Small tight sets of hyperbolic quadrics
Let \(Q^+(2n+1,q)\subset \mathrm{PG}(2n+1,q)\) denote the hyperbolic quadric of rank \(n+1\) contained in the projective space of dimension \(2n+1\) over the Galois field \(\mathrm{GF}(q)\), defined by a non-degenerate quadratic form. A tight set \(M\) of \(Q^+(2n+1,q)\) is a subset of \(Q^+(2n+1,q)\) such that the average number of points of \(M\) in the tangent hyperplanes of points of \(M\) is \[ q^n +|M|\frac{q^n-1}{q^{n+1}-1}. \] In this case the tight set \(M\) has exactly \(x(q^{n+1}-1)/(q-1)\) points with \(x\geq 0\) integer and \(M\) is an \(x\)-tight set. In general, a union of \(x\) mutually skew generators of \(Q^+(2n+1,q)\) provides an example of an \(x\)-tight set. In this article, the authors prove that if \(1\leq n\leq 3 \) and \(x\leq q\), or \(n\geq 4\), \(x\leq q-1\), and \(q\geq 71\), then any \(x\)-tight set of a hyperbolic quadric \(Q^+(2n+1,q)\) is the union of \(x\) mutually disjoint generators.
scientific article; zbMATH DE number 6168443