The geometry of projective space

Chapter 1 The geometry of projective space 1.1 Projective spaces Definition. A vector subspace of a vector space V is a non-empty subset U V which is closed under addition and scalar multiplication. In this case, we write U V. Definition. The projective space of a vector space V is the set P(V ) := {[v]: v V \ {0}} = {1-dimensional subspaces of V }. 1.2 Bases and homogeneous coordinates Definition. The dimension of the projective space P(V ), denoted dim P(V ), is given by dim P(V ) = dim V 1. 1.3 Projective linear subspaces Definition. X P(V ) is a projective linear subspace (or just subspace) of P(V ) if it is of the form X = P(U) for some vector subspace U V. In this case, we write X P(V ) and, as usual, set dim X = dim U 1. Proposition 1.1. For U 1, U 2 V, dim(u 1 + U 2 ) = dim U 1 + dim U 2 dim(u 1 U 2 ). Lemma 1.2. If U V then dim U dim V with equality if and only if U = V. Definition. Let X 1 = P(U 1 ), X 2 = P(U 2 ) P(V ). 1. The join of X 1 and X 2 is X 1 X 2 := P(U 1 + U 2 ). 2. The intersection of X 1 and X 2 is just the usual set-theoretic intersection X 1 X 2. We say that X 1 and X 2 intersect if X 1 X 2 (equivalently, U 1 U 2 {0}). Lemma 1.3. If X 1 X 2 then dim X 1 dim X 2 with equality if and only if X 1 = X 2. 1

Theorem 1.4 (Dimension Formula). If X 1, X 2 P(V ) then Theorem 1.5. Let P(V ) be a projective space. dim X 1 X 2 = dim X 1 + dim X 2 dim(x 1 X 2 ). (1.1) 1. There is a unique projective line through any two distinct points of P(V ). 2. If P(V ) is a plane (thus dim P(V ) = 2), any pair of distinct lines in P(V ) intersect in a unique point. 1.4 Affine space and the hyperplane at infinity Proposition 1.6. If V has basis v 0,..., v n then the map φ 0 : F n P(V ) \ {[v]: λ 0 = 0} (x 1,..., x n ) [1, x 1,..., x n ] is a bijection. Definition. An affine line l in F n is a subset of the form l = {x + tv : t F}, for fixed x F n and v F n \ {0}. 1.5 Projective transformations Definition. A linear map (or linear transformation) T : V W of vector spaces is a map with for all v, v 1, v 2 V and λ F. The kernel of T is ker T := {v V : T v = 0} V. The image of T is Im T = {T v : v V } W. T (v 1 + v 2 ) = T v 1 + T v 2 T (λv) = λt v, Definition. A projective map is a map τ : P(V ) \ P(U) P(W ), where P(U) P(V ), of the form for some linear map T : V W with ker T = U. τ[v] = [T v], (1.2) A projective transformation is a projective map τ : P(V ) P(W ) of the form (1.2) with T injective. In particular, if dim P(V ) = dim P(W ) then T, and so τ, is invertible. Moreover, τ 1 is then also a projective transformation (induced by T 1 ). 1.6 Points in general position Definition. Let P(V ) be an n-dimensional projective space (so that dim V = n + 1). We say that n + 2 points A 0,..., A n+1 P(V ) are in general position if any n + 1 of them are represented by linearly independent vectors. 2

