Changes in the base locus
Here we continue to explore the dependence of the base locus on the degree and multiplicities of a collection of surfaces at given points. Remember two basic observations made on the previous page:
- 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.
The following diagram illustrates a few cases of this dependence when there are only two points in space. Here's what it's showing: the point (d,m1,m2) is colored red if there are no surfaces at all which satisfy the conditions, yellow if there are such surfaces, but the collection of such surfaces has the line between the two points as its base locus (i.e. any surface with the given degree and multiplicity necessarily contains the line). A point is green if the set of surfaces with given degree and multiplicity has no base locus at all.
A choice of degrees and multiplicities, corresponding to a point (d,m1,m2), is called a divisor.
Some of the points illustrated in the diagram are examples we've considered before:
- If d = 2, m1 = 1, and m2 = 1, we are considering all quadrics which go through two given points. There are many such, and the collection has no base locus. The point is colored green in the picture.
- If d = 1, m1 = 1, and m2 = 1, this is the collection of planes through the two points. It is shaded yellow because the base locus is the line between the two points.
- If d = 1, m1 = 1, and m2 = 2, we are asking for a degree one surface (a plane) which has a point of multiplicity 2, which is impossible -- a plane never has a point of multiplicity greater than 1. The corresponding point is colored red.
Both of the trends noted above can clearly be seen in the diagram. Starting with two given multiplicities, say m1 = 1, and m2 = 1, increasing the degree moves vertically in the diagram. Starting at a red point (there are no surfaces of degree 0), we move to a yellow point (the planes through the two points have base locus the line), and then reach green point (the collection of quadric surfaces through two points has no base locus).
Similarly, if we start with a given degree and multiplicity at one point, and then increase the multiplicity at the second point, the base locus grows, and the points move from green, to yellow, to red. For example, take d = 2 and m2 = 2. If m1 = 0, the collection of surfaces has no base locus. If m1 = 1 or m1 = 2, there still exist such quadric surfaces, but the line between the two points is the the base locus. If m1 = 3, there are no such surfaces at all.
Chambers for the base locus
It is apparent that the green, yellow, and red points are not dispersed at random in the diagram: the space is divided into several regions, and within a given region, all the points have the same color.
This is a nice picture, but how to interpret it is less clear. What does it mean that the point with d = 2, m1 = 1, and m2 = 1.5 is inside the yellow chamber? It doesn't make sense to talk about surfaces with multiplicity 1.5 at a point, since the multiplicity is always an integer. One approach is to just consider what happens after multiplying everything by some integer in order to get a more sensible divisor; the base locus of d = 2, m1 = 1, and m2 = 1.5 should be regarded as that of d = 4, m1 = 2, and m2 = 3. There is a slight distinction here, which we'll ignore; looking at the base locus of multiples of a divisor really defines the stable base locus.
This definition will work as long as it's possible to clear the denominators of the degree and all the multiplicities. If the points have irrational coordinates, even this is not good enough; it's necessary to first slightly nudge the coefficients to get rational numbers, and then multiply by some integer to clear the denominators. There are two ways to go about this: we can either slightly increase the degree and slightly decrease the multiplicities (making it easier to find satisfactory surfaces), or decrease the degree and increase the multiplicity (adding restrictions, and giving a larger base locus). These two approaches yield the augmented base locus and diminished base locus respectively, which will be defined soon.
Note that the corresponding picture for surfaces going through three different points would be more complicated, because there are more possibilities for the base locus: for some values of the parameters, the base locus is the line between the first two points; for others, between the first and the third; for still others, the second and the third. It's also possible that the base locus is the entire plane through the three points. With more than three points, there are even more possibilities for the base locus.
The augmented and diminished base loci
The points which lie on the boundaries between the two different chambers pose something of an enigma. Just knowing the base locus of such a collection of surfaces doesn't tell the whole story: we'd like to know that, e.g., although a certain point is colored red, nudging it by even a tiny amount would yield a green point. To capture this new information, we name two new properties of a divisor:
- The augmented base locus of a divisor is the base locus we obtain when the divisor is nudged just a little bit in the "negative" direction: d is decreased slightly, and each of the mi is increased slightly. The augmented base locus contains the familiar base locus: decreasing d and increasing each mi means that there are fewer surfaces which meet all the requirements, and so the base locus is bigger.
- The diminished base locus of a divisor is the base locus obtained when the divisor is nudged in the "positive direction": d is increased slightly, and each of the mi is decreased slightly. The diminished base locus is contained in the usual base locus, because nudging the parameters "positively" means that it's easier for a surface to satisfy the requirements, so there are more of them and a smaller base locus.
Usually the augmented base locus and diminished base locus are the same thing, and they both agree with the regular base locus. If a divisor is in the middle of the yellow chamber in the earlier diagram, then any divisor obtained when the divisor is nudged in some direction is also in the yellow chamber, and all the nudged divisors have the same base locus.
The new information which is remembered by the augmented base locus and the diminished base locus is useful only for points which are on the boundaries between two different chambers. In essence, the augmented and diminished base locus tell which chambers a point borders, even though the point itself is contained in only one of them.
Unfortunately, the definitions above aren't quite right, because the behavior of the base loci can be more complicated than our examples reveal. When there are at least nine points involved, base loci of collections of degree d surfaces can be extremely complicated: the base locus need not be just a line or a plane, but can include other curves as well. In fact, once there are at least nine points, there are infinitely many different possibilities for the base locus, and thus infinitely many different chambers in the corresponding diagram!
Even worse, it's no longer possible to say what it means to perturb a divisor by a small amount and see what chamber it lands it. The chambers can be meet in strange ways, as in the following schematic:
Here, there are infinitely many chambers, each colored a different shade of green, which get smaller and smaller and closer and closer together as they get close to the divisor D. From the perspective of the diminished base locus, this situation is extremely strange. The diminished base locus is supposed to indicate what the base locus is when a divisor is perturbed slightly. In the example with two points in space, the precise perturbation used didn't make any difference. Here, however, it does: nudging D to the right in this diagram will put it in one of the green chambers, but which chamber we end up in depends very strongly on the amount of the perturbation. There is no way to make sense of our definition of the diminished base locus.
The base loci of the divisors in the above example are equally surprising. Moving towards D into lighter and lighter green chambers, divisors have more and more complicated base loci. In fact, the nth chamber from the left corresponds to divisors whose base locus consists of about n different curves. As a result, the smaller the perturbation of D, the more curves are in the base locus of the perturbed divisor. Our amended definition of the diminished base locus is that it's the combination of all the different base loci corresponding to the infinitely many chambers near D. The diminished base locus of D consists of infinitely many curves!
The situation illustrated in the diagram really can arise: there exists a certain divisor D, appearing when nine points are in play, for which the diminished base locus contains infinitely many curves. Making smaller and smaller perturbations of D, we find divisors with more and more curves in the base locus.
One example of such a divisor has for parameters certain irrational numbers given approximately by