Fq-linear blocking sets in PG(2,q4)

Abstract

A blocking set B in the projective plane PG(2, q), q = p, p prime, is a set of points meeting every line of PG(2, q). B is called trivial if it contains a line, and it is called minimal if no proper subset of it is a blocking set. We say B is small when its size is less than 3(q+1) 2 and we call B of Redei type if there exists a line l such that |B l| = q. The line l is called a Redei line of B. The exponent of B is the maximal integer e (0 ≤ e ≤ h) such that |l∩B| ≡ 1(mod p) for every line l in PG(2, q). In [?] T. Szőnyi proves that a small minimal blocking set of PG(2, q) has positive exponent. All the known examples of small minimal blocking sets belong to a family of blocking sets, called “linear”, introduced by G. Lunardon in [?]. Let π = PG(2, q) = PG(V,Fqn), q = p, p prime. A blocking set B of π is said to be an Fq-linear blocking set if B is an Fq-linear set of π of rank n + 1, i.e., B is defined by the non-zero vectors of an (n + 1)-dimensional Fq-vector subspace W of V , and we write B = BW . If BW is an Fq-linear blocking set, then each line of π intersects BW in a number of points congruent to 1 modulo q, hence the exponent of an Fq-linear blocking set is at least h. Also, if there exists a line l of π such that BW ∩ l has rank n, then BW is of Redei type (see [?]) and if BW has exactly exponent h, then |BW ∩ l| ≥ qn−1 + 1 (see [?], [?]). In the planes PG(2, q) and PG(2, q), the Fq-linear blocking sets are completely classified: in PG(2, q) they are Baer subplanes and in PG(2, q) they are isomorphic either to the blocking set obtained from the graph of the trace function of Fq3 over Fq or to the blocking set obtained from the graph of the function x → x (see [?]). In the plane PG(2, q) all the sizes of the Fq-linear blocking sets are known (see [?]). The next open problem is the complete classification of the Fq-linear blocking sets in PG(2, q) with n ≥ 4. An Fq-linear blocking set B of π = PG(2, q), n > 2, can be also constructed as the projection of a canonical subgeometry Σ ' PG(n, q) of Σ∗ = PG(n, q) to π from an (n−3)-dimensional subspace Λ of Σ∗, disjoint from Σ and we write B = BΛ,π,Σ. Also, if πΛ is the quotient geometry of Σ∗ on Λ, note that BΛ,π,Σ is isomorphic to the Fq-linear blocking set BΛ,Σ in πΛ consisting of all (n− 2)dimensional subspaces of Σ∗ containing Λ and with non-empty intersection with Σ. Therefore, in this paper we will use Fq-linear blocking sets BΛ,Σ in the model πΛ of PG(2, q). In this paper, we show that two Fq-linear blocking sets, BΛ,Σ and BΛ′,Σ′ , of exponent h of the planes πΛ and πΛ′ , respectively, constructed in Σ∗ (n > 2), are isomorphic if and only if there exists a collineation φ of Σ∗ mapping Λ to Λ′ and Σ to Σ′. In particular, we get that two Fq-linear blocking sets of PG(2, q), Bl,Σ and Bl′,Σ, which are not Baer subplanes, are isomorphic if and only if there exists a collineation φ of Σ∗ fixing Σ such that φ(l) = l′. In Section 4, the above result and the main theorem of [?] leads us to a complete classification of all Fq-linear blocking sets in PG(2, q).