 This algebraic geometry lecture will be on some more examples of projective varieties called projective space bundles. In particular, we will do the special cases of hits of brook surfaces and scrolls. So let me just stop by recalling what a fiber bundle is. A fiber bundle is something that looks locally like a product. So a fiber bundle is going to consist of a total space mapping onto a base. And the fibers are all going to look like some fiber F. And if you take a little open neighborhood of B, so if we select little open set in B sufficiently small, then this will look locally like the projection from F times U down to U. However globally, it might not be a product. So a simple example of this is suppose we take the base to be circle and the fiber to be the real numbers S1. And obviously we could just map S1 times R onto S1 and this would be a product which would be a special case of a fiber bundle. So this would look something like a cylinder mapping onto a circle and the fibers are all just real lines, copies of reals. On the other hand, we could twist this construction and have a Moebius band mapping onto S1. So if we've got a Moebius band, well, you can think of the Moebius band as being projected onto a circle. And then the fibers are all going to be copies of what you could make more copies of say the real numbers. And locally it looks like a product. So if you take any little piece of the circle, then the thing mapping onto this piece just looks like R times this little piece. So locally it looks like a product. Globally it obviously isn't a product because for a start it's non-orientable. So when you go around the circle, the fiber gets twisted. So fiber bundles are sort of twisted products. Quite often the fiber is a vector space. And in this case we talk about a vector bundle. All the fiber might be a copy of projective space, in which case we talk about a projective space. So what we're going to do is to give some examples of projective varieties constructed like this. So the first example will be here at Sebrouk surfaces. So what we do is we take the space a squared minus zero times a squared minus zero. And what we're going to do is we're going to act on this by the group gm times gm. So gm for the moment can be thought of as let's work over the complex numbers. You can think of it as just being the group of non-zero complex numbers. And if we take the quotient of a squared minus zero by gm, with gm taking lambda of x, y... Sorry, gm takes x, y to... lambda x, lambda y for lambda in gm. Then the quotient of a squared minus zero by gm is of course just the projective line. So we can write a squared minus naught over gm is just the projective line p1. And similarly, we could let the other gm act on this a squared minus naught by the project... act on this a squared minus naught in the same way, and the quotient be another projective line. So if we take the obvious action of gm times gm on this and take the quotient, we just get p1 times p1, which is not terribly exciting. However, we can let it act in another way. Suppose we pick a point lambda mu. This is gm times gm, so we are just non-zero numbers. And we act on a point s, t, x, y. So this is an a squared minus zero. This is an a squared minus zero. And we're going to make the action slightly different. So this is going to take s and t to lambda s, lambda t. And then I'm going to take, multiply x and y by mu. So I get mu x mu y. Well, that's just the action I've mentioned up there, but I'm now going to triddle it slightly by putting the factor of lambda to the minus a here, where a is some integer. So now I'm going to take the quotient of this space by this group action. And that's going to be called a hit-so-brook surface. So f is going to be this copy of a squared minus zero times a squared minus zero, quotient by this action gm times gm. Well, there's a map from f to the projected line, taking s, t, x, y to s colon t. Because if we act on s, t, x, y by any group element, we just multiply s and t both by the same constants. This is a well-defined element of p1. And similarly, the fiber at any point is also isomorphic to p1. And you can check that locally this is actually a product, but globally in general, it's not a product. So what we've got is a surface that is a fiber bundle of p1 with fibers that are also copies of p1. So it's almost a product of p1 times p1. Only it's kind of twisted. Well, there's no reason why we have to do this in two dimensions. We can increase the dimension of things that are actually called scrolls. So this time you take a squared minus zero times a to the n minus zero. And I'm going to act on it by gm cross gm, where this time lambda mu acting on a point s, t, x1 up to xn is now going to be lambda s lambda e. And then we get lambda to the minus a1, x1, lambda to the minus a2, x2, and so on. Where a1, a2, and so on are all integers. This time, again, we get a map. So the scroll is defined to be the quotient of a squared minus zero times a to the n minus zero by this group action. And just as before, we get a map from the scroll to p1 by taking s, t, x1, and so on to s colon t. And this time the fibres are copies of n minus one dimensional project of space. Incidentally, as growth and Dick showed, these come from essentially all possible vector bundles over the projective line. So if you've got any vector bundle over the projective line, what you can do is replace each vector space that is a fiber by the corresponding projective space, and that gives you a projective bundle over the line. And this construction essentially runs through all possible vector bundles over a line. Anyway, strictly speaking, what we haven't done yet is shown this as a projective variety. So I've constructed some quotient, but I haven't actually yet embedded this into projective space. I mean, fibres are projective space, and the bases are projective space, but we've still got to project the whole