A friend of mine from undergrad days has started up a math blog recently. He’s just getting started, but I, at least, expect him to do interesting things once he gets rolling (probably in another week or so, due to exams). So anyway, welcome Chris, and his blog Coffee and Mathematics, to the math blogosphere.

Ok, so with the overwhelming majority of one vote, the next thing I talk about will be algebraic surfaces and intersection theory.  However, first I need to do a bit of reading on this topic, as well as finishing up my coursework for the year, so for simplicity’s sake, I’m putting this blog on hiatus until I get back from the Cornell Topology Festival in early may, and when I get back, it’s back to writing this blog (which has turned out to be a rather good way of studying all of this material…)

I’ve generally been well-behaved about focusing on the math on this blog rather than going off into politics or whatever, but sometime while I was at my office today, someone went around campus and put up posters for Ben Stein’s “documentary” Expelled. This annoyed me, and so I’m printing up a bunch of admittedly plain (because I’m printing them myself and have no design abilities or colored ink) posters to stick up next to the advertisements promoting Expelled Exposed.

I’m mostly writing this post as something to do while they’re printing, so in the meantime, a list of links to places where the lies and hypocrisy of the producers of Expelled are more thoroughly debunked than I could ever hope to do.

Expelled Exposed

PZ Myers is Expelled

Lying for interviews

ERV gets angry

Collection of reviews

PhysioProf chimes in

Antisemitism in Expelled

More of the same

Greg Laden’s Expelled mini-Carnival

Expelled the Miniseries?

Hmm, without the witty commentary, I seem to have gone into a mini-carnival myself… But anyway, my flyers are done printing, and I’m going to go put them up next to the Expelled ones.

Ok, so last time, we discussed divisors. We’re going to keep going in that direction now, and now we’re going to talk about linear systems of divisors. Whenever we talk about linear systems, we’ll assume that our variety is nonsingular, so we can even talk about Weil divisors with no problem, though we’ll sometimes also use Cartier divisors due to how things will be handed to us.

Last time we took a Cartier divisor and got a line bundle. So let a divisor and the line bundle associated to the divisor. Now take . On open sets where is trivial, restricts to a regular function, so we get a Cartier divisor which we will call . This can also be taken as a formal sum of codimension one subvarieties. Now, the following are true, but I won’t prove them:

1. For any which is nonzero, the divisor is linearly equivalent to and is effective (recall that that means that the coefficients of the Weil divisor are all nonnegative)
2. Every effective divisor linearly equivalent to is for some .
3. If have , then there is a nonzero such that .

Now, this tells us that the vector space of global sections of , minus the origin, maps down to the effective divisors linearly equivalent to , but that nonzero scalar multiples get identified. This should sound familiar…mostly because it’s precisely how projective space itself is constructed. That means that there’s a natural structure of projective space on the set of effective divisors linearly equivalent to a given divisor!

So we define a complete linear system to be the set of all effective divisors linearly equivalent to some given divisor , which forms a projective space, and is denoted . A linear system is then , a linear subspace of , so it just corresponds to a vector subspace of . We define the dimension of a linear system to be the dimension of the projective space it defines.

Now, a bit more terminology: a point is a base point of a linear system if and only if for every , we have in one of the prime divisors of . In terms of sheaves, this says that for all , where is the vector space determining , we have .

Remember when we talked about how a line bundle determines a rational map to projective space? Well, in truth, it might define quite a few, though there is one into a biggest projective space arrived at this way. In fact, a morphism is the same as a linear system without base points on and a set of elements in the vector space determining it which span it.

So now when is the map a closed immersion, as defined in the comments on the post on line bundles? The conditions are that the linear system separates points and tangent vectors. The first condition is that for all points , we have a divisor in the linear section so that is contained in one of its prime divisors but is not. That is, there’s a function which is zero at one point and nonzero at the other.

Separating tangent vectors is a bit more mysterious. Take and latex . Remember that this is just a linear map . Then we want there to be a divisor such that . This tangent space makes sense, because is effective, and so gives an algebraic subset of , so . The point is that for any tangent vector, we can have it point in a direction not along some divisor.

Now that we have this definition, let’s do some examples. We’ll take , and choose our divisor to be where is a hyperplane and . It’s a fact that every divisor on projective space is linearly equivalent to one of this form. So now we look at . This will consist of all formal linear sums of hypersurfaces with . These are precisely given by the homogenous polynomials of degree . We can choose as a basis for the complete linear system the functions given by monomials of degree . Now, this system is base point free and separates both points and tangent vectors, so we get a morphism , the latter dimension being the dimension of the space of homogeneous degree polynomials after projectivizing. We’ve actually already seen this map! It’s just the Veronese Embedding, phrased with linear systems. A similar construction can be used to get the Segre Embedding, or, as mentioned, any map into projective space.

