 So this lecture is part of an online algebraic geometry course on schemes and will be about blowing up schemes. So it's like inspired by a video that Arnold Schwarzenegger made on blowing things up. So this is my version of this. So we recall how to blow up a point of a squared. What we do is we take a squared, which has coordinates x, y. And if we want to blow up the point zero zero, what we do is we take a squared times p1. So this would be points of the form x, y times a colon b. And we just take the subset of points such that x times b equals y, a. And you remember this maps onto a squared and the inverse image of any point of a squared other than the origin is just a point, but the inverse of the origin is an entire copy of p1. So it's blown up the point zero of a squared into an entire copy of the projective line. On the other hand, we also had this construction and project of s where s is a graded sheaf of quays, di-coherent algebras over a scheme. And the point is that blowing up a point or a sub variety is actually a special case of this. What we do is we take i to be a sheaf of ideals on x. And you remember a sheaf of ideals on x is essentially the same as saying a closed sub-scheme. And we blow up along the sheaf of ideals by forming a graded algebra as follows. We just take sn to be the nth power of i. So in particular, sn is i to the nought, which is just the sheaf of regular functions on x. And then project of s is called the blow-up of x along the ideal i or along the closed sub-scheme. So we should just check that this construction really is the same as our original blow-up construction. We'll just check it for this example. So here the coordinate ring of a squared is just kxy where k is the field we're working over. And the sheaf of ideals, which in this case is just the same as an ideal since we're working over an affine scheme. So the ideal is just the ideal generation by x and y. This isn't the point x, y. It's the ideal generation by x and y. Sorry, the notation is a bit ambiguous. So let's say this is an ideal, not a point. And if we call this i, we want to know what is the ring sum over i to the n. So this is an algebra over the ring r, which is kxy. And this is easy enough to work out, but the notation is a little bit confusing because we have x sort of appearing as an element of r and also as a generator of this ideal. So what we do is we map to some new variables a and b and we map r, a, b onto the sum of i to the n. And we're going to map this onto by mapping a onto x and b onto y. So we're dealing with the problem of having two copies of x and two copies of y by labeling one of the copies of x as a, if you see what I mean. So you notice this ring here is really kxyab. So it's kind of very confusing with having two different copies of x. Anyway, we notice that b times x is equal to a times y because x times y is equal to y times x and this ideal. And from this, we can see that this graded ring is really rab modulo, the ideal generated by bx minus ay. So this gives us the description of this sheaf of quasi-coherent modules, although they're really just ideals or modules. So we can now work out what proge of sum of i to the n is because we've got this explicit description of it. And you can see it's just the subset of k squared times p1 where k squared has coordinates x and y and p1 has coordinates a, b of elements such that bx equals ay. And this is just the same as our previous description of this blow-up. So blowing up along a quasi-coherent sheaf of ideals really is the same as the old version of blow-ups we had back in chapter one. Now what blow-up does, so blowing up along a sheaf i as the following effect makes i invertible or locally principal. And I want to explain what this does. So what happens is we've got a scheme x and we've got a blow-up of x along i. And over x, we've got a sheaf of ideals. And when you move the sheaf of ideals to the blow-up, it becomes invertible or locally principal. And we now run into a rather confusing technical problem which is how do we move i to x twiddle. So we've got a sheaf on x and we want to move it to a sheaf on x twiddle and it's this new sheaf that is going to be locally principal. Well, earlier on in this course, we had two ways of moving a sheaf from a space x to another space map to it. We could either take f to the minus one of i or we could take f star of i which is related to f to the minus one of i because we take f to the minus one of i and then we tensor over f to the minus one of o x twiddle. Because the problem is this is not a module on x twiddle. And in order to make it into a module on x twiddle, we have the tension with the ring of regular functions on x twiddle. So we obviously aren't going to use this as the pullback of i. So maybe we use this. So do we use this? And the answer is no. That's actually not what we mean by moving i back