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.

We’re not going to worry about what’s coming up though, at the moment.  First, we need to define what a cone is.  Let be a scheme, and let be a sheaf of graded algebras such that is surjective, is coherent, and is generated by .  We define a cone over to be anything of the form , the relative spectrum of the sheaf of algebras.  Note: every vector bundle is a cone.  This really is a good generalization of vector bundles, at least for the purposes we need.

As we go, we’ll cite the properties and definitions of cones that we need.  But right now, we’re going to start talking about the most important one: the normal cone.  Now, let be a closed subscheme define by a sheaf of ideals .  Well, one thing we can do is look at the graded sheaf of algebras .  Then we take the cone it defines, and this is , the normal cone of in .

Now, let’s take a moment to make sure that we’ve got the right definition.  What happens if is a regular embedding of codimension ? Well, in that case, we actually get a vector bundle of rank , on .  If we look back in Hartshorne, we recall that is the conormal sheaf, so then is the normal sheaf.  And this is then going to be locally free of rank , so it gives us a vector bundle.  To see that it’s the one given by the normal cone, remember that is the total space of a vector bundle.  So we have at the least, consistency of language here.

Now, part of why the normal cone is so useful is that it is connected to the notion of a blowup.  First off, recall that , the projectivization of a certain cone.  Let’s call the blowup .  Adn we’re going to look at the exceptional divisor.  That is, if we have the projection , we look at .  This is a Cartier divisor, by the universal property of blowups.  Now, which one is it? Well, it’s actually going to be the projective cone of .  What is that? It’s precisely , so we get .

So then, , with .

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 .

So first we need to figure out what is K-Theory.  There are a few approaches we can take, and I’m going to start with the most basic.  Let be the free abelian group on the vector bundles (of any rank) on , modded out by the relation that if there exists a short exact sequence .  So K-theory will ignore all extension problems, and just say that an extension is equal to the sum of the two factors.

This makes K-Theory into an abelian group, and a useful one.  What makes it into a ring, is that we have a second operation on vector bundles.  We can take the tensor product.  So we define the multiplication that way, and then extend it by linearity.

Now, the Whitney Sum formula implies that the total Chern classes multiply in short exact sequence, which means that we have as a map from K-theory to, well…something.  The what we won’t worry about just yet.  The problem comes when we try to take , which isn’t as simple.  However, we can define , the Chern character of , to be , where are Chern roots of .  Working things out, the first few terms we get in terms of the Chern classes.

Here’s what’s wonderful about the Chern character: it’s a homomorphism.  That is, and .  Now, we haven’t quite nailed down what it’s a homomorphism into, but it’s something like the dual space of , or traditionally it is with appropriate coefficients.  So in answer to “Why does K-Theory tell us something about intersections?” we have “It has a natural homomorphism into an intersection ring.”

But it gets better! K-Theory isn’t REALLY about vector bundles.  It’s just a trick of the light, really.  We look for vector bundles, so we see them.  But really, K-Theory is something we do MUCH more generally: with complexes of coherent sheaves.  So yes, we’re sitting on the Derived Category here, though for the moment, we’ll just say that if there is a map from one complex to another that induces isomorphisms on the cohomology, they’re the same.  In this case, we take K-Theory to be a free abelian group on the coherent sheaves, with the same relation by short exact sequences (which implies relations for long exact sequences).  Now, the trick for showing that they’re the same theory is that under mild hypotheses (I’m not sure exactly what hypotheses are necessary, but I know smooth is more than sufficient…is it maybe Cohen-Macaulay? Anyone out there know?) Every coherent sheaf has a locally free resolution.

This means that every sheaf is quasi-isomorphic to a complex of vector bundles.  The relations in K-Theory then say that if is the locally free resolution, we have .  So we get the same K-Theory.  Even better, using these resolutions, we can define the Chern classes of complexes of coherent sheaves, via the Chern character homomorphism!

Now, this version of K-Theory is much more clearly intersection theoretic: Let be subschemes.  Then are their structure sheaves, and their scheme theoretic intersection is given by…the tensor product! There’s some interesting work being done on making these products explicit in K-theory, working out generalizations of the classical Schubert calculus  for homogeneous spaces.  Among those working on this are Buch, Kresch, Tamvakis, Mihalcea, and others (those are just the ones that I’m familiar with).

