4 Projective space and projective varieties One of the motivations for projective geometry is to understand perspective. It is well-known phenomenon that parallel lines such as railway tracks ap- pear to meet at a point, called the vanishing point. It was not until the Renaissance that painters were able to give a geometrically correct represen- tation of objects. The painting below, The delivery of keys by Pietro Perug- ino in the Sistine Chapel (source https://en.wikipedia.org/wiki/File: Entrega_de_las_llaves_a_San_Pedro_(Perugino).jpg) is a good exam- ple of the use of perspective, the parallel lines on the ground converge towards a point near the centre of the painting. Projective geometry gives a nicer theory than affine geometry in many ways, for example, two lines in the projective plane always meet, or more generally two curves defined by degree m and degree n equations, resp., have mn intersection points counted with multiplicity. Projective geometry also has important practical applications, images are combined by applying projective transformations in photo stitching, which is used in panoramic photos, Google Street View and virtual reality. 43 4.1 Basic properties of projective space and projective varieties The definition of projective space is also motivated by the idea that if a camera is at the origin of the co-ordinate system, all points of a ray through the origin will be mapped to the same point of the image, so points of the image correspond to lines through the origin. Definition. The n-dimensional projective space over a field K, denoted by Pn(K) or by Pn if the field is understood, is the set of equivalence classes n+1 of K n f(0; 0;:::; 0)g under the equivalence relation (x0; x1; : : : ; xn) ∼ n (λx0; λx1; : : : ; λxn) for any λ 2 K n f0g. (The points of P correspond to lines through the origin in Kn+1.) n+1 The equivalence class of a point (X0;X1;:::;Xn) 2 K n f(0; 0;:::; 0)g is denoted by (X0 : X1 : ::: : Xn). X0, X1,... Xn are called homogeneous co-ordinates on Pn. Example: (1 : 2 : 3), (2 : 4 : 6) and (−1 : −2 : −3) are the same point in P2, this is a peculiarity of projective space that the co-ordinates of a point are not unique. Two ways of looking at projective space n 1. Let U0 = f(X0 : X1 : ::: : Xn) 2 P j X0 6= 0g.(X0 : X1 : ::: : Xn) = (1 : X1=X0 : ::: : Xn=X0) in U0 and X1=X0, X2=X0,..., Xn=X0 can take arbitrary values in K, so the points of U0 are in bijection with the points of n n n A . The set P n U0 = f(X0 : X1 : ::: : Xn) 2 P j X0 = 0g is clearly a copy of Pn−1. Therefore Pn = An [ Pn−1 as a set. If n = 1, then 1 X1 U0 = f(X0 : X1) 2 P j X0 6= 0g = f 1 : j X1 2 K; X0 6= 0g = f(1 : x) j x 2 Kg; X0 1 1 so we can identify U0 with A via (X0 : X1) $ X1=X0 2 A . 1 1 P n U0 = f(X0 : X1) 2 P j X0 = 0g = f(0 : X1) j X1 2 K n f0gg = f(0 : 1)g; because (0 : X1) = (0 : 1) for any K n f0g. This is the decomposition, P1 = A1 [ P0, as P0 consists of just a single point. We can identify A1 with K and denote the single point of P0 by 1, this way we can identify P1 with K [ f1g. 44 If n = 2, then 2 X1 X2 U0 = f(X0 : X1 : X2) 2 P j X0 6= 0g = f 1 : : j X1;X2 2 K; X0 6= 0g X0 X0 = f(1 : x : y) j x; y 2 Kg 2 2 so we can identify U0 with A via (X0 : X1 : X2) $ (X1=X0;X2=X0) 2 A . The complement of U0 is 2 2 2 P nU0 = f(X0 : X1 : X2) 2 P j X0 = 0g = f(0 : X1 : X2) j (X1;X2) 2 K nf(0; 0)gg; this is a copy of P1. To understand these points, consider the parallel lines x+y = 0 and x+y−2 = 2 0 in A = U0. As x = X1=X0 and y = X2=X0, we can rewrite the equations X X X X in terms of homogeneous co-ordinates as 1 + 2 = 0 and 1 + 2 −2 = 0. X0 X0 X0 X0 After multiplying by X0 we get X1 + X2 = 0 and X1 + X2 − 2X0 = 0. The solutions of this system of linear equations are X0 = 0, X1 = −X2, which correspond to the point (0 : 1 : −1) 2 P1. Therefore these parallel lines in 2 2 A = U0 intersect in P n U0. Any other line parallel to them also contains (0 : 1 : −1). y 0:1:-1 4 H L 2 x+y-2=0 x -4 -2 2 4 -2 x+y=0 -4 Similarly, all lines with direction vector (X1;X2) intersect at (0 : X1 : X2) 2 2 P n U0. 