Minimal covers of \(Q^+(2n+1,q)\) by \((n-1)\)-dimensional subspaces
The paper considers the following problem. How many \((n-1)\)-dimensional subspaces of \(\text{PG}(2n+1,q)\) lying on a fixed nonsingular hyperbolic quadric \(Q^+(2n+ 1,q)\) does one need to cover all points of that quadric? The answer is \(q^{n+1} +2q+1\), and for every set \(S\) of such many \((n-1)\)-dimensional subspaces covering the point set of \(Q^+(2n +1,q)\), there exist two (possibly coinciding) \((n-2)\)-dimensional subspaces \(U_1,U_2\) on the quadric such that each point of \(U_1\cap U_2\) is contained in exactly \(2q+1\) members of \(S\), each other point of \(U_1\cup U_2\) is contained in exactly \(q+1\) members of \(S\), and all other points of the quadric are contained in a unique member of \(S\). The authors motivate the problem they treat by saying that it is a special case of a more general problem (replacing \((n-1)\)-dimensional subspace by \(t\)-dimensional subspace) which has on its turn an important special case (the case \(t=n\) giving rise to spreads and partial spreads). I found this motivation rather weak. Instead, I think the proof of the result could be a good motivation; it uses the theory of minihypers in a nontrivial way. In the last section the authors give an upper bound for a partial \((n-1)\)-spread of \(Q^+(2n +1,q)\) (and that upper bound is \(q^{n+1}+ q-1\), \(n\geq 3)\).
scientific article; zbMATH DE number 1764795