That seems enough for now, and we’ve got a couple of options of where to go next that I’ve been thinking about, and I’m going to leave it up to you readers. Post a comment to let me know which of the following is preferred, and I’ll do it:

1. Riemann-Roch Theorem and the geometry of curves
2. Bertini’s Theorem and more about divisors, including generalizing to cycles and some intersection theory
3. Something rather different: some computational techniques, blow-ups, and the 27 lines on a cubic surface
4. Other suggestions? I know what I’d do for the other three, but if something is suggested that people want to see me make an attempt at explain, I’m open to the possibility

Some people might say that the natural place for this topic is before talk of differential forms and of the canonical bundle, but I disagree. Well, really it’s fine either way, but this is my blog, so I’m going to do it my way. Today we’re going to talk about divisors and their relation to line bundles.

First off, we need to define the local ring of a subvariety, because otherwise we’re restricted to looking at actually nonsingular varieties, when we can in fact get away with a little bit less for the nice version of divisors. If is a subvariety, we define to be equivalence classes of rational functions on where is defined somewhere on . Note that the dimension, as a ring, of will just be the codimension of in . (For our purposes, )

So we say that a variety is regular in codimension one if the local rings for every codimension one subvariety are regular. That is, for each of these rings, the maximal ideal (of functions vanishing on ) satisfies . Roughly what this means is that our variety isn’t singular along codimension one subvarieties, but it may be for higher codimension (that is, lower dimension). So a curve has to be nonsingular, but a surface can be singular at a point, and a threefold can even be singular along a curve.

So now, we define a prime divisor to be a subvariety of codimension one (thus the need for regularity). In general, a Weil Divisor is a formal integer linear sum of prime divisors. So we act like we can add and subtract subvarieties. We can write any divisor as where are prime divisors and integers. We call a divisor effective if the are all non-negative.

So now take an arbitrary rational function on which isn’t identically zero. We want to define the order of vanishing of along the prime divisor . We do this by looking at the image of in . It is a theorem that we can write as for the generator of the maximal ideal . We define , in this case. It happens that there are only finitely many subvarieties such that is nonzero. This will always be a positive number, and we call it the order of the zero along . Now, for those subvarieties on which can’t be defined, we look at , which can be extended to a function that is zero along them, and define to be along them, and call it the order of the pole along .

We call any divisor of the form a principal divisor. We note that , and so taking a function to its divisor gives a homomorphism , the group of Weil divisors on .

So now a couple of really important definitions: we say that two divisors are linearly equivalent if is principal, and we define .

Now, Weil divisors are actually not terribly good, but that’s how we’ll often be speaking of divisors, when we use them (mostly like points on a curve of curves in a surface or the like). What we REALLY want are called Cartier Divisors and for these we fundamentally need sheaf theory.

Denote by the constant sheaf taking values the field of rational functions on , and then is the group of invertible elements. Now is the sheaf of invertible regular functions. We look at the sheaf , and note that there is a sheafification that needs to be performed to get here. We define a Cartier divisor to just be a global section of this sheaf. We call a Cartier divisor principal if it is in the image of the natural map .

Now, a Cartier divisor can be written as an open cover of and on each element of the cover, a nonzero element of the function field such that on we have a regular function. Though this is a multiplicative group, we will speak as though it is additive, because we really want to pretend that these are Weil divisors. So we define a pair of divisors to be linearly equivalent if their difference (ratio) is prinicpal.

Though it isn’t in general the case, for nonsingular varieties, the notions of Weil and Cartier Divisors agree. More generally, we’ll always use Cartier divisors.

So now, let be a (Cartier) divisor. We can use this to define a Line Bundle! We do this by taking the sub--module of generated by on . Because is invertible on the overlap, the sheaves we get will coincide, and so we get a sheaf on the whole space. We call this line bundle .

The nicest fact, is that this gives a homomorphism from Cartier divisors modulo linear equivalence to the Picard group! So linearly equivalent divisors define the same line bundle, and the group operations are compatible. Even better, the homomorphism is injective. It isn’t always surjective, however, over the more general spaces often considered (like schemes), but is is true for varieties by our definition of them.

