Singularities: the multiplicity of a surface at a point
The examples we've seen already suggest that quadric surfaces have much more interesting geometry than planes do. For one, there are a lot more of them, so we can ask them to pass through more elaborate configurations of points. There's another new phenomenon which shows up for quadric surfaces which didn't show up for planes: they aren't always nice, smooth surfaces. One example of this is a cone, defined by the equation z2 = x2 + y2, which pinches off at one point. Such a point where the surface is not smooth is called a singularity.
More complicated equations can give rise to even more complicated examples of singularities, a few of which are shown below. Note that some of the equations have terms of degree bigger than two, so these are not quadric surfaces.
|z2 - 2x3 - x2 - (y - z)2/2 = 0||x3 + x2 - y2 - z2 = 0|
|x3 + x2 - y2 = 0||x2 - y3 + z5 = 0|
These illustrate several distinct ways in which a surface can fail to be smooth at a point. Some of these surfaces just look like cones, at least near the singular point. In one of the others, two sheets of the surface intersect each other; here the multiplicity is two for any point contained in this intersection. In the last example, the surface is just "pinched" at the singular point.
The multiplicity of a surface at a point is a number which measures how far surface is from being smooth there. If the multiplicity at a point is one, the surface is smooth at the point. If it's two or more, it isn't, and larger numbers correspond to even more complicated non-smoothness. Each of the examples above has points of multiplicity two on an otherwise smooth surface.
In most cases just looking at a picture doesn't give enough information to guess the multiplicity. Instead, the multiplicity of a singularity is defined using the equation for the surface. If the surface has equation f(x,y,z)=0, the multiplicity at the point (0,0,0) is equal to the smallest degree of any of the terms in f. This is perhaps easiest to define by examples:
|Equation||Multiplicity at (0,0,0)|
|x + y + z = 0||1|
|x2 + y2 - z2 = 0||2|
|x2 + y3 - z4 = 0||2|
|xy + y3 - z4 = 0||2|
|xyz + y4 - x2 z2 = 0||3|
With the multiplicity so defined, we can now consider some additional families of surfaces. Instead of merely specifying that a surface passes through a bunch of given points, we can ask that at some of those points it has a singularity with some specified multiplicity. Below is an animation of the family of surfaces of degree 3 going through 16 specified points, with multiplicity of at least 2 at the first of the points (the points aren't shown in the picture). The point with multiplicity 2 is at the center of the box; observe that none of the surfaces in question is smooth there.
Much like the previous examples, this family has a base locus containing more than just the specified points. Not unexpectedly, it is a somewhat complicated-looking configuration, shown below:
Dependence of the base locus on d and mi
The examples we've encountered indicate the main questions one may ask about a family of surfaces through specified points. Given n points in space, say p1,...,pn, a degree d, and n multiplicities m1,...,mn:
- Do there exist any surfaces of degree d, such that the multiplicity of each surface at each point pi is at least mi?
- How many such surfaces are there? Finitely many or infinitely many?
- Does the collection of such surfaces have a base locus? What does it look like?
Let's collect together the answers to these questions for the various examples encountered so far.
|Points||Degree||Multiplicities||How many?||Base locus|
|2||1||1,1||Infinite||Line through two points|
|3||1||1,1,1||One||Entire plane through three points|
|5, in plane||2||1,1,1,1,1||Infinite||Circle through points|
|8, not in plane||2||1,1,1,1,1,1,1,1||Infinite||"Wavy curve"|
Ideally, we'd like to be able to quickly calculate the answer to all of these questions whenever we're handed a degree and list of multiplicities. Unfortunately, these questions turn out to be extremely difficult in practice -- it's not even known for which d and mi the answer to the first question is affirmative. But we can still make a couple basic observations in the right direction, by thinking about how the answers change when we vary the numbers d and mi. A couple simple points jump out already:
- If d is increased, the number of surfaces increases, and the base locus shrinks.
- If any of the mi are increased, the number of surfaces decreases, and the base locus grows.
Increasing the degree gives us more freedom to choose a surface. Through three points, there is only a single plane (d = 1), but there are many quadric surfaces (d = 2). Increasing the multiplicity serves to add more restrictions, and makes it harder to find surfaces. There are fewer surfaces, and so more points are contained in all of them, giving a larger base locus.