 This algebraic geometry lecture will be about blowing up. So we will start with a simple example. Suppose we take the curve y squared equals x cubed plus x squared, which looks a bit like this. So the curve comes around like this. Now it's got a singularity at the origin and we can simplify it by making the substitution y equals tx. And then we find t squared equals t squared x squared equals x cubed plus x squared, which factorizes as x squared times t squared minus x plus one. So the result of this is in the x t plane, we now get two curves. We get the curve t squared equals x plus one, which looks like this. And we get another curve x equals zero, which looks like this. So our original blue curve has been transformed into a curve without singular points. And instead we've picked up an extra curve here, where extra line, this line here is called the exceptional curve. So we sort of simplify the singularity of our original curve at the cost of somehow introducing an extra line. So what's happening is the effect of this transformation is to leave everything in the plane more or less as it is, except this point in the plane is replaced by this entire red line. And this red line contains a point for each direction through the origin. So for example, there's a direction going like that, which corresponds to this point here. And there's a direction going like that, which corresponds to this point here. So that replaces the point by an affine line with the possible values of t. That's a bit artificial because it doesn't include a point for this vertical line. So blowing up really includes one extra point. So the general form of blowing up a point in the plane. So suppose we've got the x, y plane, a two, and we blow it up at the point zero, zero. What we do is we introduce two new variables. Let's call them x and y, which we think of as points in line P one. And now what we do is we look at the variety a squared times P one, whose coordinates can be given by x, y, and then we've got these pair of projective coordinates. And we take all the subset of this, such that x times big y is equal to y times big x. Now you notice if x, y is not nought, nought, there's just one point of P one corresponding to it. So this takes points in a squared that aren't the origin to a single point here. But on the other hand, if x, y is equal to zero, zero, then we get the whole of the affine line P one. We can have any point x, y. So if we take the subset of a squared times P one defined by this, this is called the blow-up of a two at a point. So we get a map from the blow-up back to a two, which takes x, y, x, y, just x, y. And the inverse image of a point in a two is either a single point, if we're not at the origin, or the whole of the line P one. So the point zero, zero has been changed to an entire line P one, which you can think of as being the possible directions through the point zero, zero. And P one is covered by two affine lines. We can either take x, not equal to zero, so we may as well take x equals one, or y not equal to zero, so y equals one. And if x is equal to one, then we're just making the transformation x times y equals y. And if y isn't equal to one, then we're just making the transformation x equals y times x. So in one case, we're dividing x minus y, and in the other case, we're just dividing y by x. And the blow-up is a quasi-projective variety covered by these two affine varieties. So in n dimensions, this is very similar. We take A n, which has coordinates x one up to x n. And then we take a copy of P n, whose coordinates, let's use a capital X one up to capital X n, that's P n minus one, not P n. And then we look inside A n times P n, and we take the set of all points such as x i times big x j, is equal to x j times big x i for all i and j. And again, we see that the sub-variety defined by all these points maps onto A n, and the inverse image of x one up to x n is a point if it's not equal to zero, zero, zero, and it's equal to a copy of P to the n minus one if we're at the point zero, zero. So it's replaced the origin by an entire copy of n minus one dimensional projective space. So we sort of blown up this point into an entire copy of projective space. Again, P n is cut, so P n minus one is covered by affine subsets the form x i not equal to zero. So if we look at say x one not equal to zero, we may as well take x one equals one. And then the equations we have are x, sorry, x i times x one is equal to x one times x i and since x one is equal to one, this is just x i equals x one times x i. So we just introduce a new variable big x i equals x i over x one. So the effects of blow up on this subset is that we're essentially just dividing all, introducing new coordinates by dividing all the old coordinates by x one. And of course you can do that with x one replaced by something else. So let's have a few more examples of this. What one major application of blowing up is to resolve singularities. So let's resolve the singularity of y squared equals x cubed. Okay, we haven't defined singularities yet. That comes in a few lectures, but it's fairly obvious what a singularity is. It's just somewhere where the variety goes wrong. So if we draw this curve here, it looks a bit like this and it's got some sort of singular point cusp at the origin. And now we want to blow up at the origin. Well, you remember what blowing up consists of, we can divide y by x and we can divide x by i. So let's just divide y by x. We put y equals x t and then we find we get the equation x squared t squared equals x cubed, which factors as x squared times t squared minus x. So this gives us the following picture. First of all, we've got this exceptional curve x equals zero appearing and we've also got the curve t squared equals x. So here this is t, this is the x-axis. So our original curve with the singularity of the origin and these new coordinates has been transformed into a nice parabola without any singular points. And we've picked up this extra curve here. So blowing up has resolved this singular point. So let's have an example in high dimensions. Let's look at the surface x squared plus y squared equals z squared. So this is a cone in three dimensional space. It sort of looks a bit like this. So it's just a cone. And you can see something is going wrong at the origin. It's got some sort of place where it isn't smooth. So let's blow it up at nought, nought, nought. And there are three different affine pieces we get in the blowup because we could either divide by one of these variables or one of the others. So let's divide by z rather than x and y. So we put y equals z times s and x equals z times t. So we're introducing new variables s colon t which are going to lie in the same direction. So we're going to lay it in another confine space, sorry, I shouldn't be using projective coordinates. So we get z squared t squared plus z squared s squared equals z squared. So this factors as z squared times t squared plus s squared minus one. And now if we draw things in the zst plane, we find what we're getting is this blue cone has been transformed into a cylinder. So this is just the cylinder t squared plus s squared minus one equals zero. So here we are looking at the st plane rather than