Well, really, for intersection theory, it’s true. We start with a closed subscheme, with normal cone . We’re going to construct a family of embeddings that deforms to the zero section of . Then, because intersections should vary nicely in families, we’ll have essentially reduced the problem of doing intersections to the case of normal cones.
November 13, 2009
November 11, 2009
So, last time we talked about Segre classes and cones. Now, we’re going to move ahead, and talk about a specific cone in detail, the Normal cone we defined on Monday. Let be a subscheme, and let be its normal cone. We define , the Segre class of in to be , the Segre class of the normal cone.
November 10, 2009
Leave a Comment
Last time, we talked about the Normal Cone. We’re going to go back a bit and increase the generality before coming back to it. Let be a cone over , and let be the projective closure. We define the Segre class of the cone , in to be , where is the projection.
November 9, 2009
Ok, so I took the weekend off to figure out where things are going and get a bit ahead. Will probably be doing that all month. So now, we’re going to talk about cones and normal cones, with the goal of eventually defining the intersection product itself.
November 6, 2009
Today, we’re going to construct a ring that encodes quite a lot of intersection data (though not terribly transparently) as well as some special combinations of Chern classes. A lot of modern intersection theory and enumerative geometry takes place in the K-theory ring of a scheme .
November 5, 2009
We’ve define the Chern classes now, but what about computing them, and computing with them? We have that long list of properties that will help, but there is a need to prove them, and they aren’t completely trivial. What we need is a clever trick. Vector bundles generalize line bundles, which we already understand, more-or-less, so if we can reduce computations with Chern classes to computations with the first Chern class, that would be wonderful.
November 4, 2009
Leave a Comment
So, I’ve been a bad math blogger. I’ve been identifying a bunch of different classes of things that we can really only identify on nice algebraic schemes. Things like smooth varieties (where I’ve grabbed all of my examples). There are actually three different classes of “codimension one gadgets” that I’ve been treating as interchangeable. So today I’m going to talk about them, and why they aren’t quite the same thing.
November 3, 2009
We’re going to talk about Chern classes, but first, a note on the last post. For any scheme , there’s a pairing , taken by restricting the line bundle to the curve and taking the degree (or doing the intersection as we described, and integrating). In the case of a surface, , and so we have the usual intersection pairing, as we mentioned by reproving Bezout’s Theorem in . So, at the least, our notion of cycles and intersections is recovering the basic intersection theory that we know from Hartshorne.
November 3, 2009
Today we start actually performing intersections. Fix a scheme, an inclusion of a subvariety, , and let be a divisor on . The big definition for today: in where is the support.
November 2, 2009
On Math Overflow, I just saw an “answer” to a question, given by Scott Morrison, that I just had to share with anyone who hadn’t seen it. The Message of the Day, on Oct 2, at Berkeley was the following:
Warning: Due to a known bug, the default Linux document viewer evince prints N*N copies of a PDF file when N copies requested. As a workaround, use Adobe Reader acroread for printing multiple copies of PDF documents, or use the fact that every natural number is a sum of at most four squares.