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.

Our clever trick is called the splitting principle.  The statement is that for any finite collection of vector bundles on our scheme , there exists a flat morphism such that

1. is injective
2. For all , we have a filtration by bundles such that is a line bundle.

To sketch the construction, we start with a single bundle, and we go by induction on .  Then we just projectivize , it has of the projective bundle as a subbundle, and so we can quotient by it to get a lower rank bundle.  For a collection of bundles, we just keep going and do them in sequence.

So, what does this really get us? The big bonus of splitting is that .  So Chern classes are all writeable in terms of those of line bundles, just maybe not on your original space.  We call these first Chern clases the chern roots of our vector bundle, and will write them as .

Specifically, we have that , where is the symmetric polynomial.  So then anything that we can write as a symmetric polynomial in the Chern roots will, in fact, be a polynomial in the Chern classes! We’re going to make much more use of this fact in the future.

The splitting principle lets us prove universal formulas of Chern classes of a finite set of bundles by just looking at the case of a direct sum of line bundles.  It’s a quick exercise to prove all of the following, where are vector bundles:

1. 
2. 
3. .

Before doing some applications, we make note that we’re going to abuse notation and for any a polynomial in Chern classes, we’re going to write instead of .  There shouldn’t be any real confusion.

So now, some quick applications of Chern classes to do some classical work:

1. Adjunction on a Surface: Let be an effective Cartier divisor on a complete surface .  Then, by definition, , where is the normal bundle.  Also by definition, we have .  So we have .  Thus, we have .  Setting , we get , and this becomes , the classical adjuction formula for surfaces.
2. Riemann-Hurwitz: Let be a map of smooth varieties of dimension .  We want to look at , the ramification locus, where the differential isn’t an isomorphism.  Well, that’s just the zero set of the map (take the determinant, it’s zero if and only if the map isn’t an isomorphism).  That tells us that .  Now, take and integrate both sides, wne we get , and so we recover the Riemann-Hurwitz Theorem as a special case.

This will actually often be the case: classical theorems turn out to be special cases of far more general results, which, once the correct machinery is in hand, can be proved almost effortlessly.  Note that so far, we’ve managed to recover the intersection form on surfaces, the Adjunction formula for surfaces and the Riemann-Hurwitz Theorem, all almost effortlessly from our formalism.  Things get a bit trickier when we start trying to pull Riemann-Roch out of this, however, and before then, we’ll have to talk K-theory, Chern characters and Todd classes.

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.

The first class is the most basic.  We say that a Weil Divisor on is an element of , where .  That’s it.

Next up: Cartier divisors.  These are collections such that the cover , is a nonzero function on , and on the overlaps the ratios are nowhere vanishing.  So if is a subvariety of of codimension one, we define to be for any local equation on an open set containing some point of .

Now, every Cartier divisor defines a Weil divisor by .  This is NOT an isomorphism in general, however.  It just gives a map from , where the first is the group of Cartier divisors.

Example: Let be a quadric cone in , and a line through the singular point.  Then is a Weil divisor (it’s definitely a cycle of codimension 1) but it is NOT a Cartier divisor.

The relation between Cartier and Weil divisors is part of the basic theory of algebraic geometry, as is the relation between Cartier divisors and line bundles.

The last gadget is the Pseudo-Divisor.  These are triples where is a line bundle, is a closed subset of , and is a nowhere vanishing section on on .  We call the terms the line bundle, the support and the section.  Two such objects are the same if they have the same support and there is an isomorphism of line bundles taking one section to the other.

Any Cartier divisor gives a Pseudo-Divisor by , and we say that represents this Pseudo-Divisor.  Now, note that we can have larger than .  So if , then all linearly equivalent Cartier divisors represent the same Pseudo-divisor.

Now, any Pseudo-divisor gives a Weil divisor in on its support by taking a Cartier divisor that represents (there always is one) and taking its associated Weil divisor.

Now, all these maps actually give us objects in of things, not just in .  So everything is a Weil divisor, and some things are Cartier divisors or Pseudo-divisors, which are just rigidified Cartier divisors.  So going forward, we’re going to have all of these objects floating around, but most of the time, it will be an exercise to distinguish them (when it’s not terribly important, we’ll just refer to a “divisor”).