to x. There's actually a third way of moving the sheaf of ideals i to x twiddle. And if you're thinking to yourself that there are far too many ways of moving sheaves from one space to another, I will not disagree with you. So the third way is denoted by Hart-Schulner's f to the minus one i dot o of x twiddle. And let me explain what the difference between these two is. So well, let's look at the affine case to see what's going on. So instead of having a map from x twiddle to x, what I'm going to do is I'm going to take a map from a to b. So here I'm going to take these rings and m is going to be an a module. And I want to know how to make m into a b module. And there are two things you can do. You can just take the module m. So this corresponds to the operation f to the minus one. And that's no good because this is not a b module. So what we can do is we can take m tensor over a with b. And this corresponds to the operation f star on sheaves. And this is a b module. So what's the third way of moving a module m to b? Well, it doesn't work for all modules. What we do is we take i to be an ideal of a. And then we can look at the image of i, which is going to be a subset of b. And in general, it won't be a b module. There's no particular reason why it should be. So what we do is we take b times the image of i. And this is now an ideal of b. And it's the ideal of b generated by the image of i. And this is the one that corresponds on the level of sheaves to what Hart-Shorn calls the image i.b. And what I want to do now is to discuss the relation between this module, except I guess we would take m to be i. So i tensor over a of b and this one. And they are related because there is certainly a map from i tensor over a b to i to the image of i times b by the universal property of the tensor product. And this is certainly onto, but it need not be injective. In fact, it's quite common for it not to be injective. It's not injective in some of the simplest possible cases. For instance, we can take b to be a field k, and we can take a to be the ring of polynomials over k, and we map a to b just by mapping x to 0. So this corresponds on the level of schemes to mapping spectrum of b, which is a point to the spectrum of a, which is just a line. So we're just mapping the origin into the affine line. I mean, it's about the simplest possible non-trivial morphism of schemes you could possibly have. It's not pathological or anything. And now if we work out both sides, let's take the ideal i, just to be the kernel of this map. So i is going to be the ideal x. And then the image of i is just 0 in b. So b times the image of i is the zero ideal. However, if we take i tensed over a with b, well, i is just isomorphic to a one-dimensional free module over a. So this is just isomorphic to b, which is certainly not zero. So here we have a perfectly simple and natural example where these two are not the same. And for the purposes of showing that blowing up make all ideals invertible, it's this operation on ideals that is the one we want to use, not this one. As you see at the level of rings, this is the natural way to turn an ideal of a into an ideal of b. You just take its image in b and turn it into ideal and taking tens of products is a rather odd thing to do if you're just doing ring theory, but anyway. So anyway, now that we've got that out of the way, the rest of the proof is kind of easy, especially because Hartron is already written out, so I'm not really going to bother. What we do is to show that if we've got an ideal here and we blow up according to this ideal, then the inverse image, well, it's not the inverse image, it's this thing here is now invertible. And the reason for this is that the blow up has an invertible module, sorry, an invertible sheaf O of one as as any blow up does. And secondly, this sheaf O of one happens to be isomorphic to F the minus one I times O of X twiddles. And this is this is a fairly easy piece of bookkeeping. I just sort of sketch it. So this sheaf here corresponds to the following graded module, you take the sum of I to the n, and then you shift the degrees by one. So you remember, O of one is an invertible module where you take the graded module, the first thought of and just shift the grading. On the other hand, well, this is equal to I times sum over n i to the n. And the I corresponds to this bit here. And this graded ring of algebra is just the graded ring of algebra is that you used to form this. So if you if you unravel all the definitions, you find that this this sort of strange pullback of this invertible sheaf really is just the canonical invertible sheaf on the blow up. So I just briefly mention another application of blow ups, which is resolution of singularities. So here are Mark approved that you could resolve singularities of varieties and characteristics zero by repeatedly blowing up along various