Bruck's vision of regular spreads or what is the use of a Baer superspace?
Let \(\mathbb{P}^* = \text{PG} (d,F^*)\) be a \(d\)-dimensional pappian projective space such that the field \(F^*\) admits a subfield \(F\) of index 2. Assuming \(d\) at least 3, \(\mathbb{P}\) is called a Baer subspace of \(\mathbb{P}^*\) if every point of \(\mathbb{P}^*\) is incident with at least one line of \(\mathbb{P}\), it follows that \(\mathbb{P} = \text{PG} (d,F)\). For every point \(x\) of \(\mathbb{P}^* \setminus \mathbb{P}\), there is exactly one line through \(x\) that intersects \(\mathbb{P}\) in a line \(I\). One says that the line \(I\) is induced by the point \(x\). R. H. Bruck constructed regular spreads in Baer subspaces of 3-dimensional finite projective spaces. He proved that if \(I^*\) is a line of \(\mathbb{P}^* = \text{PG} (3,q^ 2)\) disjoint from \(\mathbb{P} = \text{PG} (3,q)\), then the family of lines of \(\mathbb{P}\) induced by the point of \(I^*\) is a regular spread of \(\mathbb{P}\). Moreover he proved that for every regular spread \(\varphi\) of \(\mathbb{P}\), there are exactly two lines \(I_ 1^*\) and \(I_ 2^*\) of \(\mathbb{P}^*\) disjoint to \(\mathbb{P}\) such that \(\varphi\) is induced by the points of \(I_ 1^*\) and by the points of \(I_ 2^*\). In this paper the authors generalize this result of Bruck for infinite pappian projective spaces of odd dimension \(2t+1\). Their main result reads as follows. Let \(\mathbb{P}^* = \text{PG} (2t + 1,F^*)\) be a \(2t+1\)-dimensional pappian projective space such that the field \(F^*\) admits a subfield of index 2. If \(U^*\) is a \(t\)-dimensional subspace of \(\mathbb{P}^*\) that is disjoint to a Baer subspace \(\mathbb{P} = \text{PG} (2t + 1,F)\), then the family \(\varphi\) of lines of \(\mathbb{P}\) induced by points of \(U^*\) is a regular line spread of \(\mathbb{P}\). Conversely if \(\mathbb{P} = \text{PG} (2t + 1,F)\) admits a regular line spread \(\varphi\), then there exists a quadratic extension \(F^*\) of \(F\) such that \(\mathbb{P}\) is a Baer subspace of \(\mathbb{P}^* = \text{PG}^*\) disjoint to \(\mathbb{P}\) such that \(\varphi\) is induced by the point set of \(U_ 1^*\) and the point set of \(U_ 2^*\). If \(F^*\) is a non-separable field extension of \(F\), then there is exactly one such \(t\)-dimensional subspace \(U^*\). As an application they give a very short geometric proof of the result by Segre, that given a regular line spread \(\varphi\) of a pappian projective space \(\mathbb{P} = \text{PG} (2t+1,F)\) a pappian projective space \(\mathbb{P}(\varphi)\) of dimension \(t\), coordinatized by a quadratic field extension of \(F\) can be constructed as follows. The points of \(\mathbb{P} (\varphi)\) are the elements of the regular line spread \(\varphi\) while the subspaces of dimension \(j\) of \(\mathbb{P}(\varphi)\) are the \((2j+1)\)- dimensional subspaces \(U\) of \(\mathbb{P}\) such that \(\varphi\) induces a regular line spread in \(U\) \((j=1, \dots, t-1)\).
scientific article; zbMATH DE number 492581
scientific article