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

TL;DR

This paper investigates the intersection sizes of Hermitian varieties and certain sets in projective spaces, revealing they are congruent to 1 modulo specific powers of p, with particular results for unitals in PG(2,q^2).

Contribution

It establishes new congruence conditions for intersection sizes of Hermitian varieties with sets satisfying subspace intersection properties, including specific results for Buekenhout-Metz unitals.

Findings

Intersection sizes are congruent to 1 modulo a power of p.

For n=2, intersection size is congruent to 1 modulo √q or √(pq).

When the second unital is Buekenhout-Metz, the size is congruent to 1 modulo q.

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 $q$.