sheets of ideals. And his proof was 200 pages long. And fortunately, it's been simplified. So it's now actually proved resolution of singularities and only about 20 or 30 pages these days. But anyway, one of the frustrating things about resolution of singularities is there's there's a theorem that Hawkshaw improves that you can obtain any projective map by blowing up along a single sheaf of ideals, at least any map of a certain sort. So in order to resolve singularities of a variety, all you have to do is to think of the right sheaf of ideals. And then you could resolve the singularities in one step known as ever managed to figure out what an easy way to define the correct sheaf of ideals. So this is a sort of open problem that people have been working on for decades, try and find an easier way of resolving singularities. Anyway, the final application of blokes I want to mention is eliminating the points of indeterminacy of rational maps. And for this example, we're going to use the map from P two to P one, which I seem to be using over and over again. This is the map that takes X colon Y colon Z to X colon Y. And it's not really a morphism from P two to P one. So let's cross it out because it's not defined at the point zero zero. It's not defined at the point zero zero one, because then XY just becomes zero zero, which isn't well defined. So this is only defined on P squared minus a point. And we say it's got a point of indeterminacy at zero zero one. And you can turn it into a rational map by blowing up P squared at this point. So we get the map here. Sorry, map goes in that direction, that direction. So here we have the blow up of P two. And the blow up of P two does indeed map to P one. And the blow up of P two is just defined to be the subset. It's not defined to be it is the subset of P squared times P one, consisting of all points X colon Y colon Z and a colon B, such that XB equals YA. We should know this by now as this is about the fifth time we've had this example. So by blowing up a suitable point, you can sort of turn this this map that isn't defined anywhere into a map from this bigger space that is defined everywhere. So that's what is meant by eliminating points of indeterminacy. And you see what we're doing here is we're blowing up along the sheaf of ideals of the closed subscheme consisting of the point zero zero one. You remember closed subscheme such as being a bit sloppy here calling this point a closed subscheme, but it's fairly obvious what's meant. So you blow up along this closed reduced closed subscheme in order to resolve the indeterminacy of this rational map. So that's the simplest non-trivial example and I just finished by sketching very quickly how you use the project construction to do this in general. So now let's look at a general case. Suppose you've got a map from the scheme X to some projective space and suppose it's not defined everywhere. Well, suppose it's given as follows. Suppose we've got a line bundle on X or invertible sheaf and suppose we've got some sections S naught up to SN. And you remember taking some sections of an invertible sheaf on X defines a map from an open subset U of X to P to the N. So this isn't really a map from X to P to the N. It's really just a map from an open subset U of X to P to the N. And what we want to do is to replace this by a bigger space X twiddles mapping to X such that this bigger space does map to P to the N. And we're going to obtain X by blowing up something. So X twiddles going to be the blow-up of X along. Well, we've got to think of something to blow it up along. So what can we do? Well, we do this. So the sections S naught up to SN generate a sub-sheaf, let's call it J of L. So if they generate the whole of L, then this map would be defined everywhere. So the problem is that the sub-sheaf J may not be the whole of L. And if it was, we wouldn't have to do anything. So we've got a sub-sheaf J of L. And now we can twiddle both of them by L to the minus 1. So J tensor L to minus 1 is now a sub-sheaf of O of the ring of, is now a sub-sheaf of the sheaf of regular functions because L tensor L to minus 1 is just the sheaf of regular functions. So this is an ideal, or rather it's a sheaf of ideals. And now all we do is we blow up along the sheaf of ideals. So you can think of this as roughly speaking, it's making J into a principal, sorry, not principal, it's making J into an invertible sheaf, which is now generated by the sections S0 up to SN. And since we've now got an invertible sheaf generated by sections, this actually defines a map from X twiddle, from the blow-up X twiddle to P to the N. So we can use blow-ups to do half a dozen different interesting things. Okay, well that's enough about blow-ups. Next lecture we're moving on to study the sheaf of differentials on a scheme, which is a sort of analog of the one forms of a smooth manifold.