Small point sets of \(\text{PG}(n,p^{3h})\) intersecting each line in 1 mod \(p^{h}\) points
An \((n - k)\)-blocking set \(B\) in \(PG(n, q)\) with \(0 < k < n\) is a set of points intersecting each \(k\)-subspace non-trivially. A theorem of Bose and Burton shows that the smallest such blocking sets must be \((n - k)\)-dimensional subspaces and these kinds of blocking sets are called trivial. A point \(p\) of \(B\) is essential to \(B\) if there is a subspace of dimension \(k\) through \(p\) that meets \(B\) in \(p\) alone. A blocking set \(B\) is called minimal if it is not trivial and each point of \(B\) is essential. Further, \(B\) is called small if its size satisfies \(|B| < \frac{3(q^{n - k} + 1)}{2}\). A point set \(S\) of \(PG(n, q^t)\) is said to be linear if there is a projective space \(PG(n', q^t)\) containing \(PG(n, q^t)\) such that \(S\) is the projection of the subgeometry \(PG(n', q) \subset PG(n', q^t)\) from a suitable vertex onto \(PG(n, q^t)\). The main result of this paper whose proof involves elaborate computations proves the following special case of the linearity conjecture: The point sets of \(PG(n, q)\) with \(q = p^{3h}\) and \(p \geq 7\) of size \(< \frac{3(q^{n - 1} + 1)}{2}\) intersecting each line \(1\pmod{\root{3}\of{q}}\) points are linear blocking sets.
scientific article; zbMATH DE number 5844611
scientific article