Lemma 1.7. If A 0,..., A n+1 are in general position in an n-dimensional projective space P(V ) then there are representative vectors v 0,..., v n+1, unique up to a common scalar factor, such that n+1 v i = 0. i=0 Corollary 1.8. If A 0,..., A n+1 are in general position in P(V ), then there is a basis of V with respect to which where the 1 is in the i-th place. A 0 = [1,..., 1] A i = [0,..., 1,..., 0], i 1, Theorem 1.9. Let dim P(V ) = dim P(W ) = n and let A 0,..., A n+1 and B 0,..., B n+1 be points in general position in P(V ) and P(W ) respectively. Then there is a unique projective transformation τ : P(V ) P(W ) such that τa i = B i, for all 0 i n + 1. 1.7 Two classical theorems Theorem 1.10 (Desargues 1 Theorem). Let P(V ) be a projective space with dim P(V ) 2 and let P, A, A, B, B, C, C be distinct points of P(V ) such that the lines AA, BB, CC are distinct and meet at P. Then the points of intersection Q = BC B C, R = AC A C and S = AB A B are collinear. Theorem 1.11 (Pappus 2 Theorem). Let A, B, C and A, B, C be distinct triples of collinear points in a projective plane P(V ). Then the three points P := AB A B, Q := AC A C and R := BC B C are collinear. 1.8 Projective lines and the cross-ratio Definition. For distinct points A, B, C, D on a projective line, the cross-ratio of A, B, C, D (in that order), written (A, B; C, D), is the scalar x for which τd = [1, x] where τ is the unique projective transformation with τa = [0, 1] τb = [1, 0] τc = [1, 1]. Proposition 1.12. Let P(V ) and P(W ) be projective lines, A, B, C, D distinct points on P(V ) and σ : P(V ) P(W ) a projective transformation. Then (σa, σb; σc, σd) = (A, B; C, D). (1.3) Proposition 1.13. Let A, B, C, D be distinct points on a projective line P(V ) with homogeneous coordinates A = [a 0, a 1 ], B = [b 0, b 1 ] and so on, and inhomogeneous coordinates a = a 1 /a 0, b = b 1 /b 0 and 1 Girard Desargues, 1591 1661, was the inventor of projective geometry 2 Pappus of Alexandria, c. 290 c. 350 3

so on, with respect to some basis of V. Then Here, for example, (a c)(b d) (A, B; C, D) = (a d)(b c) det(a, C) det(b, D) = det(a, D) det(b, C). det(a, C) = a 0 c 0 a 1 c 1. (1.4a) (1.4b) 1.9 Duality Definition. The dual space V of a (finite-dimensional) vector space V is the set V = {f : V F: f is linear}. Definition. (1) For U V, the annihilator U V of U is given by U := {f V : f(u) = 0 for all u U} = {f V : f U = 0}. (2) For W V, the solution set sol(w ) V of W is given by Lemma 1.14. For U V and W V, Proposition 1.15. For U V and W V, Theorem 1.16. The map is a bijection with inverse W sol(w ). Lemma 1.17. U 1 U 2 if and only if U 2 U 1. sol(w ) = {v V : f(v) = 0 for all f W }. U sol(w ) if and only if W U. dim U + dim U = dim V dim W + dim sol(w ) = dim V = dim V. U U {U V } {W V } Definition. For X = P(U) P(V ), the dual subspace to X is X := P(U ) P(V ). (1.5a) (1.5b) Theorem 1.18. The map X X : {X P(V )} {Y P(V )} is a bijection called the duality isomorphism. For all X, X 1, X 2 P(V ), we have: dim X + dim X = dim P(V ) 1. X 1 X 2 if and only if X 2 X 1. (X 1 X 2 ) = X 1 X 2. (X 1 X 2 ) = (X 1 )(X 2 ). Proposition 1.19. The k-dimensional subspaces of P(V ) are all of the form X = {H : X H}, for some X P(V ) with dim X = dim P(V ) k 1. 4

