 This algebraic geometry video will be about a sort of bi-rational map called a flop which is sometimes used in classification of varieties of dimension at least three. An early example of this was found by Michael Atia in about 1958 and this is the example we're going to look at. So we start by looking at the hypersurface x, y equals z, t in four-dimensional affine space with coordinates x, y, z, t. And the first thing to do is what this variety looks like and it acts like a cone over p1 times p1. This is because we can also look at the corresponding variety consisting of points in projective space satisfying this condition here because this is a homogeneous equation and this is a quadric in projective space. So the set of points of x, y equals z, t as a quadric in p3 and we saw earlier that that gives us a product of two projective lines. So in projective three space, we get a product of two lines. So in affine four space, this is more or less a cone over a product of two lines. In particular, it's got a singularity at the origin at nought, nought, nought, nought, which is a sort of the singular point of the cone. And we can resolve the singularity by blowing up at the point nought, nought, nought, nought in A4. So to do this, we look at A4 times p3. So the coordinates for A4 are going to be x, y, z, t. And the coordinates for p3, let's use capital letters. So we get x, y, z, t. And the equations satisfied by the resolution, well, first of all, we have x, y equals z, t. And then we have a whole lot of equations relating these like x, y equals y, x, x, z equals z, x and so on. And we take the an image of the non-zero points in A4 under this map and then take the Zariski closure. So we also end up with the equation x, y equals z, t. So what happens is each point in A4 other than the origin corresponds to a unique point in this blow up, whereas the point nought, nought, nought, nought in A4 becomes an entire copy of p1 times p1 because it consists of all the points x, y, z, t satisfying this equation. So the singular point of this variety has now been blown up. We can check that this blow up is non-singular because along, say, x equals 1, it becomes y equals z, t. Here the coordinates are little x, big y, big z, and big t, and this is obviously a non-singular variety. So this resolves the singularity. Now the point is this exceptional variety can be blown down to p1, sorry, it can be mapped to p1 in two different ways. So there's one map to p1 on the first coordinate and one map to p1 on the second coordinate. I mean, it turns out that instead of blowing up the origin to p1 times p1, we can sort of half-heartedly blow it up to just a p1 and we can do this as follows. So what we can do is we can take this singular point with x, y equals z, t and instead of blowing it up at point zero, we can blow up along the line, y equals t equals zero. So in order to do this, we end up looking inside the variety, a4 times p1 and here we have coordinates x, y, z, t and here we're going to have coordinates x and z in projective space. And if we blow up, we get the equations x, y equals z, t and then as before, we get x, big z equals z, big x. But taking the closure of the image of a4 minus zero, we obviously get some extra equations t, z equals y, x because x over z is just t over y, except possibly the origin, so we can also add this equation. So here the blow-up is going to be given by these equations here. Now, if we say we take x equals one as one of the coordinate patches, this becomes y equals t, z and z equals x, z which and x, y equals z, t, which reduces to affine three space so it's non-singular. So in other words, if we blow up this hypersurface along the line, then we again get a non-singular, a non-singular variety. This time, the inverse image, the point zero is blown up to a copy of p1 rather than p1 times p1 as we had earlier. And instead of blowing up along the line y equals z equals nought, we could also blow up along the line x equals z equals nought and we just sort of switch y, t and x, z. So we would find another way of resolving this singularity by blowing up the origin to a line. So let's put everything together and draw a picture of what we get. What we get down here is the hypersurface x, y equals z, t. And I'm going to draw this as a sort of blob and it's got some sort of singular point here, which is the conical point. And then we can blow it up along a point. So here we blow up along nought, nought, nought, nought and what we do, what we get is a copy of p1 times p1. So I'm going to draw it like this. Here's a sort of p1 times p1. So we can see we've got these raw copies of p1 and these raw copies of p1. And somehow this point has been replaced by this. But we found two other ways of blowing it up, which sort of look like this. And we could blow up a point to just one p1 or we could blow up a point to the other p1. So all these four varieties are the same except they differ in these sort of colored points. So we've got a map here projecting all the blue lines down to points. So maybe I could guess draw points that indicate the blue lines have been contracted and here the red lines have all been contracted. And we've got regular maps going in these directions. Here is the map from here to here. This is the map directional map. And it's a bit of a strange map because it's taking a copy of p1 in this space and somehow mysteriously replacing it by p1 embedded in a subtly different way. So it's sort of changing a co-dimension to sub-variety to a different co-dimension to sub-variety. And it's otherwise an isomorphism. So what's the point of this example? Well, the point of this example is it shows there's no minimal way of resolving singularities in general. So what we would, it would be really nice if when we're given a singularity instead of a variety, so we've got some singularity. What we'd like to have is a sort of minimal resolution of it, a non-singular variety mapping nicely to it such that any non-singular variety can be factored through this. And you can do that to some extent for curves and maybe for surfaces if you're careful. But you can't do it for three files because here we've got the singularity and there are two ways of resolving singularity given by this blow-up and this blow-up, but there's no way of resolving it that these both map through. So there's no sort of minimal best way of resolving this singularity. This means if you want to find some sort of so-called minimal model, in other words, a resolution of singularities that's as small as possible, it turns out the best thing to do is to allow very mild sort of singularities in your minimal model. So instead of eliminating all singularities from your variety, it turns out to be better to allow some very mild sorts of singularities. So you'd allow singularities like this one here, x, y equals z, t. And the exact sort of singularities you allow are called terminal singularities. Okay, the next lecture will move on to discussing singularities in more detail. So for example, I'll actually explain what they are.