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Partitioning projective geometries into Segre varieties. (English) Zbl 0880.51013

Let \(m,k \in\mathbb{N} \backslash \{1\}\), \(n: =m \cdot k\), \(\Pi\) a projective space with \(\dim\Pi=n-1\) and \({\mathcal F}\) a spread of \(\Pi\) consisting of \((m-1)\)-dimensional projective subspaces of \(\Pi\). \({\mathcal F}\) is called regular (of rank \(k-1)\) if for each \((k-1)\)-dim. subspace \(\Lambda\) of \(\Pi\) meeting each member of \({\mathcal F}\) in at most one point, the set \([\Lambda]: =\cup \{X\in {\mathcal F} \mid X\cap \Lambda \neq\emptyset\}\) is an \((m-1)\)-regulus of rank \(k-1\). If \(H,K,L\) are fields with \(H\leq K\leq L\) and \([L:K] =k\), \([K:H] =m\) hence \([L:H] =m \cdot k=n\), if \(\Pi: =L^x/H^x\) is the projective space corresponding to the vector space \((L,H)\) and \({\mathcal F}: =\{a\cdot K^x/H^x \mid a\in L^x: =L\backslash \{0\}\}\) the projective subspaces corresponding to the vector subspaces \(\{a \cdot K\mid a\in L^x\}\) then \({\mathcal F}\) is a regular spread of rank \(k-1\).
The authors consider finite projective spaces, and their results can be obtained from these facts on projective incidence groups:
1. \(PG(n-1,q)\) has a regular \((m-1)\)-spread.
2. and 3. If \((m,k) =1\), \(L: =GF(q^n)\), \(K_1: =GF(q^m)\), \(K_2: =GF(q^k)\), \(H:= GF(q)\), \(\Pi: =L^x/H^x\), \(\Pi_i: =K^x_i/H^x\) then \({\mathcal F}_i: =\{a \cdot \Pi_i\mid a\in H\}\) are regular spreads and:
(i) \(\forall X_i\in {\mathcal F}_i: |X_1 \cap X_2 |\leq 1\) (ii) \(\forall a\in \Pi\), \(a\cdot \Pi_1\cdot \Pi_2\) is a Segre variety \(SV_{m,k}\) and \({\mathcal P}: =\{a \cdot \Pi_1 \cdot \Pi_2 \mid a\in \Pi\}\) is a partition of \(\Pi\) with \(|{\mathcal P} |= {(q^n-1) (q-1) \over (q^m-1) (q^k -1)}\).
4. and 5. For \((m,k)\neq 1\) a partition of \(\Pi\) into Segre varieties \(SV_{m,k}\) is not possible.

MSC:

51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
51A05 General theory of linear incidence geometry and projective geometries
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References:

[1] L.R.A. Casse, C.M. O’Keefe,Indicator Sets for t-Spreads of PG((s+1)(t+1)-1,q)and regularity, Boll. Un. Mat. Ital. B (7) 4 (1990), 13-33. · Zbl 0702.51007
[2] L.R.A. Casse, C.M. O’Keefe,t-spreads of PG(n, q)and regularity, Note Mat. 13 (1993), 1-11 · Zbl 0828.51004
[3] J.W.P. Hirschfeld, ?Projective Geometries over Finite Fields,? Oxford University Press, Oxford 1979. · Zbl 0418.51002
[4] B. Segre,Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl. 64 (1964), 1-76. · Zbl 0128.15002 · doi:10.1007/BF02410047
[5] C.-T. Yang,A theorem in finite projective geometry, Bull. Amer. Math. Soc. 55 (1949) 930-933 · Zbl 0041.47110 · doi:10.1090/S0002-9904-1949-09311-4
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