 This lecture is part of an online algebraic geometry course about schemes and in it we'll be discussing a sort of generalization of the construction of projective space where we do this construction simultaneously over every point of a variety. So first of all, let's recall the construction of a projective variety from a graded algebra s which is sum over n greater than or equal to nought of an algebra s n, sorry a module s n. So, we remember we constructed a scheme called prodge of s, projective scheme of s or something, and as an underlying set, it consists of prime ideals, not containing sum for n greater than zero of s n. And the topology was given as follows, first of all the base was given by the open sets of the form df, which was the set of prime ideals, such that f is not in the prime ideal, you can think of this as being the points where f doesn't vanish slightly informally. And we defined a sheaf on each of these open sets by saying the sheaf on the open set d of f, you get by taking the ring s you and then invert the element f, and then you take the degree zero elements of this. So we gave several examples of this showing that if s was a graded algebra, then this was, so if this was a graded algebra over a field s nought, then this was often a projective variety over the field k. And so in particular we get a map from prodge of s to the scheme spec of s zero. You also remember that this space comes with a line bundle. So roughly speaking for every graded module over this graded algebra, we get a line bundle in this, so we get a sheaf. And for the special case where we take the algebra itself and just shift the grading to get a new module, we get a special line bundle 01. Now the idea is to generalize this, instead of working over a ring s zero or an affine scheme spec of s zero, what we want to do is to work over any scheme x. So the idea is roughly, let's describe it in formal version, what we do is we pick a graded algebra for each point of x. And then this goes from each graded algebra we might be able to get some sort of projective variety for each point. And we kind of join these together somehow to get a map from some space y to x such the fibers are maybe projective varieties or schemes or something. So that's the sort of informal way of doing it, which doesn't really make too much sense if you look at it too closely. So let's be a bit more precise. What do we mean by picking a gradient algebra for each point of x? Well, a better way of describing that is we pick a sheaf of graded algebras over x. What this means is the sheaf is going to be s, which is going to be sum over n greater than or equal to zero of s i. And the sheaf is going to be an algebra so we've got a map from s, hence s to s, giving us a commutative multiplication and this preserves the grading and so on. And so the stalk of this sheaf at every point will be roughly a graded algebra over the local ring at that point. So this is that you can think of this as being something like a graded algebra for each point of x. Well, it should have various nice properties. So s should have following properties. First of all, it should be quasi-coherent if it isn't, it's a real mess. So this just says it sort of looks locally like a module over a ring. So this is the really important property. There are some other sort of optional extra properties we can have which make life a bit easier but aren't really absolutely vital. First of all, we could assume that s naught is the sheaf of regular functions on x. So this is, if we're working over a field, this would be like saying s naught is the field k. Thirdly, we can assume that s i is quite is not quite so coherent, coherent for i greater than or equal to zero. So this would be kind of saying it's sort of like a bit like a finite dimensional vector space over a field in some sense. And fourthly, we could say that s one locally generates s. So this is for a graded algebra, this would be the equivalent of saying that the degree one elements generate the whole space. In practice, these conditions are often satisfied and make things easier, but they're not really vital. So now what we do is we define a scheme proge s as follows. What we do is we pick an open affine u in x. And then if we restrict s to u, it becomes essentially a sheaf of graded algebras over the ring of u. So let's say u is a spectrum of some ring a. So it becomes a graded algebra over a. And on this, we can form the projective scheme of this graded algebra. Let's call this algebra something that's called an s a. So what's happening is x is covered by these open affine sets. And on each of these open affine sets, we're defining a sort of projective scheme mapping to the sets. And then we just glue these together. So gluing them together takes a page or two of calculations, just checking that they're compatible on these intersections and so on. But I'm simply going to omit it because it's fairly straightforward and not very interesting. Moreover, each of these has an invertible sheaf o of one that we constructed before. And we can also glue together these invertible sheaves o of one and we get an invertible sheaf over proge of s. So what we get is we get a map from proge of s to x and proge of s has an invertible sheaf denoted by o of one. So it's kind of like the constructing a projective variety from a graded algebra over a field, except that we're doing it simultaneously for all points of a scheme x. And there are several applications of this. So let's list some special cases that we will discuss in more detail. First of all, we can just construct projective varieties. Secondly, we can construct projective bundles over scheme x. And thirdly, we can blow up a subscheme or equivalently a sheaf of ideals on x. Fourthly, we can make ideals locally principal, which I will explain later. Fifthly, we can resolve singularities. And sixthly, we can make rational maps defined everywhere. Actually, these two are really both variations of the same idea. And all these four things are really variations of blowing up a subscheme. So blowing up a subscheme can make ideals locally principal. It can resolve singularities and it can make rational maps well defined everywhere. So this lecture, I'm going to just discuss examples one and two. And next lecture, we'll be just giving examples of the remaining four applications of this. So example one is really kind of trivial. All we're doing is we're taking x to be the spectrum of fields. And we're taking s to be a graded algebra over k. And then you can think of a graded algebra over k as being a graded sheaf of ideals over the spectrum of k if you want. And then the construction we've given for taking proge of the sheaf of ideals s is really the same as