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Lecture Notes Math 499/
October 27, 2004
No introductory course in Algebraic Geometry is complete without at least one days discussion on projective geometry. So far, we have been talking about affine spaces usually Rn, Cn. Today, we will mix things up a bit. Our first topic is the real projective plane, P^2 R. Recall that in the plane R^2 , any two line intersect in a point unless they are parallel. We can get rid or this ”unless” if we say that parallel lines meet in some sort of point at∞. Since we do not want lines which already intersect to intersect again at a point at infinity, there should be different points of infinity for all possible slopes of lines. Formally, we introduce an equivalence relation ∼ where L 1 ∼ L 2 if L 1 and L 2 are parallel. (One can check that this is indeed an equivalence relation.) The equivalence class [L] consists of all lines parallel to the given line L. From this discussion, we can state our first definition for the projective plane.
Definition. The projective plane over R, denoted P^2 R, is the set
P^2 R = R^2 ∪ {one point at ∞ for each equivalence class of parallel lines}.
If we let [L]∞ denote the point at ∞ of all lines parallel to L, then the set L = L ∪ [L]∞ is the projective line corresponding to L. Any two projective lines L 1 , L 2 meet in the projective plane as follows:
L 1 ∩ L 2 =
a point in R^2 if L 1 and L 2 are not parallel [L 1 ]∞ if L 1 and L 2 are parallel.
An easy way to imagine the points at infinity is to think of a straight road in the desert. As you look down the road, it appears to shrink into a point as it approaches the horizon. In the theory of perspective, the point at which that parallel sides of the road appear to be converging is called the vanishing point. Furthermore, if your road happens to have a median between the lanes traveling in different directions, then the median will shrink as you look toward the horizon and the lanes will all converge to one point. The same concept applies to any point on the horizon, i/e every point on the horizon is the infinity point of some parallel set of lines. This horizon idea points out another interesting property of the real projec- tive plane: the points at infinity form a special projective line, the line at ∞. Thus the set of lines in P^2 R are those of the form L for lines L in the plane R^2 and the line at infinity. It is a fact that any two distinct lines in the projective plane determine a unique point, and any two distinct points in P^2 R determine a unique projective line. So far, we have seen how to represent lines in the projective plane, but how do you write points. In R^2 points are specified by their coordinates, but in P^2 R points at ∞ are specified by lines. To avoid this asymmetry, we introduce new homogeneous coordinates. This requires a new definition of P^2 R. WE do this dy defining a new equivalence on R^3 by saying (x 1 , y 1 , z 1 ) ∼ (x 2 , y 2 , z 2 ) if there is a
nonzero real number λ such that (x 1 , y 1 , z 1 ) = λ(x 2 , y 2 , z 2 ). We denote [x, y, z] the class of nonzero points (x′, y′, z′) ∈ R^3 −{ 0 } which are equivalent to (x, y, z). With this relation, we can give another definition of P^2 R.
Definition. P^2 R is the set of equivalence classes [x, y, z]. We can write,
P^2 R = (R^3 − { 0 })/ ∼.
If a triple (x, y, z) ∈ R^3 −{ 0 } corresponds to a point p ∈ P^2 R, we say that (x, y, z) are the homogeneous coordinates of p.
Notice that homogeneous coordinates are not unique. Foe example (1, 1 , 1), (2, 2 , 2), and (π, π, π) are all homogeneous coordinates of the same point in P^2 R. We can now define a projective line in terms of homogeneous coordinates.
Definition. Given real numbers A, B, C, not all zero, the set
{p ∈ P^2 R | p has homogeneous coordinates (x, y, z) with Ax + By + Cz = 0}
is called a projective line of P^2 R.
Notice that this is well defined in P^2 R because if Ax + By + Cz = 0, then for all λ ∈ R∗, Aλx + Bλy + Cλz = λ(Ax + By + Cz) = 0. We should now check to make sure our two definitions of the projective plane are the same. Using the second definition of the projective plane, consider the map ϕ : R^2 → P^2 R defined by (x, y) 7 → [x, y, 1]. It is easy to check that this map is injective. Also, P^2 R \ ϕ(R^2 ) is the projective line, H∞, at infinity defined by z = 0. Thus we have found that
P^2 R = R^2 ∪ H∞.
For this to be the same as the first definition, we need to show that H∞ contains all of the points at infinity. Thus, we need to investigate the relationship between lines in the real plane and in the real projective plane. We have the following correspondence:
affine line projective line point at ∞ L : y = mx + b L : y = mx + bz [1, m, 0] L : x = c L : x = cz [0, 1 , 0]
Let’s try to understand this table. For a point (x, y) on the line L defined by y = mx + b, ϕ(x, y) = [x, y, 1] which lies on the projective line L defined by the equation y = mx + bz. Thus L is a subset of L, and the rest of the points in L come form when z = 0. This is just L ∩ H∞. At these points, we find that y = mx so get the point [x, mx, 0] = [1, m, 0] ∈ P^2 R. In the case where our line is defined by x = c, then just as above we find that we get the projective line x = cz. Then at L ∩ H∞, we find that the y-coordinate must be 1, by the three coordinates cannot all be zero at the same time in projective space. An
Definition. Let k be a field and let f 1 ,... , fs ∈ k[x 0 ,... , xn] be homogeneous polynomials. We define
V (f 1 ,... , fs) = {[a 0 ,... , an] ∈ Pnk | fi(a 0 ,... , an) = 0 for all 1 ≤ i ≤ s}.
We call V (f 1 ,... , fs) the projective variety defined by f 1 ,... , fs.
One can go further and define the ideal of a variety as before, and extend the dictionary of affine geometry to projective geometry. I will not do this here.
