John Lesieutre

Planes through given points

Given any two points in three-dimensional space, there are lots of planes which pass through both of them:

All of these planes have something more in common: they contain the entire line between the two points. Any plane which passes through both of the points necessarily also passes through every other point on the line between them. This is not particularly shocking, of course, but it's the simplest instance of a much more general idea, the base locus of a collection of surfaces. For any collection of surfaces, the base locus of that collection consists of all the points which are contained in each of the surfaces. In this case, our collection is all the planes going through both points, and the base locus is the line through the two points.

If we only specified one point for the planes to go through, the collection of planes is much bigger. Besides the specified point itself (which we won't count), there aren't any other points that all of the planes pass through; that is to say, the base locus of the collection doesn't contain anything at all.

At the other extreme, if we specify three points through which planes must pass, there's only one satisfactory plane in the first place: the collection of planes through the points has only a single item! Thus we (somewhat vacuously) must regard every point on a plane as being in the base locus of the collection.

Trying to take it further, the collection of all planes through four points doesn't contain anything at all. There's no way to find a plane that goes through all four corners of a pyramid. The base locus doesn't really make sense here.

Quadric surfaces through given points

The study of the base locus becomes much more interesting when we consider collections of surfaces other than planes. Remember that one way to think of a plane in space is as the set of points (x,y,z) which satisfy some relation ax + by + cz = d, where a, b, c, and d are numbers (different sets of numbers define different planes). If we consider more complicated equations, we get more interesting surfaces. For example, x² + y² + z² - 1 = 0 is satisfied exactly when (x,y,z) is at a distance 1 from the point (0,0,0), so the set of solutions is a sphere with radius 1. A few more elaborate examples are illustrated below.

While these surfaces don't look too much alike, they are all defined by fairly simple equations: the only things that appear are the variables x, y and z, together with their squares and the products of any two of them. A surface which is defined by such an equation (i.e. an equation which may be written in the form ax² + by² + cz² + dxy +exz + fyz + gx + hy + iz + j = 0) is called a quadric surface. Exactly what a quadric looks like depends on the various coefficients, but they are geometrically the simplest class of surfaces beyond planes.

Just as with planes, we can think about the collection of quadric surfaces which pass through some specified points. Now the study of the base locus becomes much more delicate. There are a lot more quadrics than there are planes - we need to specify ten different coefficients to define a quadric, vs. only four for a plane. Simply asking that a quadric pass through two points isn't very restrictive; like asking for a plane through a single point, there are so many such quadrics that the collection has no base locus. But when we consider the set of quadrics which pass through some larger number of points, there are various possibilities for the base locus of the collection, which take a surprising shape!

First, let's consider the simple case of specifying five points which form a regular pentagon in the xy-plane.

There are many quadrics going through all five points, a handful of which are shown below.

These pictures suggest that the collection of quadric surfaces going through the five points does indeed have a base locus: each of them contains not only the five points, but every point on the circle which goes through the five points. That circle is the base locus of the collection.

The base locus can also be understood in terms of the equations for the surfaces in question. If the five given points (x_i,y_i,z_i) are all solutions of some equation f(x,y,z) = 0 (where f(x,y,z) is a polynomial of degree 2), then every other point on the circle through the five points must be a solution of this equation as well.

As a second example, suppose we pick eight points in space. There are infinitely many quadrics which go through all eight:

It's not obvious at first glance, but there are points contained in all of these quadrics other than the eight that were specified. In fact, every surface through the eight points contains the entire wavy curve illustrated below. This curve is the base locus of the collection.

Watch these surfaces move again, this time with the curve marked.

On the other hand, if we specified just one more point in addition to these eight (one not already contained in the base locus) there would be only one quadric surface going through all of them. The base locus would then be that entire surface. If the added point were contained in the base locus, exactly the same surfaces would still go through all the points, and the base locus would not change.