Some characterizations of finite Hermitian Veroneseans
Let \(\text{GF}(q)\) be the finite field of order \(q\), and fix \(r\in \text{GF}(q^2)\setminus \text{GF}(q)\). A Hermitian Veronesean \({\mathcal H}_n\) of index \(n\) over \(\text{GF}(q)\) is the subset of \(\text{PG}(n^2+2n,q)\) given by the image of the map \(\theta: \text{PG}(n,q^2)\mapsto \text{PG}(n^2+2n,q)\), given by \[ \theta(x_0,x_1,\dots,x_n)= (y_{0,0},y_{0,1},\dots, y_{0,n},y_{1,0},y_{1,1},\dots,y_{1,n}\dots, y_{n,0},y_{n,1},\dots, y_{n,n}) \] where \[ y_{i,j}=\begin{cases} x_i^{q+1}&\text{ if }i=j\\ x_ix_j^q+x_i^qx_j&\text{ if }i<j\\ rx_ix_j^q+r^qx_i^qx_j&\text{ if }i>j.\end{cases} \] The inverse image via \(\theta\) of a hyperplane section of \({\mathcal H}_n\) is a (possibly singular) Hermitian variety in \(\text{PG}(n,q^2)\). The main aim of this paper is to provide characterisations of the Hermitian Veronesean \({\mathcal H}_n\) as the unique ovoidal embedding of \(\text{PG}(n,q^2)\) in \(\text{PG}(n^2+2n,q)\), in the sense of the following theorem. Theorem. Let \({\mathcal S}=(P,L,I)\) be the point-line geometry of a finite projective space of dimension \(n\) and order \(q^2>4\), where \(q\) is a prime power. Suppose that \(P\) is a subset of the projective space \(\text{PG}(d,q)\) with \(d\geq n^2+2n\). Suppose also that \(P\) is not fully contained in any hyperplane of such a space. If the points of \(\text{PG}(d,q)\) incident with each line \(l\) of \({\mathcal S}\) form an ovoid in some \(3\)-dimensional projective subspace \(S_l\) of \(\text{PG}(d,q)\), then \(d=n^2+2n\) and \(P\) is projectively equivalent to the Hermitian Veronesean of \(\text{PG}(n,q^2)\). For \(n=2\) this result yields that \({\mathcal H}_2\subseteq \text{PG}(8,q)\) is characterised by the following three properties: (1) \(| {\mathcal H}_2| =q^4+q^2+1\); (2) each hyperplane of \(\text{PG}(8,q)\) meets \({\mathcal H}_2\) in either \(q^2+1\), \(q^3+1\) or \(q^3+q^2+1\) points; (3) for each solid \(\Sigma\) of \(\text{PG}(8,q)\), \( | \Sigma\cap{\mathcal H}_2| \geq q+3 \Rightarrow | \Sigma\cap{\mathcal H}_2| =q^2+1. \)
scientific article; zbMATH DE number 2139259