Hyperovals arising from a Singer group action on \(\mathcal{H}(3, q^2)\), \(q\) even
Let \(H(3,q^2)\) be a generalized quadrangle associated with a non-singular Hermitian surface in \(\mathrm{PG}(3,q^2)\). A hyperoval (or local subspace) of \(H(3,q^2)\) is a set of points with the property that each line of the generalized quadrangle meets the set in either 0 or 2 points. The hyperovals of \(H(3, 4)\) have been classified by \textit{A. A. Makhnev} [Discrete Math. Appl. 7, No. 4, 419--435 (1997; Zbl 0964.51004); translation from Diskretn. Mat. 9, No. 3, 101--116 (1997)]. Suppose \(q\) is even. Let \(W_1=\mathrm{PG}(3,q)\cap H(3,q^2)\) be a symplectic subquadrangle of \(H(3,q^2)\). If \(\mathcal{S}\) is a partial spread of \(\mathrm{PG}(3,q^2)\) such that \(\mathcal{S}\) induces on \(W_1\) a Desarguesian spread, then there are \(q-1\) symplectic subquadranlges \(W_1,W_2,\dots ,W_{q-1}\) of \(H(3,q^2)\) such that \(\mathcal{S}\) induces on \(W_i\) (\(i=1,2,\dots,q-1\)) a Desarguesian spread. Let us choose two symplectic subquadrangles \(W\) and \(W'\) among the subquadrangles \(W_1,W_2,\dots,W_{q-1}\). If \(G\) is a Singer cyclic group of \( \mathrm{PO}^-(4,q)<\mathrm{PSp}(4,q)\), we can suppose that \(\mathcal{S}\) is \(G\)-invariant. Let now \(\Omega\) and \(\Omega'\) be two elliptic quadrics in \(W\) and \(W'\), respectively, which are invariant under \(G\). The authors prove that \(\mathcal{I}=\Omega\cap \Omega'\) is an hyperoval of \(H(3,q^2)\). This implies that there exists an infinite family of hyperovals of \(H(3, q^2)\), \(q\) even, admitting a Singer cyclic group of order \(q^2+ 1\).
scientific article; zbMATH DE number 6854775
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