the construction we gave earlier for constructing a projective variety from x. So application one, we've sort of already done. Slightly more interesting variation of this is taking projective bundles over a scheme x. And we do this as follows. Let's take s1 to be some locally free sheaf. So this looks locally like o of u to the n for u and open affine subset of x. And then we can put s to be the symmetric algebra of s1 over s0. So for vector spaces, we can define the symmetric algebra, which is just the sum of the nth symmetric power of s1. And you can do exactly the same thing for sheaves just by doing it locally. So this gives us a symmetric algebra and we can then form proge of this symmetric algebra sum over s to the ns1. And locally, this looks just like constructing projective space over open sets u. So locally, this just maps some projective space over u to some open set u. But globally, it might be a sort of twisted version of this. So the point is the projective spaces at different points can't necessarily be identified with each other. So we can probably easiest to see the simplest example of this. See what's going on. Let's just take x to be one dimensional projective space. So s1 is going to be some locally free sheaf. Well, we classified the locally free sheaves over p1. This is growth index theorem or maybe Burkov's theorem. And they're all sums of line bundles o of ni. And let's just take the dimension of s1 to be two. So we're going to take o of m plus o of n. And we're going to form the sheaf that is the sum of the symmetric powers of o of m plus o of n. And we're going to take proge of s and see what it looks like. So what we're getting is over each point. This will look roughly like a two dimensional vector space. So the corresponding projective space will just be one dimensional projective space. So we get a map from proge of s to p1. And the fibres are just copies of p1. So we say it's a p1 bundle over p1. Obviously it could be p1 times p1 mapping to p1, but it can also be some other things as well. So these things are called Hitzebrook surfaces. Hitzebrook studied them in his first ever paper published in the early 1950s. And first of all, let's see how many we get. So obviously if we start with o m plus o n, this is going to give us the same as if we did o n plus o n. So we can just switch m and n without making any difference. But also if we take the surface of o m plus k plus o n plus k, this is isomorphic to the surface from o n plus o n. And the point is that this thing here is just o m plus o n. It's twisted by a line bundle. Now if you've got a vector space v, and we tensor it with some one-dimensional vector space l, then there's a canonical identification from the projective space of v tensor with l. I mean, these are obviously isomorphic projective spaces, because they're the same dimension. But the point is this isomorphism here is canonical and doesn't depend on choosing a basis for l or whatever. And because it's canonical, this means if we've got a locally free sheaf s1 and a locally free sheaf s1 tensor l, where this is an invertible sheaf, then the projective space bundle we get from s1 is isomorphic to projective space bundle we get from s1 times l. Because since all these isomorphisms over open sets are canonical, they're compatible whenever you take intersections of open sets. So the projective space bundles of these two are actually isomorphic. So the hits of Brooks surface of this two-dimensional bundle is isomorphic to the hits of Brooks surface of this two-dimensional bundle because you're just twisting by a line bundle. Incidentally, there is one difference between the hits of Brooks surfaces you get from these because the construction gives us not only a surface, but also a line bundle over the surface. And we do, in fact, get different line bundles over the hits of Brooks surface from these two different line bundles over p1. And so let's look at a few hits of Brooks surfaces and see what they look like. So every hits of Brooks surface is isomorphic to a hits of Brooks surface sigma n, which comes from taking the line bundle O nought plus O n over some integer n greater than or equal to zero because if we're allowed to switch these and add a constant to them we can make one of the entries zero and the other entries some non-negative integer. So sigma nought is easy. This is just p1 times p1 because what we're doing is taking the trivial line bundle O of zero plus O of zero over p1 the corresponding projective bundle and obviously we're just getting something canonically isomorphic to p1 because the fiber of this at each point is just canonically a two-dimensional space. So sigma nought is just a copy of p1 times p1. Sigma one turns out to be a little bit more complicated. It's actually isomorphic to p2 blown up at a point. And this is a little bit confusing to see. It's not difficult, but it's confusing. So let me try and get it right. So first of all, we take the copy of p1. So you can think of this as having coordinates x colon y. And now we're looking at the line bundle O of zero plus O of one. And let's think about what this line bundle looks like. Well, this line bundle means we're assigning a two-dimensional vector space for each point of p1. And we want to take an element to this two-dimensional vector space for each point of p1. So how do we do this? Well, O of zero is easy because that just means we take a complex number. On the other hand, if we want to take an element of the one-dimensional vector space corresponding to p1 it doesn't mean we take a complex number. It means we take an element of the line corresponding to p1. So really we have to take points B and C such that xc equals yb. So if we take a non-zero element of this, what we're really doing is taking a point of form A colon B colon C with, again, xc equals yb. So in other words, the elements of the projective bundle over p1 corresponding to this can be thought of as follows. They consist of pairs xy, A colon B colon C, such that yb equals xc. So we're taking a subset of p1 times p2 satisfying this condition here. And this is just equal to the blow-up of p2 at the point 1, 0, 0. So sigma1 is isomorphic to p2 blown up at a point. So here's a book showed that, in fact, all the surfaces, sigma n for n greater than or equal to nought, are distinct and not isomorphic to each other. However, sigma n is homeomorphic to sigma m, if and only if m is congruent to n mod 2. So the surfaces sigma nought and sigma 2 give examples of two surfaces that are birational and homeomorphic, but not actually isomorphic as algebraic surfaces. OK, next lecture we'll give some examples of using this construction to blow up ideals and some of the things you can do with that.