A universal curve

What follows is a rambling example attempting to explain some mathematics. I didn't really have a well defined audience in mind when I wrote it (or rather, I had a number of well defined audiences in mind and all of them pulled in different directions) but hopefully it is interesting and intelligible to some anyway. I clearly need to draw some more pictures and insert them to break up the ramble, but until I get around to it, I'll leave that bit of work as an exercise for the reader.

I'll probably try to work this idea into a general audience talk, perhaps one introducing the minimal model program, so if anyone has any thoughts, I'd be glad to hear them.

Families of curves are rather interesting objects. Consider the set of all curves in a given geometric space Z and call this set Curves(Z). Then we can consider the universal curve, which is the subset of the Cartesian product Z × Curves(Z) containing pairs of a point in Z and a curve containing that point, namely

UCurve(Z)={(z,[C]) ε Z × Curves(Z) | z is in C}.

It turns out that with some reasonable restrictions on the type of thing that we want to consider as a curve, that Curves(Z) and UCurve(Z) have the structure needed to be considered as geometric objects in their own rights. A family of curves in Z is a subset of Curves(Z) that fits into the same language of describing geometric things that we use to describe Z, the curves on Z and Curves(Z).

A simple example of a family of curves is the set of lines through a given point in the Euclidean plane, E. Call this family H. It is a subset of Curves(E). H is by definition the real projective line, i.e. the circle, parametrized by slope, where a vertical line is said to have a slope of ∞. The universal curve, which will have the name F and sit inside UCurve(E), is relatively easy to see - it is quite similar to the plane itself. Using polar coordinates centered at our given point to describe the plane is almost the same as using the universal curve. The problem with polar coordinates is that when the radial coordinate is zero, the angular coordinate is meaningless. And standard polar coordinates use angle instead of slope, which forgets that we are interested in lines through a point, not rays coming out of that point. To get lines, we need to remind ourselves that x denotes the same line as x+180, which is a bit ad hoc.

So what we are looking at is a parametrization of the plane E by a slope coordinate and a line coordinate. For non vertical lines, we might as well use the symbols y/x for slope and the r for the line coordinate, reminding ourselves that y/x=∞ and negative values for r are okay. When we use these coordinates for the plane, there is ambiguity at the points where r=0. But the points (0,y/x) are on different curves, so they are not the same point on the universal curve. This difference between the universal curve and the plane is the key to what is going on. The universal curve coordinates work fine to describe the plane, except for the point where all the curves that are collected into the universal curve come together. On the plane, this is one point, but in the universal curve, there is a different point for each curve in the family - that is, there is a copy of H sitting inside of F such that the whole thing gets contracted to a point. This phenomenon is called blowing up a point, and F is called the blowup of E at a point. The copy of H that gets collapsed is called the exceptional curve.



F is topologically a Mobius strip. There is a picture of this in Hartshorne, conveniently reproduced on Davis's backpack for all you Seattle people without a copy or two lying around. But better than looking at a picture, draw your own. Draw a disk around the point that gets blown up instead of the whole plane and make a Mobius strip with a stapler or some tape. Draw the circle that follows the middle of the strip of your Mobius band (this whole circle is going to correspond to the point) and and a few lines across the strip perpendicular to the central circle, in both cases, press hard enough or use dark enough ink or use tracing paper so that you can see the line on both sides. You should think of the line that you draw being equally visible on both "sides" of the strip, so when you draw it, the ends will join through the paper. Right now this is just the way we'll do things. We aren't doing noncommutative geometry yet. Label these lines with slopes starting with ∞, then move into big negative numbers, increase up to 0 and then up to positive slopes and back to ∞. Trace your way around the central circle and mark the correspondingly sloped lines on the flat plane to convince yourself of what I'm rambling on about.

The Klein bottle shows up if we look at the family of complete geodesics through a fixed point on a sphere, say the south pole on the earth, in which case we get the meridians. The universal family is again very similar to the space on which the curves live, in this case the sphere. In this case, there are two places where we need to resolve the indeterminacy about which line we are on at the north and south poles all the lines come together. We do this by by blowing up these two points.

This corresponds to what is going on topologically when we look at a world map, for example with a Mercator or cylindrical projection. Latitude and longitude provide the location at any point other than the poles, but at the poles, the longitude is irrelevant information. But it is still a coordinate that we can read off of our map, so it must parameterize something. These somethings are the exceptional curves, one middle circle for each of the Mobius bands of the Klein bottle. At first glance, this comparison should seem sketchy, since we all know that a map makes the upper edge of the map collapse to one point - the north pole, the sides stitch together, and the bottom edge collapses to the south pole. So why bother with this intermediate step that seems to needlessly complicate things by introducing a Klein bottle? The advantage of the intermediary step is that one keeps an intrinsic picture of the geometry at the poles.

Without a labeling indicating that the meridian at x extends through the pole as the meridian at 180+x, there is no way to tell what the geometry is at the poles. This labeling is exactly the labeling that identifies the map as a Klein bottle.

So what is the point of this example? I thought of it in the context of two connected ideas. The first is the minimal model program (MMP) in algebraic geometry, and it relates to the interpretation of a map that appears to be a Klein bottle as in fact depicting a sphere. One way to understand the goal of the MMP is that it seeks to understand the right geometric object that corresponds to a given map. In this example, running an appropriate version of the MMP would tell us to take our Klein bottle map and blow down the two exceptional curves and recover a sphere with all the geometric data at the poles and elsewhere described as the mapmaker intended.

The second idea is the construction of the map as a universal curve. First of all, this has nice philosophical implications, because the way that a map is drawn is by first exploring the territory along paths. Perhaps a more vivid example comes from the knowledge we have that comes from astronomy. All this information has essentially coming straight at us along some sort of geodesic through spacetime. Sitting here on Earth can do little more than collect the information as it converges at our south pole of the universe. But philosophy aside, this idea of using a universal curve for a family of relatively simple curves through a fixed point as a map of some object to be studied is central to the proofs of many results in algebraic geometry. In particular this idea can be used to prove a result due to Shigefumi Mori, one of the key architects of the minimal model program, namely that a variety with ample tangent bundle is projective space.

It is here that what I am researching makes connection to this example. I believe that families of (rational) curves through some fixed point that are as simple as possible, given that I want my families to contain a curve pointing in every possible direction, should make a good enough map to reconstruct what sort of space these curves are living in. My particular tack is to try to simplify the problem even further by working with approximations to the family of curves, approximations that I call minimally indicating rational arcs. The trouble is, I'm not sure if these approximations actually make anything any simpler, but hopefully we'll see.