Chapter 2 Quadrics 2.1 Symmetric bilinear forms and quadratic forms Definition. A bilinear form B on a vector space V over a field F is a map B : V V F such that B(λv 1 + v 2, v) = λb(v 1, v) + B(v 2, v) B(v, λv 1 + v 2 ) = λb(v, v 1 ) + B(v, v 2 ), for all v, v 1, v 2 V, λ F. (Thus B is linear in each slot separately.) B is symmetric if B(v, w) = B(w, v), for all v, w V. B is non-degenerate if, whenever B(v, w) = 0, for all w V, v = 0. degenerate. Otherwise, we say that B is Definition. Let B be a non-degenerate, symmetric bilinear form on a vector space V and let U V. The polar U of U with respect to B is given by U := {v V : B(v, u) = 0, for all u U} V. (2.1) Proposition 2.1. Let B be a non-degenerate, symmetric bilinear form on V. Then dim U + dim U = dim V, U 1 U 2 if and only if U 2 U 1, (U 1 + U 2 ) = U 1 U 2, (U 1 U 2 ) = U 1 + U 2, (U ) = U, for all U, U 1, U 2 V. Definition. Let V be a vector space over a field F. A quadratic form on V is a function Q : V F of the form Q(v) = B(v, v), for some symmetric bilinear form B on V. 5

2.2 Quadrics Definition. A quadric is a subset S P(V ) of a projective space of the form where Q is a non-zero quadratic form. S = {[v] P(V ): Q(v) = 0}, We define the dimension of S by dim S := dim P(V ) 1. Definition. Let S P(V ) be a quadric defined by a quadratic form Q with polarisation B so that S = {[v] P(V ): B(v, v) = 0}. Let β : V V be the associated map with β(v)(w) = B(v, w). A = [v] S is a non-singular point of S if β(v) 0 and a singular point otherwise. The set of singular points is called the singular set of S. S is non-singular if all its points are non-singular and singular otherwise. If A S is non-singular, the tangent hyperplane to S at A is the hyperplane A given by A := P([v] ) = P(ker β(v)) = {[w] P(V ): B(v, w) = 0}. Proposition 2.2. Let S P(V ) be a quadric defined by a quadratic form Q with polarisation B. Then S is non-singular if and only if B is non-degenerate. Proposition 2.3. Let S be a quadric on a projective line P(V ). Then one of the following holds: 1. S = 2 and then S is non-singular. We say that S is a point-pair. 2. S = 1 and then S is singular. We say that S is a double point. 3. S = and this case is excluded if F is C or any other algebraically closed field. Lemma 2.4. Let S P(V ) be a non-singular quadric with dim S 1 (so that dim P(V ) 2) and let H P(V ) be a hyperplane. Then S H is a quadric in H. Proposition 2.5. Let S P(V ) be a non-singular quadric with dim S 1 and let H P(V ) be a hyperplane. Then A H S is a singular point of H S if and only if H = A, the tangent hyperplane at A to S. Corollary 2.6. Let S P(V ) be a non-singular quadric with dim S 1 and let H P(V ) be a hyperplane. Then H S is a singular quadric in H if and only if H is a tangent hyperplane to S and, in this case, H S has a unique singular point. 2.3 Conics Definition. A conic is a 1-dimensional quadric, that is, a quadric in a projective plane. Proposition 2.7. If C 1, C 2 are non-empty, non-singular conics in P(R 3 ), there is a projective transformation τ : P(R 3 ) P(R 3 ) with τc 1 = C 2. Proposition 2.8. Let C be a non-empty, non-singular conic in a projective plane P(V ) and L P(V ) a line. Then exactly one of the following holds: 6

