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Henryk Oryszczyszyn, Krzysztof Prazmowski on PROJECTIONS IN DEMONSTRATIO MATHEMATICA Vol. XXXI No 4 1998 Henryk Oryszczyszyn, Krzysztof Prazmowski ON PROJECTIONS IN SPACES OF PENCILS Abstract. In this paper we continue investigations of projections in line spaces de- fined over a vector space. The aim of the paper is to characterize projectivities and pro- jective collineations in spaces of pencils. The main result is an expected one-projective collineations are those determined by "linear" transformations over underlying vector space. Basic notions, definitions and results are taken from [3]. Introduction The aim of this paper is to characterize the group of projective collinea- tions of the space Pfc(V) of /¡¡-pencils defined over a vector space V. For k — 1 i.e. for projective spaces the situation is quite clear. While arbitrary collineation is determined by a semilinear bijection of V, a linear bijection cp corresponds to a projective collineation / = <p*, i.e. to a collineation / such that every restriction of / to a line is a composition of projections. This old and classical result can be considered both as a characterization of projective collineations (defined geometrically), and of "linear" collineations, defined analytically. Note that it makes sense to define and investigate projections in ar- bitrary (even partial) line space 0; in a consequence, we can investigate projective collineations of 0. In the paper we are mainly concerned with a particular case of (5 = Pfc(V), for arbitrary (finite) k. It turns out that the situation is similar as in the case of k = l.A general analytical characteriza- tion of collineations is known. The only complication is that, besides those determined by semilinear bijections, for 2k = dim(V) the space Pfc(V) may contain a collineation determined by a sesquilinear form £ on V (i.e. by a correlation in Pi(V)). Then the class of projective collineations coincides with the family of those, which are determined by linear bijections ip (and bilinear forms £). It is worthwhile to note that the result, though quite natural, is not quite evident. Let us mention two examples. Among partial line spaces there 826 H. Oryszczyszyn, K. Prazmowski are also Grassmann spaces Gfc,*+i(V) = (Subk(V), Subk+\(V), C). It is seen that both (5 = G^+^V) and Pi (V) have essentially the same (analytically) collineations, so one can define linear collineations of On the other hand, for k > 1,0 contains no projection (as a total map from a line to a line), so it looses sense to consider projective collineations, or the definition of a projection must be complicated. Another example is a multiprojective space (the Segre product of pro- jective spaces). In such a case the classical projections exist, but we must generalize the notion of projection (in comparison with the one adopted in the paper), since otherwise identity would be the only one projective collineation of such a structure. 1. Basic definitions and preliminary results As previously, let V be a vector space over a field J with J Z2, let 0 < it < dimV and let 9Jt = P*(V) be the space of ¿-pencils. For WUW2 € S(V) we set [WuW2]k := [WUW2] D Sk(V). We write p{WuW2) for [Wi,W2)k if VFi C W2, dimWi = jfc - 1, dimW2 = k + 1 and call the set p(Wi, W2) a ¿-pencil. Then^(V)= {p^,!^) : WUW2 € S(V)}and the space P/t(V) of pencils is the structure P*(V)= <Sfc(V),7\(V)). The space of pencils Pjt(V), as defined above, is sometimes called a Grass- mann space (more precisely: a Grassmann space representing k— 1-dimensio- nal subspaces of the projective space Pi(V), see [4]). In the sequel we shall investigate projective collineations of P^(V). Let us begin with some general commonly known facts: • For every semilinear bijection <p of V the map if* defined by <pm{U) = <p(U) = {tp(u):ueU} a maps Sk(V) onto itself and preserves ¿-pencils; thus V*st(v) collineation ofPit(V). • Every nondegenerate sesquilinear form £ on V determines a correlation : 5(V) — 5(V) defined by v e Ki(c/)iff t(v,u) = o. Then dim(£/) = codim(K^(U)), so if dim(V) = 2k then «{^(y) a collinea- tion