the xy plane up here. And we've also got the plane as z equals nought. So we kind of get an extra red. The entire plane z equals nought is now an exceptional curve. It's an exceptional plane. So just as with blowing up in two dimensions, our original variety, the singular point gets expanded into something less singular. And we also sort of pick up an extra exceptional variety. Let's have a look at a slightly more complicated example. Now let's look at y to the eight equals z to the five, which is some sort of higher order cusp. And let's blow it up. Well, we can blow it up once. So we can either divide y by z or we can divide z by y. So if we divide y by z, we can put z equals y times t. And then our curve becomes z to the, sorry, y to the five, t to the five equals y to the eight. So we get y to the five, t to the five minus y cubed equals zero. And y equals nought is going to give us our exceptional curve and t to the five minus y cubed still has a singularity. So we take this and we blow it up again. So how can we blow it up again? Well, we can put y equals t times s. So we find t cubed s cubed minus t to the five equals nought as what happens if we blow up twice at a point, which obviously gives us t cubed times s squared minus t squared equals zero. That should be an s cubed. And t cubed is going to give us our exceptional curve and we've got an s cubed minus t squared. Well, we've had that before. We can blow it up by putting t equals s times u and then we get s cubed equals s squared u squared. So we find s squared equals nought or s equals u squared. And now finally, we've managed to transform our curve into a curve without a singularity. So here it's just a parabola. So we had to blow it up three times. We start with this curve. If we blow it up once, we get this curve. If we blow it up twice, we get this curve. Sorry, if we blow it up twice, we get, where's it gone? Sorry, I should have said this one here. And if we blow it up three times, we get this curve here. So now let's have a look at a pinch point, x, y squared equals z squared. This is called the Whitney umbrella. And if you draw a picture of it, you can see y. So a picture of it looks something like this. If I choose my axes like this, so here's the x-axis and here's the y-axis and here's the z-axis, then the set of points like this kind of looks something like this. There's a sort of parabola coming in here. It's definitely like that. So what happens is that the plane has sort of, we've got a plane, but it's kind of been folded in on itself. So it's got a sort of double line here where the plane actually crosses itself. In fact, the entire axis here is also in the curve. So this is the line y equals z equals zero. You may think this looks like a combination of a one-dimensional line and a two-dimensional surface, but it's really an irreducible two-dimensional surface if you look at it in the complex plane. So the axis is the set of singular points. So these are all singular points, obviously because this plane is folded. And these are actually singular points and you can't see this because we haven't drawn the complex points. So you can think of this as being the handle of the umbrella and this bit here is the umbrella that Whitney apparently used to keep the rain off him or something. Now, what we want to do is to blow it up in order to make it better. Now, if you look here, it's got a line of singular points, but there's one singular point that seems to be even worse than all the others, which is the origin because here we've just got two planes crossing each other transversely and here we've got something more complicated. So we can say the worst point is zero, zero, zero. So we're going to blow it up. Well, let's blow it up. What we do is we introduce new variables. So we're going to look at, remember, we take a copy of affine space and then we take a copy of two-dimensional projective space. Let's call these coordinates x, y, z. And we choose one of these three coordinates to be zero to get one of the affine pieces covering it. So if we divide by x, for example, we put y equals to x, s, sorry, x, y, and z equals x, z. So let's figure out what's going on. Well, then we get x times x squared, y squared is equal to x squared, z squared. And this simplifies as x squared times x, y squared minus big z squared. So we've transformed this singularity into an exceptional curve and another singularity. And now we sort of notice that this new singularity is in fact exactly the same as the singularity we started with, which is a little bit unfortunate. So blowing up this singular point of the Whitney umbrella, the worst singular point, doesn't actually make it any better. We get same singularity all over again. So what can we do to make this better? Well, I've been talking about blowing up along a point, but we can also blow up along an entire line. So let's try blowing up along a line and see what happens. So we're going to take the Whitney singularity and blow it up along a line, whatever this means. So we take coordinates x, y, z. Sorry, these shouldn't be projective coordinates, they should be ordinary coordinates. So we take a cubed and then instead of taking a copy of p two, we're just going to take a copy of p one and I'll give this coordinates s colon t because if I call them a subset of x and y and z, I'll get muddled up. And we're going to add in the conditions that y times t is equal to z times s. So what we're doing is we're replacing each point on this line by a projective line of, you can think of it as being all directions that are normal to this line in some sense. So let's see what happens where we've got x, y squared equals z squared. So if you blow up along this line, well, we should choose s or t to be equal to one. Let's choose s equals one. So we have y times t equals z. So we're putting z equals yt and we get x, y squared equals y squared t squared. And now this is a big improvement because we get y squared equals naught or x equals t squared. And x equals t squared is a nice and unsingular surface. So blowing up along a point doesn't work, but a more complicated operation of blowing up along a line does get rid of all the singularities. So in general, we would like to get rid of all singularities of varieties by repeatedly blowing up along various subvarieties. And an obvious way of trying to do this is you say at each point, you find the points with the worst possible singularity and they form a subvariet and you might try blowing up along that. And this sort of works, but you have to be really careful about how you define worst singularity. So you see in the Whitney Umbrella, it looks at first sight as if this point is the worst singularity, but blowing up along that point doesn't really help at all. So somehow the subvariety of worst singularities must be this entire line. So you have to define worst singular point in a very subtle way. So this point here is no worse than all the other points, even though it seems to be. Okay, so that gives some examples of blowing up along points and occasionally lines. And in the next lecture, we'll talk about a bit about blowing up along more complicated objects such as subvarieties or ideals or even sheaves of graded algebras.