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.

Now, let be a line bundle, and let a subvariety.  Then is given by some Cartier divisor, which gives us a well defined element of .  We’re going to denote this element by .  Extending linearly, we have (after using inclusions, of course).  This has the nice property that if is a divisor representing , then , so we get a lot of the same properties we had before.  But we also get some other things, here’s a list of basic properties:

1. Commutativity:
2. Projection:
3. Flat Pullback:
4. Additivity: and .

So, why don’t we just define the Chern class itself, rather than the map it induces on ? Well, really, what we have is just a map.  The Chern classes are somewhat cohomological objects, and so they pair with homological objects.  We’ll find the Chern class naturally once we have a cohomological object to work from.

We’re going to stop after defining all the Chern classes of a vector bundle, and will start playing with them later.  So first, we need to define the Segre clases.  Let be a rank vector bundle on , and the projectivization, with the projection from .  This bundle has a natural line bundle, .

We define by .  Now, the specific properties here aren’t too important, because we won’t be using them very much.  (And, if I’m wrong, they’re pretty much the standard properties)

So now, define as a formal power series with coefficients endomorphisms of .  Then the Chern polynomial is , and the coefficient of is .  These are the Chern classes.  Now, we’re mostly going to forget this (it’s important to construct them) and use their nice properties when we’re doing math:

1. Vanishing: For all , we have $c_i(E)=0$.
2. Commutativity:
3. Projection:
4. Pullback:
5. Whitney Sum: For any exact sequence , we have .
6. Normalization: when is a line bundle with .

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.

More generally, let be a cycle, then is the intersection class.  Let’s go through a list of all sorts of nice properties that this intersection product satisfies:

1. 
2. 
   The first two are fairly self explanatory: intersection is distributive on both sides.
3. Let proper, a divisor on and a cycle on , and set to be the restriction of to .  Then we get the Projection Formula .
4. Let flat, the appropriate restriction, and we get
5. If then .
6. .

Now, what does this get us? Look in , and let and be curves with no common components, of degrees and .  Because is a surface, we can view and both as being divisors and being -cycles.  So, we look at .  This is going to be a zero cycle on , and so we look at .   We want to compute this number.  Now, we make use of equivalence.  A divisor is linearly equivalent to , where is a line, so we have .  However, by symmetry, we have , and using linear equivalence again, we get , which is easily seen as .

So our new formalism contains Bezout’s Theorem, and generalizes it: let be divisors in , then where .  So this formalism should be useful as a beginning to intersection theory proper.  That seems like a good place to stop, so either there will be a second post later today, or a longer one tomorrow (or two tomorrow).

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.

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.

That question turns out to be less difficult than expected.  The key condition we’re really going to need is that we have fibers all the same dimension.  We’re going to assume that we actually have a FLAT morphism, and that the fibers are all of dimension .  This will give us pullbacks to open subschemes, to fiber bundles, to products with other schemes, and from a curve to a variety with a dominant map to the curve, so this will work out in a lot of the cases we care about.

So, very simple, we define .  This gives us a map .  In fact, this definition will actually work for any subscheme, when we interpret as the inverse image scheme of under .  That is, the fiber product of the scheme with over , both using the map .  It’s a fact, though a bit technical to prove, that this map descends to .

Now, we’ve got that is a covariant functor for proper morphisms and a contravariant functor for flat morphisms.  Now, we can use the pushforward and pullback operations to get an exact sequence: let be a closed subscheme of , and , with the inclusion maps and .  Then we have an exact sequence for all given by .  This is just the statement that cycle classes on are just cycle classes on which aren’t cycle classes on , which is something that is known to be true for divisors, that is, if is a variety, an effective divisor, and the complement, then the Picard group of is just that of after killing the class of .

Now, we look at affine bundles.  These are fiber bundles with fiber which trivialize in the Zariski topology.  Here, there’s a theorem: flat pullback is surjective for all .  To prove it, we take to be an open subset of on which an affine bundle is trivial, and the complement.  Applying the exact sequence, and then getting the maps via flat pullback, we can reduce to checking the problem on the inverse images of and .  Induction on dimension takes care of , and on , we have just .  As the projections factor, we can even reduce to the case of , and this case is manageable.

