
.
: the polynomial
-rational lines in
. This is the case considered in the paper.