Journal of Geometry Vol. 57 (1996) 0047-2468/96/020058 -05 $ t. 50+0.20/0 (c) Birkhfiuser Verlag, Basel PARTITIONING PROJECTIVE GEOMETRIES INTO SEGRE VARIETIES In ricordo di Giuseppe Tallini Arrigo Bonisoli, Antonio Cossidente and Donato Saeli Using suitable subgroups of Singer cyclic groups we prove some properties of regular spreads and Segre varieties, which in turn yield a necessary and sufficient condition for partitioning a finite projective space into such varieties. Assume n = m. k with 1 < m, k < n. With the notation of [2] let Sl2,~,k be a Segre variety with indices m and k in PG(n - 1, q). Denote by ~m-1 the family of (m - 1)-dimensional projective subspaces lying on SVm,k. L.R.A. Casse and C.M. O'Keefe in [2] call 7~m-1 an (m- 1)-regulus of rank k - 1 in PG(n- 1, q). In the same paper they define an (m- 1)- spread 5 r of PG(n - 1, q) to be regular (of rank k - 1) if, for any (k - 1)-dimensional subspace A of PG(n- 1, q) meeting each member of 5 ~ in at most one point, the members of F actually meeting h form an (m - 1)-regulus of rank k - 1. In the paper [1] by the same authors it is shown that the above concept of a regular spread is equivalent to the following generalization of the notion of a linear elliptic congruence, see also w in [4]. Let A0, A1, ... , A,~-I be pairwise skew (k - 1)-dimensional subspaces of PG(n - 1, qm) which span PG(n - 1, qm) and are conjugate over GF(q) with respect to GF(q m) (that is Ai = Ag ~, where a denotes the collineation of PG(n - 1, qm) arising from the Frobenius automorphism GF(q m) -+ GF(qm), x F-~ xq). For any point Po c A0 set Pi = Pff'. Then P0,..-, P,~-I are conjugate points over GF(q) with respect to GF(qm). ~rthermore, Po,..., Pro-1 span an (m - 1)-dimensional subspace H of PG(n - 1, q'~) intersecting PG(n - 1, q) in an (m - 1)-dimensional subspace II. The set of all such subspaces 1I is a regular (m - 1)-spread in PG(n - 1, q). Research performed within the activity of G.N.S.A.G.A. of C.N.R. with the support of the Italian Ministry for Research and Technology