COLLINEATIONS OF PROJECTIVE PLANES

Transcription

1 COLLINEATIONS OF PROJECTIVE PLANES TIMOTHY VIS 1. Fixed Structures The study of geometry has three main streams: (1) Incidence Geometry: the study of what points are incident with what lines and what can be derived about a geometry. (2) Analytic Geometry: the study of coordinates of a geometry and the algebra derived, as well as geometric properties involving this algebra. (3) Transformational Geometry: the study of collineations of a geometry and the way structures can be moved around. All three areas have been covered so far in class. Incidence Geometry was used to give axiomatic descriptions of projective planes and projective spaces, and to prove results such as the fact that all lines in a finite projective space are incident with a constant number of points. Analytic Geometry was used in our study of field planes and in the development of coordinates when we coordinatized an arbitrary plane. Transformational Geometry was used when we developed the Fundamental Theorem of Field Planes. We are now going to study Transformational Geometry in an arbitrary plane and see what we can say about collineations in general. Definition 1. A collineation of a projective plane π is a bijective map on the points such that the images of collinear points are collinear. That is, P, Q, R are collinear if and only if P σ, Q σ, R σ are collinear. Given a line l and points P, Q on l, we define l σ to be the line P σ Q σ. One of the most fundamental properties studied about collineations is the property of which points and lines are mapped to themselves. Definition 2. If σ is a collineation of a projective plane π, (1) A point P is a fixed point if and only if P σ = P. (2) A line l is a fixed line if and only if l σ = l. (3) A line l is fixed pointwise if and only if P σ = P for all points P l. (4) A point P is fixed linewise if and only if l σ = l for all lines l containing P. Obviously any line that is fixed pointwise is a fixed line and any point that is fixed linewise is a fixed point, but the converse is not necessarily true. Example 1. Consider the collineation of the Fano plane shown in Figure 1. Notice that the points A, F, and G are fixed and that the lines ABC, ADE, and AFG are fixed. Since all points on the line AFG are fixed, AFG is fixed pointwise, and since all lines through A are fixed, A is fixed linewise. Date: April 23, 29. 1

2 2 TIMOTHY VIS A A B D F C E F C E G B D G Figure 1 We ll focus somewhat on the applications of these ideas to field planes; however they apply equally well to arbitrary projective planes. Recall the Fundamental Theorem of Field Planes, which states that every collineation of a field plane is the product of an automorphic collineation and a homography. That is, if σ is a collineation, we have σ : X AX α where α is an automorphism of the underlying field. We also have for lines σ : L L α A 1. Suppose that α is the identity collineation. We then have X σ = AX and L σ = LA 1 or (L σ ) T = ( A 1) T L T. If a point X is fixed by σ, we then have AX = λx, while if a line L is fixed by σ, we have ( A 1) T L T = µl T. Thus the eigenvectors of A are the fixed points of σ, while the eigenvectors of ( A 1) T are the fixed lines of σ. Example 2. Consider the homography given by A = a 1 where a F \ {, 1}. The fixed points are the eigenvectors of A, while the fixed lines are the eigenvectors of ( A 1 ) 1 T a = 1. To find the eigenvectors of A, we determine the characteristic polynomial of A. λ a λi A = λ 1 λ 1 = (λ a)(λ 1) 2.

