Normal Cones

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.

 

Chern Character and K-Theory

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 .

 

Chern Classes: Part 2

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.

 

Some Technical Points

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.

 

Chern Classes: Part 1

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.

 

Intersections with Divisors

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.

 

Random Thing

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.

 

Manipulating Cycles

Yesterday, we defined cycles, cycle classes, and the abelian groups in which they live.  Today, we’re going to fiddle with them a bit.  We’ve got a proper pushforward map, so today, we’ll start by figuring out when we have a pullback.

 

The Chow Groups

And today, we start intersection theory.  So, to establish notation a bit, we’re only going to be talking about algebraic schemes.  These are separated schemes of finite type over our base field .  That is, they admit finite open affine covers such that each affine is the spectrum of a finitely generated -algebra.  We’re already almost to varieties, the only thing left is to assume reduced and irreducible (equivalently, integral) to get there, but we won’t do that unless necessary.  Nor will we assume smooth if we don’t have to, so we’re hoping to get a theory that will work, at least somewhat, for singular varieties.

 

And the winner is…

Intersection theory! So in November, I will be attempting to post at least once a day on things like Chow groups, Chern classes, normal cones, positivity, intersection products, degeneracy loci, Grothendieck-Riemann-Roch, etc.  Thanks to everyone who voted.  Let’s see how well this experiment works.