Spreads admitting net generating regulizations
Let \(S\) be a spread of a Pappian projective 3-space (not necessarily finite). The author defines a regularization \(\Sigma\) of \(S\) to be collection of reguli contained in \(S\) such that each element of \(S\), with the possible exception of at most two lines, is contained either in exactly one regulus of \(\Sigma\) or in all reguli of \(\Sigma\). The collection of lines obtained by taking the union of complementary reguli to the reguli of \(\Sigma\), together with any exceptional lines as noted above, is called the complementary congruence of \(S\) with respect to \(\Sigma\), and is denoted by \(S^c_\Sigma\). Any exceptional line of \(S\) with respect to \(\Sigma\) is considered an improper regulus of \(\Sigma\), and it is also considered to be self-complementary. If \(S^c_\Sigma\) happens to be a nondegenerate linear congruence of lines (hyperbolic, parabolic, or elliptic), then \(\Sigma\) is called net generating regularization (hyperbolic, parabolic, or elliptic). In the finite case it is well known that flocks of a hyperbolic quadric, an elliptic quadric, or a quadratic cone yield spreads with a net generating regularization, and vice-versa [for instance, see \textit{M. Walker}, Geom. Dedicata 5, 135-146 (1976; Zbl 0356.50022)]. In the infinite case a flock of an elliptic quadric is defined to be a collection of (proper or improper) disjoint conics which partition the elliptic quadric and which contain \(\underline{\text{at most}}\) two improper conics (rather than exactly two improper conics). Examples are given (over the field of rational numbers) of flocks of an elliptic quadric with zero, respectively one, improper conic. Flocks of a hyperbolic quadric or a quadratic cone are defined analogously to the finite case. It is then shown in the paper under review that there is a one-to-one correspondence between the spreads of a general Pappian 3-space with an elliptic (respectively, hyperbolic or parabolic) regularization and the flocks of an elliptic quadric (respectively, hyperbolic quadric or quadratic cone). Other interesting examples are given over the field of real numbers. For instance, it is shown that the nonregular, topological spread \({\mathcal B}_{72}\) of \(PG (3,\mathbb{R})\) given by \textit{D. Betten} [Math. Z. 128, 129-151 (1972; Zbl 0237.50017)] admits an infinite number of elliptic and parabolic regularizations, but admits no hyperbolic regularization. It is proven that if a spread of \(PG (3,\mathbb{R})\) admits either two distinct hyperbolic regularizations or both a hyperbolic and a parabolic regularization, then the spread is necessarily an elliptic linear congruence. It is noted that this theorem does not remain valid for spreads admitting two distinct parabolic regularizations since the Betten spread \({\mathcal B}_{72}\) is nonregular.
scientific article; zbMATH DE number 929895
scientific article