Let A and B be two points of PG(2, q n ), and let be a collineation between the pencils of lines with vertices A and B. In this paper, we prove that the set of points of intersection of corresponding lines under is either the union of a scattered GF(q)-linear set of rank n + 1 with the line AB or the union of q − 1 scattered GF(q)-linear sets of rank n with A and B. We also determine the intersection configurations of two scattered GF(q)-linear sets of rank n + 1 of PG(2, q n ) both meeting the line AB in a GF(q)-linear set of pseudoregulus type with transversal points A and B.

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The study of the intersection of two Baer subgeometries of PG(n, q), q a square, started in Bose et al. (Utilitas Math 17, 65-77, 1980); Bruen (Arch Math 39(3), 285-288, (1982). Later, in Svéd (Baer subspaces in the n-dimensional projective space. SpringerVerlag) and Jagos et al. (Acta Sci Math 69(1-2), [419][420][421][422][423][424][425][426][427][428][429] 2003), the structure of the intersection of two Baer subgeometries of PG(n, q) has been completely determined. In this paper, generalizing the previous results, we determine all possible intersection configurations of any two subgeometries of PG(n, q).

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