Classification of Veronesean caps (Q2468010)

The authors start from the following assumptions: In a projective space \(\mathbf{P} := \text{PG}(M,K) \) of dimension \(2 < M \in \mathbf{N} \) over a skew field \(K\), let \(X\) be a spanning point set of \(\mathbf{P} \) and let \(\Pi\) be a collection of planes such that for any \(\pi \in \Pi\), the intersection \(\pi \cap X\) is an oval. Then they call \((X,\Pi)\) a Veronesean cap if: {\parindent=9mm \begin{itemize}\item[(A1)] \(\forall x,y \in X\), \(x \neq y\) \(\exists_1 \pi \in \Pi : x, y \in \pi\) (denoted by \([x,y] := \pi\)). \item[(A2)] \(\forall \pi_1, \pi_2 \in \Pi\), \(\pi_1 \neq \pi_2 : \pi_1 \cap \pi_2 \subseteq X\). \item[(A3)] If \(x \in X\), \(\pi \in \Pi \) with \( x \notin \pi\) then there is exactly one plane \(\pi' \in \Pi\) denoted by \(T(x,\pi)\) such that for each \(y \in \pi \cap X \) the tangent line \(T_x([x,y])\) touching the oval \(X \cap [x,y]\) in the point \(x\) is contained in the plane \(\pi' = T(x,\pi)\). \end{itemize}} Then they show: \(K\) is a commutative field, there exist a projective space \(\mathbf{P}' = \text{PG}(n(n+3)/2,K) \) with \(n \geq 2 \) containing \(\mathbf{P}\), a subspace \({\mathbf R} \) of \(\mathbf{P}'\) skew to \(\mathbf{P}\) and a Veronesean variety \( V_n \) in \( \mathbf{P}'\) such that \( {\mathbf R} \cap V_n = \emptyset \) and \(X\) is the projection of \( V_n\) from \({\mathbf R}\) onto \(\mathbf{P}\). If \({\mathbf R}\) is empty then \(X\) is projectively equivalent to \( V_n\).