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 / =
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)= 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 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 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( 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. Then Y G Sk(V). In such a case /(Y1) = Y1 and by (o), f\p = Now let Yi be an arbitrary element of ). Choose YQ G Sk-\{W) arbitrarily and let q = As a consequence of the above G f2(k). Since [0,X\]k is a projective (sub)space of Pfc(V), there exists a sequence <7i,...,<7„ of elements of A4(k) such that for h = gn o ... o g1 it holds dm(h) = p, rg(h) = q and dm(<7j) C Sk(X 1) for i = 1,.. .n. Note that / is a collineation from Pfc(Xi) onto Pfc(X2), so h* G f2(k) and then 1 f ftp = (h~ ) oflqohei2(k). (ii) is a direct consequence of (i). • Finally, we come to the main theorem of this section - linear maps de- termine projective collineations. Corollary 2.4. PG(k) C Aut(Pk(V))n{k). Proof. Let / G PG(k). Let p = p(Xi,X2) G Vk(V). Then / = ip*, for some 3. Projectivities determined by collineations In this section we shall be concerned with collineations of Pfc(V) deter- mined by (projective) collineations of Pi(V). Lemma 3.1. Let p{ = p([//,i7/') G PFC(V), / = G M(k), then there exists a linear map (p: U" —• such that the following holds f(U[+ Proof. By the assumption we must consider two cases (i) U[ = U!yCZ and U[' C W' + Z or (ii) Z C U{' = UV and U'2 n Z C U[. Assume (i). Consider any Y € S\(Z) such that Y n U[ = 6. Set g = {v(i; by 2.3(o), g £ 0(k, 1) and / = By 2.2 from [3], there exists a linear monomorphism f{U[+ Now assume (ii). We can also assume U[ / U2. Set W = U[ fl Z, then W = U2 n Z, and dim(H') = k - 2. Consider any Q £ 52(V) with W © Q = Z, then U" = U{ © Q = U2®Q. Thus there are vectors e\,e2 such that U[ = W© < e; >; in particular, i/j" = t^' = Zffi < e, > for i =1,2. Every u € i/" \ i/j' can be uniquely decomposed u = u; + fia + x, where w £ IV, i € <5> so for X\ = u[+ < u >£ pi we have A'i = W+ < ei > + < x >. Set X2 = W+ < e > > + < x >, then U2 C X2 C U2 , so we get A'2€p2, and A",, A'2, Z £ p( W+ < x >, W+ < ex, e2 > + < x >), thus /(A'i) = A'o. Consider the linear map if defined on U" = U" by the conditions (p^z = id, v?(ei) = e2; from the above f(U[ + ) = U2 + < LEMMA 3.2. Let p, = p(t/,',t/,") £ Vk(y), f = \\\ € M{k), then there exists a linear map ift:U"/U[ —• U2/U2 such that for any X £ p\ holds 1>(X/U[) = f(X)/Ui. Proof. It follows immediately from the above. Indeed, just consider a linear map LEMMA 3.3. Let p, = p(Ui¡,U[')€ Vk(V), /£ f2(k), dm / = pu rg/ = Pi, then there exists a linear map f(U[+ THEOREM 3.4. QPG(V,k)n(k) Q PG{k). Proof. Let / G QPG(V, k)n(k), so / = 4. Collineations induced by correlations Now we pass to the case dim V = 2k, for some k > 2. Then «¿(¿¡¿(V)) = 5fc(V) for every nondegenerate sesquilinear form Set / = re^^^. Note that a pencil p = p(U{, U[') G Vk(V) is mapped onto f(p) = p(U^U!{) U where U'2 = 2 = thus / G Aui(Pfc(V)). THEOREM 4.1. Let£ be a nondegenerate sesquilinear form, f — KtjSkW If f\p G i2(k) for some p G "Pfc(V) then £ is bilinear and the field J is com- mutative. Proof. Let £ be over an antiautomorphism \i of Consider, by Lemma 3.3, a linear map if) such that f{U[+ In view of 4.1 if we consider A(k)p^k) / 0 we can restrict ourselves to the case of J being commutative and the forms in questions being bilinear. From now on we adopt the above assumptions. THEOREM 4.2. LEI dim V = 2k, then there exist: a nondegenerate bilinear form a linear map (p and p 6 7\(V) such that Proof. Let e\,..., ek, e^+i,... ,e2k be a basis of V. Set U[ =< ci,.. .,ek-i >, U" =< ei,. ..,ek,ek+i >, then pi = p(U[,U") £ Vk(V). Let £ be a form defined in the above basis by the condition for i = j otherwise for 1, j = 1,..., 2k. Then U2 := *i(i/i) =< ek, ek+i,. ,.,e2k>, U'2 := Kt(U[') =< ek+2,...,e2k > p2 := K(PI) = p(U2,U2). Let X\ €pi, then X\ = U[-\- < eka + ek+i0 > for some a, p. Clearly X2 = € p2, so .Y2 = Uo+ < eki + ek+i6 > for some 7,<5. It holds {(Xi,X2) = 0, thus eka + ek+1/3,eky + ek+1S) = 0. This gives a~/ + 06 = 0. Up to proportionality this gives: 7 = ¡3,6 = —a. Now consider a linear map <¿5 given by the conditions ¥>*(*!) = f'{U[+ < eka + ek+1/3 >) = < -ek+1a + ek(3 > = U2 + < e*7 + ek+1S >= X2. Thus = m THEOREM 4.3. If dim V = 2k then (i) there exist a bilinear form f and a pencil p € Vk(V) such that € i?(Jb); (ii) there exists a form £ such that for / = the following holds f € AutPk{V)n{k) and f € AL{k). Proof, (i) follows from 4.2 and 2.4; (ii) follows from (i) and 2.2. • Projections in spaces of pencils 833 Clearly, the following is true. If /i, /2 G AL(k), g G PG(k) then /2 o fr G PG(k) and 5/1,/iff 6 AL(k). From this we obtain Lemma 4.4. Let f — K^^y). Assume that f G AuiP^V)^/.) and / £ AL(k). Let tj be a sesquilinear form. Set g = ^j^s,, (v) • Then g G AL(k) iff fogePG(k). As a corollary we obtain Theorem 4.5. = AL(k). Proof. Consider / = KifS)t(V) as 3-3(ii). Let g G AL(k). Clearly g e A(k)-, moreover gf € PG(V), so gf G Aut{Pfe(V))i?(fc). From / G ¿^(P^V))^ we deduce g G A«i(Pfc(V))n(fc). Thus g G ¿(*)n(fc)- Now let g G ¿(fc)n(k)> then G Aut(Pk(y))n{k), fg G Atti(Pfc(V))0(fc) and fg G PG(V). Thus g G AL(fc). • Theorems 4.5, 3.4 and 2.4 can be summed up in the following complete analytical characterization of the group of projective collineations of P^(V). a am _jPG(V,h;) if 2k ^ dim V Aut{fk{V))n(k)- <^pG(y,k)[jAL(V,k) otherwise. References [1] E. Artin, Geometric Algebra, New York: Interscience 1957. [2] J. Dieudonne, La geometrie des groupes classiques, Berlin, Springer, 1971. [3] H. Oryszczyszyn, K. Prazmowski, On projections in projective spaces, Demon- stratio Math. 31 (1998), 193-202. [4] G. Tallini, Partial Line Spaces and Algebraic Varieties, Symp. Math. 28 (Roma 1983), 203-217. INSTITUTE OF MATHEMATICS UNIVERSITY IN BIALYSTOK ul. Akademicka 2 15-267 BIALYSTOK, POLAND Received November 17, 1997.