On smallest covers of finite projective spaces (Q1359064)

Let \({\mathcal P}= \text{PG} (d,q)\) be a projective space of dimension \(d\) and order \(q\). A set \({\mathcal S}\) of \(t\)-dimensional subspaces of \({\mathcal P}\) is called a \(t\)-cover of \({\mathcal P}\) if and only if every point of \({\mathcal P}\) is contained in at least one element of \({\mathcal S}\). \textit{A. Beutelspacher} [J. Geom. 12, 10-16 (1979; Zbl 0386.05023)] proved that the cardinality of such a \(t\)-cover is at least \(| {\mathcal S} |= q^{r+1} \cdot {q^{k(t+1)} -1\over q^{t+1} -1}+1\). The author sketches Beutelspacher's construction and proves that all examples of minimal cardinality ``look like'' the examples given by Beutelspacher.