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