Just a little bit more, and then we’re done for the day. We call a Cartier divisor effective if it can be represented by regular functions on the open sets. Then it defines a subvariety, which then has ideal sheaf . We want to be able to say that is isomorphic to , but there is one problem. Divisors can have multiplicity along subvarieties. So if the regular functions vanish only to order one, then we’re fine. Otherwise, we’ll need a more general notion of a subvariety, which I would want to call a nonreduced variety, but we’re not going to get into this (at least not yet, they’re easiest to describe as schemes, but I’m trying to avoid that word).

And as we finish, one last definition. A divisor corresponding to the canonical bundle will be called a canonical divisor, and all canonical divisors are linearly equivalent. We will most definitely use this in the future.

We’re going to need to start out the day with a bit of algebra, because we’re going to talk about differential forms. Once we have forms, we’ll make a sheaf out of them, and then we’ll use this sheaf to construct other things.

We start out by needing the notion of a derivation. Given a ring , an -algebra (that is, a ring which is also an -module) and an -module , we define an -derivation to be a function such that for all , , and for all . (We say that an element is in if it is of the form for using the module structure.)

So really, what this is is just a function that acts like taking a derivative. The product rule is there, we think of as the set of constants, so they have derivative zero, it’s additive, etc. There is, in fact, a universal such derivation. Specifically, there’s an -module called and a derivation such that any other derivation is given by composing with a module homomorphism . Even better, we can describe it.

We take the collection of symbols for and take formal finite sums of the form where . These are the elements, but we do identify some of them. Specifically, we identify with , with and with zero for . We define the universal derivation to be which takes . We call this module the module of Kähler differentials

To get a bit of a sense of this, we’ll note that if and , so we’re just looking at affine space, then is where the are polynomials. For those who have done a bit of differential geometry, this should be looking familiar: it is an algebraic analogue of 1-forms.

Now, Kähler differentials have an extremely nice property: they commute with localization. That is, if is an -algebra and is multiplicatively closed we have . And by the way, I do apologize for the notation, using for an -algebra is standard, as far as I know, and in the localization post I used it for a multiplicatively closed set.

So now we must come up with a sheaf theoretic version of this, because varieties are only locally equivalent to rings. Let be a topological space and sheaves of rings, with a homomorphism of sheaves of rings. This makes into an -modules. We now define to be the presheaf taking to . The restriction maps are given by taking , which is then an -derivation, and so factors through , giving is the desired restriction maps. Now, once we’ve done all of this, we sheafifiy, and now we have a sheaf which we will call , the sheaf of relative differentials.

So now, we can specify to varieties. Let be a morphism of varieties. By definition we have a map . But we want the sheaf to be on . So recall that when we talked about morphisms of sheaves, we briefly mentioned the inverse image sheaf. Using this, we get a morphism , which makes into a -module. We define to be , and call it the relative cotangent sheaf.

As a special case, we define to be the cotangent sheaf of , and as an immediate consequence of the above definition, we can see that if is a morphism of affine varieties, then is the sheaf associated to , and this fact implies that is always a coherent sheaf.

So now, the cotangent sheaf gives us a new way to test for a point being nonsingular. If has dimension , and , then is nonsingular at if and only if is isomorphic to . This tells us that is nonsingular if and only if is a locally free sheaf of rank !

So this gives us a cotangent bundle! Now, we’re going to change notation slightly, and call this bundle (and the sheaf associated to it) . We can get the sheaf of -forms by looking at , where this is the exterior algebra construction. So in that case, we see that if has dimension , then is zero, so we only get of these sheaves. Aside from , there is another special one among these: . This one always turns out to be a line bundle, and we denote it by , because it is so important. In fact, we call it the canonical bundle, and it will be showing up in the future regularly.

Before we stop for the day, we’re going to use to define one more object that is very important. We define , the dual, to be the tangent sheaf . If is a vector bundle, so is , and it is called the tangent bundle. We’ve now brought over a lot of valuable objects from differential geometry, and they’ll be quite important in our further study of varieties.

We’ve now talked about vector bundles and locally free sheaves, we’re going to specify to the nicest case: rank 1. We’re generally going to ignore the distinction between the sheaf and the line bundle.

We start by noting an alternative name for locally free sheaves of rank one: invertible sheaves. The reason for this is because given an invertible sheaf , we can define , the sheaf hom. This turns out to be a locally free sheaf itself, and has rank one. Now, if we look at the sheaf (recall that we do this by taking tensor products over open sets, and then sheafifying) we get . So in a sense, is the inverse for , as their tensor product is not just locally free, but is in fact free.

