Saturday, March 5, 2016

The Projective Plane: An Algebraic Exploration I

This is the second post in a series on projective space. For the first, see here.

The idea of "adding points at infinity" to the plane introduces new behavior to the study of the intersections of lines and curves. Since there are different points of infinity for each direction in the affine plane (as discussed in the last post) and parallel lines intersect at infinity, it is reasonable to suppose that certain lines and curves also intersect at infinity (see below).

For example, the hyperbola above is given by the equation xy = 1. Away from the origin, the two branches of this curve approach the x- and y-axes, defined by the equations y = 0 and x = 0, respectively. Since the distance between the curve and these lines (called asymptotes of the curve) approaches 0 far from the origin, it makes sense to suppose that the hyperbola intersects with these asymptotes at infinity. On the other hand, for other curves that clearly "go off to infinity" like the cubic curve y = x3 shown below, there is no asymptote. What point at infinity, if any, does the cubic intersect?

Answering this question is difficult in our as yet fuzzy picture of the structure of the projective plane. However, the algebraic definition of the projective plane provides the tools necessary for solving this and many other related problems. Introducing this machinery is the purpose of this post.

Lines, parabolas, cubics, hyperbolas and many other curves in the plane may be expressed in the following algebraic form: f(x,y) = 0, where f is a polynomial function of x and y. This means that it is a sum of terms of the form a*xiyj where i and j are nonnegative integers and a is a constant coefficient. For example, the equation for the hyperbola written above may be written xy - 1 = 0 and the equation for the cubic y - x3 = 0. Any curve defined by an equation of this form is known as an affine variety.

The previous post introduced the projective plane as the set of lines through the origin in three-dimensional space. It then illustrated two different ways in which certain representatives may be chosen from the lines to get a "picture" of the projective plane in three-dimensional space. We show that for equations of certain forms, it does not matter which representative we choose from a given line through the origin. First, let P = (x,y,z) be a point distinct from the origin in three-dimensional space (that is, at least one of x, y, and z is nonzero). Then since any two distinct points determine a line, P determines a unique line L through the origin. The point (ax,ay,az) is then on L for any constant a and every point on L is of this form. In other words, only the ratio of the coordinates to one another is required to determine on which line through the origin a given point lies. This fact can more easily be seen in two dimensions, as in the figure below.

The line above has equation y = 2/3*x. It passes through the origin and has slope 2/3, so any point (x,y) for which y/x = 2/3 is on the line (as demonstrated by the construction of a suitable triangle, as above). With this fact in mind, we introduce the concept of homogeneous coordinates. Homogeneous coordinates (x:y:z), where at least one is nonzero, define a point of the projective plane, with the understanding that only the ratio of x, y, and z matters. Thus (1:2:3) = (3:6:9), for example. With these identifications in mind, every point in the projective plane may be assigned homogeneous coordinates (though in many equivalent ways).

Next we consider projective varieties, i.e. certain types of curves in the projective plane. As before, they are defined as the set of points satisfying a certain polynomial equation, but in three variables instead of two: F(x,y,z) = 0. However, in light of the equivalence between points with different x-y-z coordinates, we must consider only polynomial equations that have the same points of the projective plane as solutions for any coordinate representation of the given points. These are called homogeneous polynomials. A polynomial is homogeneous if each one of its terms, or monomials, is of the same degree, meaning that the sum of the exponents in each term are the same. For example, F(x,y,z) = x2yz3 is trivially homogeneous of degree 6 because it has only one term and the sum of its powers are 2 + 1 + 3 = 6. F(x,y,z) = xy2 + z3 is homogeneous of degree 3 because the sum of the exponents of the xy2 term is 1 + 2 = 3 and is obviously also 3 for the second term, z3. The crucial property of homogeneous polynomials is that if F(x,y,z) = 0, then F(ax,ay,az) = 0 for any constant a:

The crucial fact used in the proof (click to enlarge) is that the exponents of each term (the p's, q's, and r's) must always add up to the same degree n. The an term can then be factored out, confirming that F(x,y,z) = 0 always implies F(ax,ay,az) = 0. This means that for any point in the projective plane, a homogeneous polynomial that is zero on one representative is zero on all. Conversely, if F(ax,ay,az) = 0, then the same proof (using 1/a) shows that F(x,y,z) = 0 so long as a is not zero. All this manipulation distills down to the following crucial statement: it is meaningful to say that a homogeneous polynomial is zero at a point in the projective plane since any representative gives the same result. We can thus denote projective varieties by the equation F(x:y:z) = 0 in homogeneous coordinates.

It follows that a homogeneous polynomial in three coordinates has a solution set of points (a curve) in the projective plane. These solution sets are the projective varieties. The next post (coming soon) continues to fill in the algebraic picture of the projective plane and relates affine varieties to projective ones, ultimately answering the questions posed at the beginning of this post.


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