Small minimal blocking sets and complete \(k\)-arcs in PG\((2,p^3)\)
The author deals with blocking sets in \(\text{PG}(2,q)\). The theory in the case \(q=p^3\) is especially developed with results describing both the structure and the size of such blocking sets, and allowing consequences on the complete arcs in \(\text{PG}(2,p^3)\). A blocking set in the projective plane \(\text{PG}(2,q)\), \(q=p^n\), is a set \(B\) of points which intersects every line. If \(B\) contains a line it is called trivial and if no proper subset of it is a blocking set, it is called minimal. Let \(B\) be a non-trivial blocking set. For every line \(\ell\), the obvious inequality \(|B\setminus\ell|\geq q\) holds. If there exists a line \(\ell\) such that \(|B\setminus\ell|=q\), then the blocking set is called of Rédei type. A small blocking set is a non-trivial blocking set of size less than \(3(q+1)/2\). \textit{T.~Szönyi} [Finite Fields Appl. 3, 187-202 (1997; Zbl 0912.51004)] proved that the size of a small minimal blocking set lies in an interval depending on \(q\) and an integer \(e_0\) such that \(1\leq e_0\leq n/2\). The first result in this paper is an improvement of the upper bound given by Szönyi. More precisely, if \(e\) is the largest integer such that each line intersects \(B\) in 1 modulo \(p^e\) points, then asymptotically \(|B|\leq q+(q/p^e)+(q/p^{2e})+2(q/p^{3e})+\cdots\). Then it is proved that for every small minimal blocking set \(B\) in \(\text{PG}(2,p^3)\) with \(p\geq 7\), either (i)~\(B\) is of Rédei type, or (ii)~for any line \(\ell\) it holds \(|B\cap\ell|\leq 1+(3+c)p\), where \(c\) is such that \(c\in\{0,1\}\) and \(|B|=q+p^2+cp+1\). The proofs use the theory of lacunary polynomials. In a subsequent paper to appear the author excludes case (ii). A \(k\)-arc \(K\) in \(\text{PG}(2,q)\) is a set of \(k\) points, no three of them collinear. \(K\) is complete if there is no \((k+1)\)-arc containing it. The secants of a complete arc form a dual blocking set. A.~Blokhuis conjectured that for every complete \(k\)-arc, \(k>\sqrt{3q}+1/2\); this is known to be true for \(q=p\) and \(q=p^2\) [cf. \textit{S. Ball}, Discrete Math. 174, 29-34 (1997; Zbl 0891.51006)]. In the paper under review, such conjecture is proved for \(q=p^3\) by means of the above results.
scientific article; zbMATH DE number 1369802