Embeddings of hermitian unitals into Pappian projective planes (Q2273625)

The authors study the ways a generalised Hermitian unital embeds in projective planes defined over fields, and determine the full automorphism group of such a unital, their only restriction being that each block of the unital has at least four points. More precisely: A generalized Hermitian unital \(\mathcal{H}(C|R)\) -- where \(C|R\) is any (possibly inseparable) quadratic extension of fields -- is a point-block incidence structure in the Pappian projective plane \(\mathsf{PG}(2,C)\), whose point set \(U\) consists of triples \([X,Y,Z]\) satisfying a certain condition and whose blocks are given by the intersections of \(U\) with secant lines (lines of \(\mathsf{PG}(2,C)\) containing at least two points of \(U\)) (cf. Definition 2.1). Provided that \(|R|>2\), the main theorem of this paper states that, if \(\mathcal{H}(C|R)\) embeds into a projective plane \(\mathsf{PG}(2,E)\) over a field \(E\), then this is in the standard way: \(\mathsf{PG}(2,E)\) contains a subplane isomorphic to \(\mathsf{PG}(2,C)\), in which \(\mathcal{H}(C|R)\) is embedded naturally. The condition \(|R|>2\) is necessary since \(\mathcal{H}(\mathbb{F}_4| \mathbb{F}_2)\) is isomorphic to an affine plane \(\mathsf{AG}(2,\mathbb{F}_3)\), and this affine plane embeds not only into its projective closure \(\mathsf{PG}(2,\mathbb{F}_3)\) but in many other Pappian projective planes, of arbitrary characteristic. The above is used to determine the full automorphism group of \(\mathcal{H}(C|R)\) (cf. Theorem 5.2).