Splitness of the Veronesean and the Taniguchi dual hyperovals
Let \(V\) be an \((n+m)\)-dimensional vector space over the finite field \(\mathbb{F}_q\). A set \(\mathcal{H}\) of \(\frac{q^n-1}{q-1}+1\) \(n\)-dimensional subspaces (\(n \ge 3\)) of \(V\) is called a dual hyperoval of \(V\) (DHO in short) if \(\langle \mathcal{H} \rangle = V\), \(\dim(X \cap Y) = 1\) and \(X \cap Y \cap Z = {0}\) for any three distinct members \(X , Y , Z \in\mathcal{H}\). A DHO \(\mathcal{H}\) is said to be of split type if there exists a subspace \(Y\) of \(V\) such that \(Y \oplus X = V\) for all \(X \in\mathcal{H}\). In the nice paper under review, the case \(m = n(n-1)/2\) and \(q = 2\) is investigated. There are four known DHOs, called the Huybrechts, Buratti-Del Fra, Veronesean and Taniguchi DHO. It is known that the first two DHOs are of split type. Here, the author shows that the Veronesean and Taniguchi DHOs also are of split type.
scientific article; zbMATH DE number 7004813