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The Betten-Walker spread and Cayley’s ruled cubic surface. (English) Zbl 1126.51003

The authors give the following elegant construction of the Betten-Walker spread of \(\text{PG}(3,K).\) The Cayley’s ruled cubic surface \(F\) of \(\text{PG}(3,K)\) has a line \(g_{\infty}\) of double points and any point of \(F\) not on \(g_{\infty}\) is incident with a unique osculating tangent line. The set \(\mathcal O\) of all the osculating tangent lines of \(F\) at a point not incident with \(g_{\infty}\) is a maximal partial spread of \(\text{PG}(3,K)\) if and only if \(\operatorname{Char} K \neq 3\) and the polynomial \(x^2+x+1\) has no root in \(K.\) If \(K \) has characteristic different from \(3\) and each element of \(K\) has a third root in \(K,\) then \(\mathcal O\) is a spread, which is exactly the one associated with the Betten-Walker plane.
Moreover, there is a pencil of lines \(\mathcal L\) such that \({\mathcal O}\cup {\mathcal L}\) is an algebraic set of lines.

MSC:

51A40 Translation planes and spreads in linear incidence geometry
14J26 Rational and ruled surfaces
51M30 Line geometries and their generalizations
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