1. C L = 2; 2. C L = 1 and L is the tangent line A where A = C L; 3. C L = (this is not possible if F is C or another algebraically closed field). Theorem 2.9. Let C P(V ) be a non-empty, non-singular conic. Then there is a bijection between C and a projective line. Corollary 2.10. Let C P(V ) be a non-empty, non-singular conic, A C and L P(V ) a line with A / L. Then there is a bijection ˆα : C L with X, ˆα(X), A collinear, for each X C. 2.4 Polars in projective geometry Definition. Let S P(V ) be a non-singular quadric with corresponding symmetric bilinear form B. For X = P(U) P(V ), the polar subspace (with respect to S) of X is X := P(U ) where U V is the polar of U with respect to B. Lemma 2.11. Let X, Y P(V ). Then X Y if and only if Y X. Theorem 2.12. Let C be a non-singular conic in a complex projective plane. Then (a) if A C then A is the tangent line to C at A; (b) if A / C, the polar line A meets C at two points whose tangents intersect at A, see Figure??. Lemma 2.13. Let S P(V ) be a non-singular quadric and X P(V ) a projective subspace. Then X S if and only if X X. Corollary 2.14. Let S P(V ) be a non-singular quadric and X P(V ) a projective subspace. If X S then dim X 1 2 dim S. 2.5 Pencils of quadrics Definition. A pencil of quadrics is a projective line in P(S 2 V ). Proposition 2.15. Let A 0, A 1, A 2, A 3 be four points in general position in a projective plane P(V ). Then the set of conics through all four points is a pencil. Definition. The base locus of a pencil of quadrics L P(S 2 V ) is Z := {[v] P(V ): Q(v) = 0, for all [Q] L} = S. Lemma 2.16. For any distinct S 0, S 1 in a pencil, the base locus is given by S 0 S 1. Proposition 2.17. A pencil of quadrics in an n-dimensional projective space P(V ), at least one of which is non-singular, contains at most n + 1 singular quadrics. Corollary 2.18. A pencil of conics, at least one of which is non-singular, contains at most 3 singular conics. Proposition 2.19. Let Q 0, Q 1 be linearly independent, non-degenerate, simultaneously diagonalisable quadratic forms on C 3. Then the pencil of conics through [Q 0 ] and [Q 1 ] has either: or three singular conics, all line pairs, and four points in general position as base locus, S L 7

two singular conics, a line-pair and a double-line, and two points in the base locus. Theorem 2.20. Let Q 0, Q 1 be linearly independent quadratic forms and suppose that there are exactly three singular conics in the pencil through [Q 0 ] and [Q 1 ]. Then, (i) The pencil consists of all conics through four points in general position; (ii) The singular conics are the three line-pairs though the four points; (iii) The vertices of the line-pairs give a basis which simultaneously diagonalises Q 0 and Q 1. 8

Chapter 3 Exterior algebra and the space of lines 3.1 Exterior algebra Definition. Let V be a vector space of dimension n + 1. The exterior algebra of V is a vector space V, with V V, equipped with an associative, unital, bilinear product such that: 1. v v = 0, for all v V. It then follows that : V V V v σ(1) v σ(k) = ( 1) σ v 1 v k, for all v 1,..., v k V and permutations σ Σ k. 2. Let v 0,..., v n be a basis for V. Then a basis of V is given by 1 along with all products v i1 v ik, for {i 1 < < i k } {0,..., n}. In particular, dim V = 2 dim V. The k-vectors k V V are the span of all products of k elements of V. 3.2 Lines and 2-vectors Lemma 3.1. Let U V with basis u 0, u 1. Then v U if and only if u 0 u 1 v = 0. Corollary 3.2. The Klein correspondence k is injective. Definition. a 2 V is decomposable if a = v w, for some v, w V. Proposition 3.3. k : {U V : dim U = 2} K := {[a] P( 2 V ): a is decomposable} is a bijection. Theorem 3.4. a 2 V is decomposable if and only if a a = 0. 3.3 The Klein quadric Theorem 3.5. There is a bijection k : L l between lines L in a 3-dimensional projective space P(V ) and points l in the 4-dimensional quadric K P( 2 V ) given by L = P(U) l = 2 U = [u 0 u 1 ]. 9

Proposition 3.6. Two distinct lines L 1, L 2 P(V ) intersect if and only if the line l 1 l 2 joining the corresponding points l i = k(l i ) of K lies in K. Proposition 3.7. For X P(V ) a point, the set {k(l) : X L} is a plane in K. Definition. A plane in K of the form {k(l): X L}, for a fixed point X P(V ) is called an α-plane. Proposition 3.8. Let Y P(V ) be a plane. Then the set {k(l): L Y } is a plane in K. Definition. A plane in K of the form {k(l): L Y }, for a fixed plane Y P(V ) is called an β-plane. Theorem 3.9. Any plane in K is either an α-plane or a β-plane. 10