thing into projective space. Let's map this to projective space. Well, let's assume that all the AI are greater than zero, which is fairly harmless, because if you shift all the AIs by a constant, it doesn't really make a whole lot of difference. And what we can do is we can then consider the monomials s to the i, t to the aj minus i, xj, where j is 1, 2, opt to n, and i is less than or equal to aj. So there's a finite number of monomials. You can see that you can take these as the coordinates of a point in projective space of dimension, sum of all the aj's plus 1, minus 1, because this is the number of different variables we have. And if you do this, you find that each of these fibres is mapped to a linear subspace of this large projective space. And strictly speaking, classically, when you talk about a scroll, you don't talk about this fibre bundle. What you talk about is its image in projective space. Well, it's a little bit clumsy to have to map everything into projective space. So here you see we've constructed a perfectly good algebraic object that's fibred by p to the n minus 1 over p1. And there's no real reason why we should be forced to embed it into projective space. For this reason, it's sometimes useful to talk about, to use abstract algebraic varieties. So abstract varieties were invented by André Vey. So the reason André Vey wanted them is improving the Vey conjectures for curves over finite fields. He needed to do a construction known as taking the Jacobian of a curve, which he wanted to be an algebraic variety. And unfortunately, it wasn't at all obvious at the time that these things could be embedded in projective space. And that turned out later they could be. So he came up with this concept of abstract varieties, which meant he could construct varieties without having to embed them into projective space. So let's summarise how you do this. First of all, let's look at differentiable manifolds. So differential manifold, an old view, might be that it's a subset of Euclidean space defined by some equations. For instance, a sphere might be the set of points x squared plus y squared plus c squared equals 1 or 3. However, this isn't really a differential manifold. It's really a differential manifold together with an embedding into Euclidean space. And nowadays, you don't think of a differential manifold as coming with an embedding into Euclidean space. So you can say the new view. We might, for example, cover S2 with charts. For instance, we might take two copies. If we've got a copy of S2, we might cover it with two pieces. We might cover it with a sort of top piece. And we might also cover it with a bottom piece, which might look like this. Now, each of these two pieces just looks like an open disk in Euclidean space. So we might construct S2 by taking two open disks in Euclidean space. And then we take this sort of ring-shaped bit and we glue these bits together. So we glue this ring to this ring here. And we can do exactly the same thing for an algebraic geometry. So we have an old view. A variety is a subset of project and PN. Well, the new view, Andre-Veys' view, is that you take some affine varieties and we glue them together. For example, in the old view, you might just take projective space. Say, let us take the projective line. We might consider that to be a subset of projective space. Well, that's not very difficult. We can just take it to be a subset of itself. Alternatively, we can glue it together as follows. We can take two copies of the affine line. So we take a copy of the affine line A1 and A1 and we glue them together. So let's see how to do that. Well, if we look at projective space with coordinates x, y, we have one copy of the affine line of the set of points 1, y with x none 0. And another copy of A1 consists of the points x1 with y none 0. So how do we glue these together? We've got two copies of the affine line. Well, the first copy, we look at the subset A1 minus 0 and the second copy of A1, we again look at its subset A1 minus 0. So here we're going to have points denoted by x and here is a point denoted by y. And we glue these two copies of A1 minus 0 together by mapping x to 1 over y, which you can see if you think about it. You can ask actually, what happens if you glue them by mapping x to y? Well, then you get something really funny because the first copy of A1, you've got one copy of A1 which is A1 minus 0 and a point 0 and you've got a second copy of A1 minus 0 together with a point 0 and you're gluing all the none 0 elements together with each other, but then you've got these two points left over. So if you sort of carelessly glue them by mapping x here to y there, what you get is something that looks like a line with two origins. That's a famous example of a non-Hasthorff manifold if you're doing topology. It also gives an example of an abstract variety that isn't a projective variety because projective varieties can't have this funny phenomenon that there are two points in the same place. So abstract varieties really are more general than projective varieties. Actually, this example is so annoying that people usually modify the definition of abstract varieties to exclude things like this and eliminate these non-Hasthorff sorts of varieties. It turns out that all reasonable abstract varieties, in other words, if you eliminate this sort of phenomenon in dimension one are automatically projective varieties. In two dimensions, if they don't have singularities, they're also projective varieties, but there are some three more examples of abstract varieties that are non-singular but not projective. All abstract, at least all complete abstract varieties turn out to be quite close to projective varieties. There's a lemma called Chow's Lemma saying that if you've got an abstract complete variety, it's pretty close to being a projective variety. So next lecture, we'll give a few examples of abstract varieties called Toric varieties.