### About this blog

Rigorous trivialities is a web log about mathematics, but especially geometry, broadly construed. Contributors will be Charles Siegel, Jim Stankewicz and occasionally Matt Deland. Charles specializes in algebraic geometry, topology and mathematical physics. Jim specializes in arithmetic algebraic geometry. Matt has transitioned from algebraic geometry to work in industry.

Header is taken from the larger work by fdecomite under the creative commons license.

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# Category Archives: Intersection Theory

## Everything is a Normal Cone

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

Posted in Intersection Theory, MaBloWriMo
3 Comments

## Segre Classes of Subschemes and their Applications

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

Posted in Intersection Theory, MaBloWriMo
2 Comments

## Segre Classes of Cones

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

Posted in Intersection Theory, MaBloWriMo
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## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
1 Comment

## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
4 Comments

## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
3 Comments

## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
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## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
3 Comments

## 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.

Posted in Intersection Theory, MaBloWriMo
1 Comment

## 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 … Continue reading

Posted in Intersection Theory, MaBloWriMo
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