Classification of finite Veronesean caps.
Let \(K\) be any field and let \(n\geq 1\). The quadric Veronesean \({\mathcal V}_n\) of index \(n\) is the set of points of \(PG(n(n+3)/2,K)\) with generic element \[ (x_0^2,x_1,\dots,x_n^2,x_0x_1,x_0x_2,\dots,x_0x_n,x_1,x_2,\dots,x_1x_n,\dots,x_{n-1}x_n), \] where \((x_0,\dots,x_n)\) are homogeneous projective coordinates of a point of \(PG(n,K)\). In a paper of 1984, Mazzocca and Melone formulated three geometric properties that should characterize \({\mathcal V}_n\) and call objects satisfying these three properties ``Veronesean caps''. In this nice paper the authors prove a slightly stronger result by weakening one of the axioms of Mazzocca and Melone (ovals instead of conics!), proving the following theorem: Let \(X\) be a set of points in \(\Pi:PG(M,q)\), \(M>2\), spanning \(\Pi\), and let \(\mathcal P\) be a collection of planes of \(\Pi\) such that for any \(\pi\in{\mathcal P}\), the intersection \(X\cap\pi\) is an oval in \(\pi\). For \(\pi\in{\mathcal P}\) and \(x\in X\cap\pi\), let \(T_x(\pi)\) denote the tangent line to \(X\cap\pi\) at \(x\in\pi\). Assume that: (Q1) Any two points \(x,y\in X\) lie in a unique member of \(\mathcal P\) denoted by \([x,y]\); (Q2) if \(\pi_1,\pi_2\in{\mathcal P}\) and \(\pi_1\cap\pi_2\neq\emptyset\), then \(\pi_1\cap\pi_2\subseteq X\); (Q3) if \(x\in X\) and \(\pi\in{\mathcal P}\) with \(x\not\in\pi\), then each of the lines \(T_x([x,y])\), \(y\in X\cap\pi\), is contained in a fixed plane of \(\Pi\), denoted by \(T(x,\pi)\). Then there exists a natural number \(n\geq 2\) (called the index of \(X\)), a projective space \(\Pi':=PG(n(n+3)/2,q)\) containing \(\Pi\), a subspace \(R\) of \(\Pi'\) skew to \(\Pi\), and a quadric Veronesean \({\mathcal V}_n\) of index \(n\) in \(\Pi'\), with \(R\cap{\mathcal V}_n=\emptyset\), such that \(X\) is the (bijective) projection of \({\mathcal V}_n\) from \(R\) onto \(\Pi\). The subspace \(R\) can be empty, in which case \(X\) is projectively equivalent to \({\mathcal V}_n\).
scientific article; zbMATH DE number 2056162