3 COLLINEATIONS OF PROJECTIVE PLANES 3 Thus A has eigenvalues a and 1. Corresponding to a are eigenvectors determined as vectors in the null space of ai A. That is, a 1 x y = (a 1)y =. a 1 z (a 1)z Now since a 1, we must have both y = and z =. Thus, the only fixed point corresponding to the eigenvalue a is (1,, ). Corresponding to 1 are eigenvectors determined as vectors in the null space of I A. That is, 1 a x y z = (1 a)x = Again, since a 1, we must have x =. Thus, the fixed points corresponding to the eigenvalue 1 are all points of the form (, y, z). On the other hand, notice that A 1, being diagonal, is self-transpose. So ( A 1) T has the same eigenvectors as A 1, which are precisely the eigenvectors of A. Thus, the fixed lines have the same coordinates as the fixed points: [1,, ] and [, y, z]. Now notice that every line through (1,, ) is of the form [, y, z], and each of these lines are fixed, so that (1,, ) is fixed linewise. Similarly, every point on [1,, ] is of the form (, y, z), and each of these points are fixed, so that [1,, ] is fixed pointwise. We now prove some results regarding what possibilities exist for fixed points and lines. We shall assume the following easily verified facts (which you prove in your homework): Proposition 3. Given a collineation σ, (1) If P and Q are distinct fixed points of σ then PQ is a fixed line of σ. (2) If l and m are distinct fixed lines of σ then l m is a fixed point of σ. These results allow us to prove several other results. Proposition 4. Let σ be a collineation of π fixing l pointwise. If distinct points P and Q in π \ l are fixed by σ, then σ is the identity collineation (σ fixes all points of π. Proof. Suppose R is some point of π that does not lie on either l or PQ. So PR and QR are distinct lines through R meeting l in points P and Q respectively. But PR = PP and since both P and P are fixed, PR is fixed. Similarly, QR is fixed. But then PR QR = R is fixed. Now let S be a point of PQ other than P and PQ l. Let R be any point on neither l nor PQ. Then P, R, S are non-collinear points, and both P and R are fixed. By the last argument then, S is also fixed. So every point of π is fixed, and σ is the identity collineation. Collineations that fix a line pointwise or a point linewise play a central role in the study of projective planes. One of the most fundamental results regarding such collineations is the following theorem that states that any collineation fixing a line pointwise fixes a point linewise. The converse of this statement is, of course, also true by the principle of duality..

4 4 TIMOTHY VIS Theorem 5. If σ is a collineation of π fixing l pointwise, σ fixes some point V linewise. Proof. If σ fixes a point V not on l let m be a line through V and let m l = W. Then m = V W, and since both V and W are fixed, m is also fixed. So V is fixed linewise. Suppose then that no point of π \ l is fixed by σ. Let P be a point of π \ l and consider the line PP σ. This line intersects l in a point V. So PV = P σ V = (PV ) σ and is fixed by σ. Let Q be any point of π \ (l PV ). In the same manner as for P, QQ σ is a fixed line. So PP σ QQ σ is a fixed point and must then lie on l. So QQ σ l = V as well. Now consider any line m through V and let R be any point on this line. Then RR σ is a fixed line through V, so that RR σ = m and m is fixed. So V is fixed linewise. Corollary 6. If σ is a collineation of π fixing V linewise, σ fixes some line l pointwise. Proof. This is the dual to Theorem 5. With these results in hand, we are ready to make several definitions. Definition 7. A collineation σ fixing a point V linewise and a line l pointwise is called a (V, l)-perspectivity or central collineation. The point V is called the center of σ and the line l is called the axis of σ. If V l, σ is called an elation, while if V l, σ is called a homology. Example 3. Two of the most familiar motions in the Euclidean plane give us examples of central collineations of the Extended Euclidean Plane. Consider a translation of the Euclidean plane. No points of the Euclidean plane are fixed, but a translation fixes the slopes of all lines in the Euclidean plane. These slopes correspond exactly to the points at infinity, so that every point at infinity is fixed by a translation, and thus, l is the axis of every translation. Furthermore, the lines parallel to the direction of the translation stay in place, so that these lines are fixed. But then every line through the point at infinity corresponding to this slope is fixed, so that this point is the center of the translation. Since this point is on l, a translation is an elation with axis l. Now consider a reflection over the line l of the Euclidean plane. Every point on l is fixed by this reflection, so l is the axis of the reflection. The only lines that are fixed by a reflection, however, are the lines perpendicular to l, so the point at infinity corresponding to the slope perpendicular to l is the center of the reflection. So a reflection is a homology of the Extended Euclidean Plane. Example 4. The homography determined by the matrix a 1 was shown to have the point (1,, ) fixed linewise and the line [1,, ] fixed pointwise. Since (1,, ) does not lie on the line [1,, ], this homography defines a homology.

