The sizes of the intersection of two unitals in PG$(2,q^2)$

Abstract

We show that the size of the intersection of a Hermitian variety in $\PG(n,q^2)$, and any set satisfying an $r$-dimensional-subspace intersection property, is congruent to 1 modulo a power of $p$. In particular, in the case where $n=2$, if the two sets are a Hermitian unital and any other unital, the size of the intersection is congruent to 1 modulo $\sqrt q$ or modulo $\sqrt{pq}$. If the second unital is a Buekenhout-Metz unital, we show that the size is congruent to 1 modulo $q$.