Now, this implies that for , we have zero, except for , as it is an affine bundle over a point.  Now, note, we’re talking about affine bundles, not vector bundles.  The fibers are torsors rather than vector spaces.  When we have the additional structure of a vector space (and thus can projectivize and keep working) we’ll be able to prove that flat pullback is an isomorphism.

Here’s an example by Chow.  Let has a cellular decomposition.  Say, like this one.  Then, we actually get that is finitely generated, and by the closures of the cells! So now, we know the Chow groups of the Grassmannian.

Finally, for our basic manipulations before we start trying to really do some intersections, let be schemes and their product.  We have an exterior product on Chow groups given by .  This is a priori defined only on cycles, but, as always, preserves rational equivalence, and so descends to the Chow groups themselves.  Additionally, both pullback and pushforward distribute over this product, and the product is associative.  So we have a lot of nice properties for a product .  This should give us some hope there there is a ring structure nearby.  A quick guess would by that if , the diagonal, is flat (which it is) and of relative dimension zero, we should be able to get a graded ring structure by .

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.

So first we need to introduce the basic objects of our study, a collection of groups that will eventually be turned into a graded ring.  We’re going to start with them acting somewhat more homologically, but later, they will act more cohomologically.  These are called the Chow Groups or the Cycle Groups of our scheme .

Put succinctly, a -cycle is a -linear combination (formal, of course) of subvarieties of dimension .  We’ll call this group .  Now, we want an equivalence relation on -cycles to get down to, in some sense, the fundamental ones.

For a dimensional subvariety , and any nonzero rational function on it, we define , just as we do for divisors, except now in the larger group of cycles on .  Then we say that a -cycle is rationally equivalent to zero if there are finitely many and finitely many rational functions such that .  These form a subgroup of , which we denote by .

And now, we get to the big definition: .  We’ll denote the direct sum by or , and we call a general element a cycle.  We say that a cycle is positive if its coefficients are, and a class is positive if it can be represented by a positive cycle.

Example: Here’s a quick and easy example.  Look at .  Then, for each , we get , where is a -plane.

A few other things: Chow groups don’t differentiate scheme structures on the same underlying set.  Anything has the same Chow groups as its reduction does.  Additionally, Chow groups add over disjoint union, so when studying Chow group, we can assume connected and reduced without losing any generality.  Finally, and this will be useful to reduce to the case of a variety, we have a short exact sequence: let be closed subschemes of .  Then we have .

Before we talk about what we can do with cycles, let’s associate a cycle to every subscheme of .  We start by defining the cycle of our ambient scheme .  Now, each irreducible component of gives us an Artin local ring , and we define to be the geometric multiplicity of in .  Then, we define to be the cycle in .  To get this to work for any subscheme, we note that there is an inclusion for any a subscheme of , and so we just take the image of in .

So now, we want to know what we can do to push these cycles around.  We’re thinking of them like homology at the moment, so our first thought should be that there should be a pushforward functor.  For this, we can’t work with just any morphism.  We’ll be needing a proper morphism.  Let be one.  We can define the pushforward for cycles on the subvarieties of , so let be one.  We define , where , and is the degree of the field extension if and are the same dimension, and zero otherwise.  It is straightforward to check that this gives a map on , not just .

In fact, we have a more precise formulation: assume also that the map is surjective.  Then, if has dimension smaller than , for every , we get the pushforward of to be zero.  If the same dimension, we get instead the divisor of the norm of , that is, the determinant of the -linear map .

Now, given a complete scheme, that is, one proper over a point, we have a natural map , which we’ll denote by or by .  This actually gives a map , which we call the degree of a cycle on .  Degree is preserved by proper morphisms, and so we’ll often just write

So now, this does all generalize a few things we know about.  For a -dimensional scheme , we have , so for a curve, we have is the Picard group, and the kernel of the degree map gives us back the Jacobian of the curve.

Eventually, we’re going to switch to cohomological viewpoint (essentially, we’ll reverse the grading) and introduce a product structure that encodes intersections.  Next time, we’ll talk about when we can take pullbacks, work with them for a bit, and talk about a few more things regarding general Chow groups before we start talking about divisors, line bundles, chern classes and vector bundles.

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.