45 n n−1 This is true in general for arbitrary n, the points of P nU0 = P correspond n to directions of lines in A or equivalence classes of parallel lines. (0 : X1 : ::: : Xn) is the point where all the lines with direction vector (X1;X2;:::;Xn) in An meet. (The single point of P0 corresponds to the only line in A1.) For this reason, these points are often called points at infinity, but this does not mean that they are intrinsically different, the disctinction depends on the choice of co-ordinates. The youtube video https://www.youtube.com/watch?v=q3turHmOWq4 (also linked to from the Animations and videos section of the course website) contains a good explanation of the projective plane. 2. The other approach is is to consider the sets n Ui = f(X0 : X1 : ::: : Xn) 2 P j Xi 6= 0g; 0 ≤ i ≤ n. Each one is a copy of An and their union is Pn, since there is no point (0 : 0 : ::: : 0). These n + 1 copies are glued together by identifying their points via rational maps 'ij : Ui 99K Uj so that P 2 Ui is the same n point of P as 'ij(P ) 2 Uj. For example, if n = 1, let t = X1=X0 be the co-ordinate on U0, and u = X0=X1 the co-ordinate on U1. We have the rational maps '01 : U0 99K U1, t 7! 1=t, and '10 : U1 99K U0, u 7! 1=u. A point t 2 U0 is the same point in 1 P as '01(t) 2 U1, whenever '01(t) is defined and similarly, a point u 2 U1 is 1 the same point in P as '10(u) 2 U0, whenever '10(u) is defined. If n = 2, then let x = X1=X0, y = X2=X0 be the co-ordinates on U0, u = X0=X1, v = X2=X1 the co-ordinates on U1 and s = X0=X2, t = X1=X2 the co-ordinates on U2. Now we have u = 1=x and v = y=x, which give the rational map '01 : U0 99K U1,(x; y) 7! (1=x; y=x), so that (x; y) 2 U0 2 and '01(x; y) 2 U1 are the same point in P whenever the latter is defined. The point (2 : 3 : 4) 2 P2 is the same as (1 : 3=2 : 2), (2=3 : 1 : 4=3) or (1=2 : 3=4 : 1), therefore (3=2; 2) 2 U0, (2=3; 4=3) 2 U1 and (1=2; 3=4) 2 U2 correspond to the same point of P2 and they get identified in the gluing process. n In general, if F 2 K[X0;X1;:::;Xn] and P = (X0 : X1 : ::: : Xn) 2 P , F (P ) cannot be defined, since F (λX0; λX1; : : : ; λXn) will give different values for different values of λ 2 K nf0g. However, if F is homogeneous of degree d, d then F (λX0; λX1; : : : ; λXn) = λ F (X0;X1;:::;Xn), so we can tell whether F (P ) = 0 or not, and this is enough to define projective algebraic varieties. Definition. A polynomial is called homegeneous if and only if all of its terms have the same degree. 46 E. g., x3 + 2y2z − 3xz2 is homogeneous of degree 3, while x3 − y2 is not homogeneous. Definition. An ideal I/K[X0;X1;:::;Xn] is called homogeneous if and only if it can be generated by homogeneous elements.( Warning: This does not mean that all elements of the ideal are homogeneous polynomials.) Definition. Let I/K[X0;X1;:::;Xn] be a homogeneous ideal. The projective algebraic variety defined by I is the set n V(I) = f(X0 : X1 : ::: : Xn) 2 P j F (X0;X1;:::;Xn) = 0; 8F 2 I;F homogeneousg: Two ways of looking at projective varieties 2 2 1. Let's consider the variety V = V(hX1X2 − X0 i) ⊂ P . If X0 6= 0, we can 2 2 divide X1X2 − X0 = 0 by X0 to get (X1=X0)(X2=X0) − 1 = 0, which can be written as xy − 1 = 0 in terms of the affine co-ordinates x = X1=X0 and 2 y = X2=X0 on U0 = A , so V0 = V \U0 is a hyperbola. The points V nV0 are 2 the points of V with X0 = 0. By substituting X0 = 0 into X1X2 − X0 = 0 we obtain X1X2 = 0, therefore the points at infinity are (0 : 1 : 0), which corresponds to lines parallel to the x-axis, and (0 : 0 : 1), which corresponds to lines parallel to the y-axis.