On the stability of Baer subplanes

Abstract

A blocking set in a projective plane is a point set intersecting each line. The smallest blocking sets are lines. The second smallest minimal blocking sets are Baer subplanes (subplanes of order q). Our aim is to study the stability of Baer subplanes in PG(2,q). If we delete q+1−k points from a Baer subplane, then the resulting set has size q+k and (q+1−k)(q−q) 0-secants. If we have somewhat more 0-secants, then our main theorem says that this point set can be obtained from a Baer subplane or from a line by deleting somewhat more than q+1−k points and adding some points. The motivation for this theorem comes from planes of square order, but our main result is valid also for non-square orders. Hence in this case the point set contains a relatively large collinear subset.