Suppose that we have a compact Riemann surface (complete complex curve) and we remove one point from it. Then we can easily recover the pair from . As a result, different communities of mathematicians have different opinions as to whether or is the more fundamental object. People from mathematical physics like to talk about curves with punctures, adopting the first perspective, while algebraic geometers like to talk about curves with marked points, adopting the latter. What I want to do in this post is explain why we algebraic geometers are correct — if you allow us to choose all the definitions :). (Perhaps AJ will write a post explaining why the physicists are correct if you let them choose the definitions.) Specifically, I want to explain the following fact which, when I was first informed of, I could not believe:
Let , which should be considered as an infinitesimally small neighborhood of the origin in the complex plane. Let be the trivial family of projective lines over and let and be two closed subsets of : namely, we take and . (More precisely, and .) Geometrically, and should each be viewed as four infinitesimal discs, each projecting down isomorphically to . There are three discs which and share in common. The fourth disc of is different from the fourth disc of , but tangent to it over the origin .
Now, it turns out that there is no automorphism of taking to . However, and are isomorphic! Thus, the operation of “filling in punctures”, which is so common when working with individual Riemmann surfaces, can not be done in a well defined manner in families over schemes like . If you are presented with the family , you have no way of knowing whether I obtained this family as or as . So this is why algebraic geometers talk about marked points, rather than forgetting information by passing to the family of punctured curves.
Below the fold, I will work out this example carefully. I will then use it to advertise many of the further subtleties which arise in deformation theory. I should say that this post was entirely inspired by Prof. Hartshorne’s course on deformation theory in spring 2005; the notes from which are still online. I came up with this example in the course of trying to understand his far more subtle ones.
First, let me sketch why there is no automorphism of taking to . It will be convenient to introduce the notation for the ring . One can check that the Picard group of is still . Therefore, the line bundle on can be identified from the fact that its space of global sections is so any automorphsim of must pull back to a bundle isomorphic to . From here, one obtains in the usual way that the automorphism group of is , where is the group of matrices with entries in whose determinant is a unit of . The in comes from the choice of isomorphism between the pulled-back copy of and itself. So, all the autmorphisms of look like Mobius transformations , with , , and in and a unit of . A little work will check that only the identity Mobius transformation fixes three points, even over this strange ring .
The argument in the previous paragraph is just the standard argument that crossratio is a well defined invariant of four points on , done carefully enough to work over the ring . So you shouldn’t find it surprising. What you should find surprising is that is isomorphic to . The easiest way to see this is so slick that you’ll think I’m cheating. is simply . But, in this ring, is already invertible; it’s inverse is . So . Similarly, .
This feels like cheating to me, so let’s redo the argument in the coordinate . In this coordinate, and are and respectively. But, in the former ring, is already a unit, with inverse . So
. Yup, the argument is still right!
Of course, we can mix punctures and marked points together. For example, suppose that we delete from , leaving behind the affine line . Let’s mark and . Is there an automorphism of taking to ? Find out in the next installment!