Some \(p\)-ranks related to Hermitian varieties (Q1361718)

Let \(\mathcal S\) be the incidence structure consisting of the points of a nondegenerate Hermitian variety of \(PG(n,q ^2)\) and of all hyperplanes of \(PG(n,q ^2)\), where \(q = p ^e\) and \(p\) is prime. The author shows that the \(p\)-rank of any incidence matrix of \(\mathcal S\) equals \(\left[{p + n -1 \choose n}^2 - {p + n -2 \choose n}^2 \right]^e + 1\). As a corollary, he obtains some new bounds for the size of caps and the existence of ovoids in finite unitary polar spaces.