Abstract: It was conjectured that the smallest minimal point sets of PG$(2s,q)$, $q$ a square, that meet every $s$-subspace and that generate the whole space are Baer subgeometries PG$(2s,\sqrt q)$. This was shown in 1971 by Bruen for $s=1$, and by Metsch and Storme [MS] for $s=2$. Our main interest in this paper is to prepare a possible proof of this conjecture by proving a strong theorem on line-blocking sets in projective spaces (see Theorem 1.1). We apply this theorem to prove the conjecture in the case $s=3$. The general case will be handled in a forthcoming paper by the first author.
[MS] K. Metsch; L. Storme: $2$-blocking sets in PG$(n,q)$, $q$ square. Beitr. Algebra Geom., submitted.
Keywords: smallest minimal point set; Baer subgeometry; line-blocking sets in projective space
Classification (MSC2000): 51E20; 05B05
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