4-dimensional compact projective planes with a 5-dimensional nilradical
The automorphism group \(\Sigma\) of a 4-dimensional compact projective plane \(\mathcal P\) is at most 16-dimensional, and if \(\dim \Sigma \geq 9\), then \(\mathcal P\) is the classical complex plane and \(\Sigma \cong \text{PSL}_3 \triangleleft\), [the reviewer et al., Compact Projective Planes, Berlin (1995), 73.22]. All planes \(\mathcal P\) with \(\dim \Sigma = 8\) or 7 have been determined explicitly. If \(\dim \Sigma = 8\), then \(\mathcal P\) is a translation plane, [loc. cit.]. For \(\dim \Sigma = 7\) the plane \(\mathcal P\) is either isomorphic to a (dual) translation plane or to a so-called shift plane [the author, Geom. Dedicata 36, No. 2/3, 151-170 (1990; Zbl 0717.51013)]. The present paper deals with the case \(\dim \Sigma = 6\). In addition, the author assumes that \(\Sigma\) has an open orbit in the space of flags, i.e., that \(\mathcal P\) is flexible. If \(\Sigma\) is not solvable, then its connected component \(\Delta\) is isomorphic \(\gtreqless^2 \cdot GL_2 \gtreqless\), [\textit{R. Löwen}, Geom. Dedicata 36, No. 2/3, 225-234 (1990; Zbl 0712.51011)]. Planes with such a group are semitranslation planes as will be proved in a forthcoming paper by H. Klein, N. Knarr, and R. Löwen. Thus, one may further assume that \(\Delta\) is solvable and it can be shown that \(\Delta\) fixes a flag \((\infty, W)\). Now several cases can be distinguished according to the dimension of the nil radical \(\mathcal N\) of \(\Delta\). \(\dim {\mathcal N} = 6\) leads to a single shift plane [the author, Mitt. Math. Ges. Hamb. 12, No. 3, 741-747 (1991; Zbl 0749.51015)]. If \(\dim {\mathcal N} = 5\) one proceeds by considering the orbits of \(\Delta\) on the line \(W\) and on the line pencil through \(\infty\), [the author, J. Geom. 42, No. 1/2, 30-40 (1991; Zbl 0745.51009)]. The possible four cases for such orbits are treated separately in this paper and as a result the author obtains four new families of 4-dimensional compact projective planes which are neither translation planes (or duals thereof) nor shift planes. The other possible cases \(\dim {\mathcal N} = 4\) and \(\dim {\mathcal N} = 3\) have also been treated in the meantime. Thus the present paper is an essential step towards a complete classification of all flexible 4-dimensional planes.
scientific article; zbMATH DE number 826786
scientific article