The Projective Plane: An Algebraic Exploration II

This is the third post in a series discussing the projective plane. For the first, see here.

The previous post explained how certain types of polynomials, namely the homogeneous polynomials, define curves in the projective plane called projective varieties. This post will explicate the relation between projective varieties and affine varieties (typical curves in the plane) and indicate how projective varieties are in a way the extensions of affine varieties to include their points at infinity.

First, we consider how projective varieties naturally give rise to normal affine plane curves. Consider the projective variety defined by the equation F(x:y:z) = 0, where F is a homogeneous polynomial. In the first post of this series, we saw that the plane z = 1 in three-dimensional space can represent the subset of the projective plane that corresponds to the normal affine plane (i.e., without the points at infinity). We repeat the image from the first post for convenience, where each line through the origin (a point of projective space) is represented by the point at which it intersects the plane.

Since we obtain the affine plane by setting z = 1, it seems reasonable that we should be able to "collapse" projective varieties algebraically by setting z = 1 in the equation F(x:y:z) = 0. This is indeed the case, since substituting z = 1 yields a polynomial in only two variables: f(x,y) = F(x,y,1). For example, if F(x:y:z) = x²y + 2yz² - 5z³ (note that F is homogeneous and therefore defines a projective variety), then f(x,y) = F(x,y,1) = x²y + 2y*1² - 5*1³ = x²y + 2y - 5. The projective variety F(x,y,z) = 0 therefore corresponds to an affine variety f(x,y), as desired.

There is also an algebraic process that does the reverse by taking a polynomial f(x,y) and producing a corresponding homogeneous polynomial in three variables, F(x:y:z). The process works as follows:

1. Add up the powers of x and y in each term of f and let n be the greatest degree that appears
2. Multiply each term by z^n-k, where k is the degree of the term (this ensures that the resulting polynomial is homogeneous)

Let us take the polynomial f(x,y) = x²y + 2y - 5 from above and apply the new process. The degrees of the three terms are 3, 1, and 0, respectively (-5 = -5x⁰y⁰). The maximum of these is 3, so we multiply the first term by z^3-3 = z⁰ = 1, the second by z^3-1 = z², and the third by z^3-0 = z³. The resulting polynomial is F(x:y:z) = x²y + 2yz² - 5z³, exactly the same as the original polynomial! It is easy to verify that these two processes are mutually inverse in general, except in certain special cases such as when F is only a function of z.

Now we may apply these algebraic tools to solve the problems introduced in the last post that cannot be solved visually. First, regarding the hyperbola, algebra confirms our intuition. To see this, take the equation xy - 1 = 0 and transform it into the corresponding projective variety. The result is easily calculated as xy - z² = 0. The asymptotes x = 0 and y = 0 to this hyperbola (see image in previous post) are unchanged by the process since they only have one term, clearly of maximal degree.

Next, recall that the points at infinity in the projective plane are those for which z = 0 in the homogeneous coordinates (x:y:z). This can be seen in the above visualization, where only points of the form (a:b:1) belong to the affine subset of the projective plane. Now any (x:y:z) can be scaled to this form by multiplying each component by 1/z (remember, only the ratio of the coordinates matters), but only when z is nonzero. Therefore, we substitute z = 0 and solve the equations to see which points at infinity each curve intersects. For the hyperbola, this gives xy = 0, so x = 0 or y = 0. Therefore, the two points at infinity the hyperbola intersects are (0:1:0) and (1:0:0). Other coordinate triples satisfying xy = 0 such as (3:0:0) differ only by a scale factor from one of the two solutions above and therefore define the same point in the projective plane. It follows that (0:1:0) and (1:0:0) are the only solutions. But the asymptotes x = 0 and y = 0 hit exactly the same points, (0:1:0) and (1:0:0), respectively! This confirms our intuition: a hyperbola and its asymptotes really do intersect at infinity.

The cubic y - x³ = 0 has no asymptote, but clearly goes off to infinity in some manner. We may use our algebraic tools to investigate the function's behavior in the projective plane. The highest degree term is x³, of degree 3, so we must multiply the other term (namely the expression y, of degree 1) by z^3-1 = z². The projective variety corresponding to the cubic is therefore defined by the equation yz² - x³ = 0. Substituting z = 0 yields x³ = 0, which has the single point (0:1:0) in the projective plane as a solution (since z is already set to 0). Note that even though the cubic goes to infinity in both the positive and negative directions, it meets only one point at infinity because opposite directions are identified (see the representation of the projective plane in a sphere in the first post). This indicates that the projective variety induced by the cubic meets that induced by the y-axis with equation x = 0 at infinity. Indeed, this makes some intuitive sense: as x becomes very large, it becomes insignificant relative to y = x³ and therefore the point (x,y) is "close" to the y-axis x = 0 (this can also be seen by zooming out a graph of the cubic - the graph eventually becomes nearly indistinguishable from the y-axis). We can also visualize the projective variety yz² - x³ = 0 that extends the cubic on the sphere (see below).

This image shows the cubic curve in the affine plane as well as its projection (via lines through the origin, the center of the sphere) onto the surface of the sphere. It differs slightly from our earlier sphere representation since the plane is below and not above the sphere, but this makes little difference. At the bottom of the sphere, the origin of the plane touches the sphere (which is a point on the curve). At first, the path veers away from the y-axis (the grid line from top to bottom through the origin), but notice how when the curve approaches the equator of the sphere (infinity), it comes back to hover above the y-axis. Images like these help to interpret the results of our algebraic manipulations.

The projective plane has very elegant geometric properties (every two lines in the plane intersect in exactly one point, for example) and gives us a sturdy mathematical grounding for the slippery concept of behavior "at infinity." Generalizations of this concept are crucial in the study of polynomial curves and their corresponding equations.

Sources: http://voltage.typepad.com/.a/6a00e55375ef1c8833014e610f8df7970c-pi