### November 2009

Well, really, for intersection theory, it’s true.  We start with $X\subset Y$ a closed subscheme, with normal cone $C$.  We’re going to construct a family of embeddings that deforms $X\subset Y$ to the zero section of $C$.  Then, because intersections should vary nicely in families, we’ll have essentially reduced the problem of doing intersections to the case of normal cones.

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 $X\subset Y$ be a subscheme, and let $C_X Y$ be its normal cone.  We define $s(X,Y)$, the Segre class of $X$ in $Y$ to be $s(C_X Y)$, the Segre class of the normal cone.

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 $C$ be a cone over $X$, and let $P=P(C\oplus 1)$ be the projective closure.  We define the Segre class of the cone $C$, $s(C)$ in $A_*(X)$ to be $s(C)=q_*\left(\sum_{i\geq 0} c_1(\mathscr{O}(1))^i\cap [P]\right)$, where $q:P\to X$ is the projection.

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.

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 $X$.

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.

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.

We’re going to talk about Chern classes, but first, a note on the last post.  For any scheme $X$, there’s a pairing $\mathrm{Pic}(X)\times A_1(X)\to \mathbb{Z}$, 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, $\mathrm{Pic}(X)\cong A_1(X)$, and so we have the usual intersection pairing, as we mentioned by reproving Bezout’s Theorem in $\mathbb{P}^2$.  So, at the least, our notion of cycles and intersections is recovering the basic intersection theory that we know from Hartshorne.

Today we start actually performing intersections.  Fix $X$ a scheme, $j:V\to X$ an inclusion of a subvariety, $\dim X=n, \dim V=k$, and let $D$ be a divisor on $X$.  The big definition for today: $D\cdot [V]=[j^*(D)]$ in $A_{k-1}(|D|\cap V)$ where $|D|$ is the support.

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.