Relative blocking sets of unions of Baer subplanes

Abstract

We show that, for small t, the smallest set that blocks the long secants of the union of t pairwise disjoint Baer subplanes in $$\hbox {PG}(2,q^2)$$ has size $$t(q+1)$$ and consists of t Baer sublines, and, for large t, the smallest such set has size $$q^2+q+1$$ and is itself a Baer subplane of $$\hbox {PG}(2,q^2)$$ . We also present a stability result in the first case.