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Dual partial quadrangles embedded in \(PG(3,q)\). (English) Zbl 1044.51002

A known example of a dual partial quadrangle embedded in a projective space can be obtained as follows. Let \(\mathcal H\) be a non-singular Hermitian variety in \(PG(3,q)\), \(q\) a square, and let \(L\) be any line on \(\mathcal H\); an incidence structure \({\mathcal S} = ({\mathcal P},{\mathcal B},I)\) is defined by taking as point set \(\mathcal P\) the point set of \(\mathcal H \setminus L\), as line set \(\mathcal B\) the set of lines of \(\mathcal H\) minus all the lines concurrent with \(L\), and as incidence the inclusion. This incidence structure, which is denoted by \(H(3,q)^\ast\), is actually a dual partial quadrangle, and is embedded in \(PG(3,q)\). This example is particularly interesting since no projective full embedding of a proper partial quadrangle is known.
In the paper under review the authors prove the following result: If \(\mathcal S\) is a dual partial quadrangle of order \((s,t)\) with \(s=q\), fully embedded in \(PG(3,q)\), then \(\mu \leq q - (q / (t+1))\). If equality holds, then \(\mathcal S\) is uniquely determined, namely either \(q=2\) and \(\mathcal S\) is \(W(2)\) minus a spread, or \(q\) is a square and \(\mathcal S = H(3,q)^\ast\).

MSC:

51E12 Generalized quadrangles and generalized polygons in finite geometry
51A45 Incidence structures embeddable into projective geometries
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