Category Archives: Intersection Theory

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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.

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 … Continue reading →