of Pjt(V). Projections in spaces of pencils 827 We define Clearly AL{k) C A(k) and PG{k) C QPG(k). The following is known (see [2], Ch III, §2, [1]): every collineation of Pjt(V) is determined by a collineation of Pi(V) or by a correlation of Pi(V): if 2k ± dimV U A(V ,k) otherwise. Note, that Pfc(V) is a partial line space. In an arbitrary partial line space (S = (5; C) one can investigate relations £(mi,a, m2) defined for a € 5, mi, 7772 € C, a g mi U m-i as follows: x(£(mi,a,m2))y iff x (E mi, y G mi, x, a, y 6 m for some m £ C. Then £(mi,a,m2) C m\ x m^ C 5x5. Such relation is called a. projection if it is a bijection with domain mi and range m2- If £(mi, a, m2) is a projection = then we write {^(z) V instead of z(£(mi, a, m2))j/. Let be the set of all projections in <3 and A(<3) = {fj ° •.. ° /i: /, € S((5)} be the set of all sensible compositions of projections in (9. Then is a concrete category with objects in C. THEOREM 1.1 (Bachmann). If for every m\,rri2E£ there exists g £ A{<£>) with dm(g) = mi, rg(g) = then the following conditions are equivalent for f e Aut(<3): (i) f E Aut(<3)A(v), (ii) there exists k 6 C such that f^ (E Proof. Of course A(<&) is invariant under Aut(<5). Thus the thesis fol- lows from general Bachmann theorem (1.1 in [3]). • Let Pi = p(C//, un e 7MV), Z e Sk(V). If (*) u[ = U'2CZ, U[' CU'l + Z or ZC U[' = U^ U'2 n Z C U[, then £(pi,Z,p2) is a projection in P^(V). 828 H. Oryszczyszyn, K. Prazmowski are Let us recall that if , W2 subspaces of V such that dim W\ = k - 1 and dim W2 = k + 1 then the sets [V^i, V]jt and [0, are strong subspaces of Pfc(V), hence they are projective spaces. Under assumption of (*), in the first case the set p\ U P2 U {Zj is a subset of [U'2, U2 +Z]k, which is a plane in [U2,V]k- the second case p\ Up2 U {Z} C [Z fl U^U^k which is a plane in [0, U2]k- In each of the above cases the projection £(pi, Z,p2) is a com- monly known projection in the corresponding projective space ([t^Vjk or = [0, U2]k)- As we said, if £(pi, P2) is a projection then we write f instead of x(£(pi, Z, P2)y- Let M{k) be the set of all projections fz^ with Pi, Z, P2 satisfying the condition (*). Set n(k) = {fmo...of0-.f,eM(k), dm(/,) = TgUi-i) for 1 = 1 ... m}. One can see that S7(k) is a concrete category with objects from Vk(V). The aim of this paper is to characterize the group Aut(Pk(V))n(k) of projective collineations of Pjt(V). 2. Projectivities in spaces of pencils LEMMA 2.1. For arbitrary p^,p2 € Vk(V) there exists g € Q{k) such that dm{g) = pu rg(g) = p2. Proof. Let p, = p(i//, i/t") for i = 1,2. Then <E Sk+i(V). By / 3.8 of [3] there exists a sequence yo>--->} n such that Yo = UYn = U2 and dim(yi_i D Y. ) = k for i = 1,..., n. Let X{ £ Sk-i(Yi-i n V.) for i = 1,..., 71. Set Xo = U[, Xn+\ — U'2. We consider the following sequence of pencils: 92i+i = p(X,, yt), q2i+2 = p(A"i+i,Yi) for i = 0,.. .,n; note that <7, = p\, q2n+2 = P2- Each two consecutive pencils of the above sequence are contained in a strong subspace of P*(V). Indeed, 92.+1 U92.+2 C [0,y,]fc and q2i+2 U q2i+3 C [Xi+i,V]*. Hence for j = 1,.. .,2n + l, there exist gj £ fl(k) with dm(<7j) = qj, rg(</j) = qj+1. The map g = g2n+\ o ... o ji is a required map. • THEOREM 2.2. Let f E j4ui(Pfc(V)). If f[p e /?(*) for some p e Vk(V) then f G ¿u<(Pfc(V))„(Jt). Proof. The thesis follows from 1.1 and 2.1. • LEMMA 2.3. (O) Let u[', UZ e Sk+i(V), u[ nu? = we sk(V), u e St_i(W); Projections in spaces of pencils 829 consider v G (U{' + \ (U{' U and put Z =< v >, Y = U + Z, Pi = P(U,U('). Setf = \r\,g= f^. ThenfeM(k),gee(k,l,k+l), and g^Pl = /. (i) If f £ £{k, l,k+l),p£ Vk(V), and p Cdm(f), then /rp € i2(k). (ii) If f £ A(k, 1, k + 1), p Cdm(f), then /rp G Q(k). Proof, (o) follows immediately from linear algebra and the definitions of corresponding projections, (i) By the definition, / = }V, where dimXi = k + 1 = dim X2, dim Z = 1, and X\ + Z = X\ © Z = Xi © Z. By the as- sumption, p = p(Yi,Xi), for some Y\ G Sk-i(Xi). Clearly f{p) = p(/(Yi), f(X 1)). Denote W := X1 n X2; W G Sk(V). Assume first that W G p, so Yi C W; set Y = Yl + Z.
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