5 COLLINEATIONS OF PROJECTIVE PLANES 5 Example 5. Consider the homography determined by the matrix A and its inverse transpose ( A 1) T as given. A = 1 1 ( 1 1 A 1 ) T = The characteristic polynomial of both matrices is (λ 1) 3 and the only eigenvalue of either is 1. A has eigenvectors (x, y, ) and ( A 1) T has eigenvectors [x, x, z] Thus, the fixed points are the points of the form (x, y, ), while the fixed lines are the lines of the form [x, x, z]. Notice that the fixed points all lie on the line [,, 1] so that this line is fixed pointwise. Similarly, the fixed lines all contain the point (1, 1, ). Since (1, 1, ) [,, 1], this homography defines an elation. A significant property of central collineations is that they are very easily determined. Theorem 8. A central collineation is completely and uniquely determined by its axis l, center V, and the image P σ of any point P distinct from V not lying on l. Proof. Let R be any point of π \ (l PV ) and let RP l = W and RV = m. Then R = PW m so that R σ = (PW m) σ = P σ W σ m σ = P σ W m. Thus, R σ is completely and uniquely determined. Now let S PV and let R π \ (l PV ). So S π \ (l RV ) and since R σ is completely and uniquely determined, S σ is completely and uniquely determined using V, l, and the image R σ of R as argued when determining R σ. A word of warning here: this theorem applies only when a central collineation actually exists with the appropriate property. There may not be any collineations (other than the identity) with a particular axis, with a particular center, with a particular axis, center pair, or mapping a given point to any other. This theorem does not state that such a collineation exists; it only states that when one exists, it is unique. In order to show that one exists, we need particular properties of a plane. For example, in a field plane, every possible central collineation exists, although we will not prove this. Proposition 9. If σ is a (V, l)-perspectivity and τ a collineation, then τ 1 στ is a (V τ, l τ )-perspectivity. Proof. Let m be a line through V τ. Then m τ 1 necessarily contains V and is fixed by σ. m τ 1 στ = ((m τ 1) σ) τ = (m τ 1) τ = m.

6 6 TIMOTHY VIS So m is fixed by τ 1 στ and V τ is a center of τ 1 στ. Similarly, if P is a point on l τ, P τ 1 lies on l and is fixed by σ. P τ 1 στ = ((P τ 1) σ) τ = (P τ 1) τ = P. So P is fixed by τ 1 στ and l τ is an axis of τ 1 στ. So τ 1 στ is a (V τ, l τ )- perspectivity. Example 6. Suppose in a field plane there is a ((1,, ), [1,, ])-perspectivity σ such that (1, 1, 1) σ = (1, b, b). What is the equation of this perspectivity? We know that as a collineation, σ must be the product of an automorphic collineation and a homography. We also know that every automorphic collineation fixes every point of the fundamental quadrangle. Notice now that the points of the triangle of reference are all fixed by σ, and thus by the associated homography A ((1,, ) is the center and both (, 1, ) and (,, 1) are on the axis). Thus, a a 1 a 2 a 1 a 11 a 12 a 2 a 21 a = ρ x ρ y ρ z Since (1, 1, 1) is fixed by automorphic collineations, we must have (up to scalar multiples) ρ x = 1, ρ y = ρ z = b. So the homography of this collineation is A = 1 b. b Now notice that for any point X on [1,, ], X = (, y, z). But AX = bx then, so that X is fixed by A. Further, notice that for any line L through (1,, ), L = [, y, z] and LA 1 = 1 bl, so that L is fixed by A. So A is the desired ((1,, ),[1,, ])-perspectivity. In the last example, we did not need any automorphic collineation to obtain the appropriate perspectivity. In fact, in a finite field plane, every (V, l)-perspectivity is a homography; that is, every (V, l)-perspectivity has the identity as associated automorphic collineation. Theorem 1. Every (V, l)-perspectivity of a finite field plane is a homography. That is, every (V, l)-perspectivity of a finite field plane has the identity as the associated automorphic collineation. Proof. By Proposition 9 and the transitivity of homographies on ordered quadrangles in PG(2, q), we need only show the result holds for all perspectivities with a given axis. Other axes are then obtained by conjugation by an appropriate homography. Suppose then that l = [1,, ]. Let α be the associated automorphic collineation and A = (a ij ) be the associated homography, and let σ be the collineation in question. Now since both (, 1, ) and (,, 1) are fixed both by σ and α, we know that a 1 = a 2 = a 12 = a 21 =. Let a 11 = ρ y and let a 22 = ρ z. Now consider (, 1, u) σ..

7 COLLINEATIONS OF PROJECTIVE PLANES 7 Since this point lies on the axis, it must be fixed. But (, 1, u) σ = A(, 1, u α ) = (, ρ y, u α ρ z ). It follows that u = u αρ z ρ y = u αρ z ρ y u for all values of u in GF (q). If q = p h, we need p h 1 zeroes to a polynomial of degree α. But α p h 1, a contradiction unless u α ρz ρ y u is the zero polynomial. But this can only be the zero polynomial if α = 1, so we necessarily have α = 1, and σ is a homography.