More is true, in fact. is always invertible if both factors are. Now we make note of the following, assuming are all line bundles.

1. 
2. 
3. .
4. 

These isomorphisms should look familiar: they are precisely the axioms for an abelian group! This group is called the Picard group of , and denoted . Because of this group structure, we will denote by . We will, however, keep the notation for higher rank locally free sheaves.

The fun with the Picard group will REALLY start next week, when I introduce divisors, but there is a bit more we can do before then: like use line bundles to define maps of varieties into projective spaces.

Let be a line bundle. If our variety is defined over the field , then will be a -vector space. Now, choose a basis (if the vector space is trivial, we call the map the empty map to the empty projective space, and so we assume positive dimension ).

So now we remember that these are not just sheaves, but the sheaves of sections of a geometric vector bundle. We fix an isomorphism of each fiber with the field , it will turn out not to matter which isomorphism we choose, we just need one. Then any global section defines a regular function on . So we get a map into . This map, however, depends on the isomorphisms chosen for the fibers, but we can get rid of this dependence by projectivizing. So now we have a rational map . It’s only a rational map, because we don’t know in advance if the have any common zeros, and the map can’t be defined on those points. And finally, the choice of basis for doesn’t matter, because different bases will give maps that differ only by a linear transformation of the projective space, so we won’t worry about that ambiguity.

So now, we have a rational map for any line bundle. If the map is in fact a morphism, then we call very ample. A line bundle is ample if some tensor power is very ample. There are lots of other characterizations of these line bundles. some of which we’ll encounter.

That’s all for now, next time, we’ll construct a couple of specific vector bundles on nonsingular varieties.

Last time, we talked about sheaves of modules, and focused on the correspondence between sheaves of ideals and subvarieties. We were talking about the internal geometry of the variety. Today, we’ll talk a bit about more external geometry. Specifically, we’ll talk about how sheaves give us new varieties with maps to whose fibers are all vector spaces. In fact, they’ll look locally like open sets of times a vector space. Such objects are called vector bundles, and are rather closely tied to the theory of sheaves of modules on .

We call a sheaf of modules free if it is isomorphic to a direct sum of copies of . We call the number of copies of the rank of the free sheaf. Now, we will call a sheaf locally free if there exists a cover of by open sets (we don’t require them to be affine) such that is a free -module. Note that a locally free module is automatically quasi-coherent, and is coherent if it has finite rank. If is connected, then the rank is the same everywhere.

A quick algebraic comment is in order: the coherent sheaf associated to a finitely generated module is locally free if and only if it is a projective module, that is, there exists such that is free. Though beware, if we were working with schemes instead of varieties, we need to be more careful about this. But we’re focusing on varieties right now, so everything’s ok.

Now let’s look at things from the other direction. As mentioned earlier, if is a variety, a vector bundle over is a variety with a map such that the following conditions hold:

1. For each , we have is isomorphic to for some
2. There exists a cover of such that is isomorphic to .

We note quickly that if is connected, then we can use the same at each point, and say that the vector bundle has rank .

So now lets do some examples of each. For any variety , there is a vector bundle for each positive . This is called the trivial vector bundle. Other vector bundles are easiest to describe by using locally free sheaves, once we’ve described the correspondence. But first, some examples of locally free sheaves. Fix an integer . Then we define the locally free sheaf of rank one on called to be the one taking each open set to the set of ratios of homogeneous functions such that is nonzero on and . This is locally free, because if we restrict to a standard copy of affine space, we are then looking at is assigned rational functions defined everywhere on , that is, we get precisely .

So on to the correspondence. If we take a vector bundle, there is a relatively simple way to construct a sheaf. Assign to the collection of morphisms such that is the identity on . This sheaf will be locally trivial, and the open sets we use can be the ones for which . This is because on these sets, we’re looking at functions which compose with the projection to give the identity. This is the same as looking at morphisms , which is just a list of regular functions on . So a vector bundle defines a locally free sheaf.

So now we start with a locally free sheaf of rank . Pick an open cover of such that is free for each in it. We can choose the open cover to be finite, because varieties are quasi-compact. So now we take the disjoint union of for all . So now we have one isomorphism and another . Restricting each of these to (we will use this convention from here on out) we get two different isomorphisms , and we will denote them by . We then get , an automorphism of . Now, by the isomorphism with , we can identify this with an matrix of regular functions on .

