\((q^2+q+1)\)-caps of class \([0,1,2,3,q+1]_2\) and type \((1,m,n)_4\) of PG\((5,q)\) are quadric Veroneseans (Q1887653)

In a paper to appear in [J. Comb. Theory, Ser. A], \textit{J.~A.~Thas} and the reviewer show that, if a \((q^1+q+1)\)-cap \(K\) of PG\((5,q)\) has the property that every plane meets \(K\) in \(0,1,2,3\) or \(q+1\) points (but not necessarily all these values must occur) and also that every hyperplane meets \(K\) in either \(1\), or \(m=q+1\), or \(n=2q+1\) points (and all of these values do occur), then \(K\) is isomorphic to the quadric Veronesean \(\mathcal{V}_2^4\). In the paper under review, the author shows that the assumptions \(m=q+1\) and \(n=2q+1\) are superfluous, and that one can start with any different values \(m,n\) (both distinct from \(1\)) to reach the same conclusion. The proof consists of clever counting.