So now we glue. We take and and identify them along by the map which takes to . So now we perform this for all , and call this object , and it comes with a map by forgetting the vector coordinate on any point. So the fibers are now copies of and by construction, around each point there’s a neighborhood on which the space is . So all we need to do in order to check that this is a vector bundle is to check that it is a variety. It certainly has an open cover by affine varieties, again by construction, and in fact this cover is finite. The rest follows from the fact that is the identity map. So we’ve now established a correspondence between locally free sheaves and vector bundles.

Next time, we’ll further investigate a special class of these, the line bundles. That is, vector bundles of rank one. The associated locally free sheaves are called invertible sheaves, because we’re going to make a group out of them, which means that each will have an inverse.

I’ve decided to start experimenting with wordpress’s themes in order to get a different look from the other math blogs.  What do people think of this one? There’s a few others that I’ve been considering as well, and I may continue experimenting throughout the month.

So now that we have abstract varieties on hand, we’re going to do a bit more with sheaves, leading to some of the intimate connections between sheaf theory and geometry. Sadly, this often gives students a lot of trouble (I know I had a bit of trouble with it at first) because things are presented very algebraically and the geometry gets lost. So we’ll be making a point of the connections between the geometry and the algebra.

As usual, we start out with a definition. We take to be a ringed space (say, a variety) with structure sheaf . Then we say that a sheaf is an -module (or sometimes a sheaf of -modules) if for each , we have is an -module, and the restriction maps on are compatible with the module structures induces by the restriction maps in . A morphism is a homomorphism of -modules if over each open set it is a module homomorphism.

Now, pretty much anything we can do with -modules we can do with -modules. We can take kernels, cokernels, images, quotients and the like and all we get are more -moduels. We can talk about exact sequences of -modules by just looking to see if they are exact as sequences of sheaves. In fact, we can even make a sheaf out of morphisms of -modules so that this sheaf is one too, by defining it as , where restriction to just means that we consider it as a sheaf on and so only care about open sets contained in .

We can even define the tensor product of two sheaves, though we do have to perform a sheafification, because it won’t generally be a sheaf itself.

However, now, we’re going to focus for a moment on what these can tell use geometrically. To start with, we will look at an affine variety , which has coordinate ring . Now, any closed subset of is defined by an ideal . In fact, this gives an ideal in every localization of , which is to say, for each open subset of , we have an ideal in the coordinate ring of that subset. Now, ideals are always modules, so in fact, this collection of ideals forms a sheaf of modules!

We can, in fact, generalize this. Let be a closed subset of a variety. Then, to each open, we assign , that is, on each open set, we assign the regular functions that are zero on . This gives a sheaf of modules (in fact, this example is so important that we call these sheaves of ideals) and it is determined by the closed subset. So if we took a subvariety, that is, an irreducible closed subset of a variety, we’d even get a sheaf of prime ideals.

Now, we should note that the correspondence doesn’t quite go both ways. A closed subset of a variety defines a unique sheaf of ideals. However, if we are given a sheaf of ideals , and then we take the set of points in each where the elements of vanish, we might not even get a closed subset. There are sheaves of ideals which bear little to no resemblance to the example of an ideal for an affine variety, so we will fix this with a condition called quasi-coherence. We say that a sheaf of modules is quasicoherent if we can cover with open affine subsets such that when we restrict the sheaf to each of these sets we get particularly nice sheaves of modules, which we will now describe:

Given an affine variety and a module over its coordinate ring, we get a sheaf of modules called on , by looking at the localizations of the modules. We will call a sheaf of modules quasi-coherent if it is locally isomorphic to sheaves of this form. If we can even take the modules to be finitely generated, we will call the -module coherent.

So what makes the ideals we want special? We only care about quasi-coherent sheaves of ideals, that is, sheaves of ideals that locally look like they come from the localizations of actual ideals. In fact, we can do better, because the Hilbert Basis Theorem tells us that ideals are finitely generated for polynomial rings and their quotients. Thus, the sheaves of ideals that we want are coherent.

So now given a coherent sheaf of ideals, we do in fact get a closed subset. However, many coherent sheaves of ideals give the same one, just as many ideals give the same closed subset of an affine variety. We can fix this by taking the radical ideal over each open set. This still gives us a coherent sheaf of ideals, and one which is equal to its radical will be called reduced. But now we get the same correspondence we used to have: closed subsets of are in one-to-one correspondence with reduced coherent sheaves of ideals. Subvarieties, in fact, correspond to reduced coherent sheaves of prime ideals, and any other operation on ideals can still be performed, taking all the notions from affine geometry with us into this new land of abstract algebraic geometry.