 This algebraic geometry video will be about cubic surfaces and whether or not they are rational. So I'm going to give a somewhat informal argument to suggest that cubic surfaces are rational. So for cubic curves, such as x cubed plus y cubed equals z cubed in the projected plane, we have seen that these are not rational. So we can rewrite this as x cubed plus y cubed plus z squared equals naught, if you like, because then if you look at a cubic surface, w cubed plus x cubed plus y cubed plus c cubed equals naught in p three. You can see that these are very similar. And your first instinct might be to say that, you know, this cubic surface is almost obviously not rational. For instance, if you set w equals naught, you're getting a cubic curve in p two, which isn't rational. Therefore, this isn't rational. Well, that argument just doesn't work. There's no reason why a section of a rational surface needs to be rational. In fact, this is rational. And I'm going to give two arguments for this. And the first one is we observe that this contains two non-intersecting lines. And it's quite easy to write down these lines explicitly. One line could be given by w plus x equals naught, y plus z equals naught. So this line obviously lies on this surface. Another line might be w plus y equals naught, x plus omega z equals naught here, where omega is a cube root of one, and omega does not equal one. You should choose a slightly bad notation because this omega here is nothing to do with this w here. So it's pretty trivial to check these lines of no points in common. You may think that zero, zero, zero, zero is a common point, but that's not a point of p three. And now if we've got two non-intersecting lines, we can draw them. So here I'm going to draw a red line and a blue line. These are non-intersecting lines in p three. And now if I take a point on each of these lines, if I take a point here and a point here, I can draw the line through these two points and it will generally, generally a straight line will intersect the cubic surface in three points. And here are two points. There's probably going to be a third point where it intersects the cubic surface. Now there's actually a bit of a problem here because it's possible that this line might lie entirely in the cubic surface and every point will lie in the cubic surface, but let's ignore that possibility for a moment. So what this is giving us is a map from a point x on p one and a point y on another copy of p one and we're getting a point on the cubic surface. So what you see is we've sort of construct the map from p one times p one to the cubic surface. Well, the problem is this map isn't defined everywhere because if x and y happen to line a line that lies entirely in the cubic surface, then this point won't be defined. Now most points x and y, this line won't line the cubic surface, but there will be a sort of smaller dimensional subset where it does. So we see this map isn't actually defined anywhere, everywhere and is in fact a rational map. So it's very definitely not a regular map. There's no regular map from p one times p one to the cubic surface. Now we see these are both two dimensional and you can convince yourself in various ways that it's very plausible this map is almost onto. Again, it's not going to be onto. There will be some points on the cubic surface that don't arise from this construction, but most of the time, if you take a point on the cubic surface and kind of wave a line round through a varying point on this red line, there will usually be one red point you can find where it intersects this blue line. So by sort of waving your hands a bit, it makes it very plausible there's a rational map from p one times p one to the cubic surface. Now, does this map have a rational inverse? So it's possible that it's say a three to one map, maybe every point from the cubic surface arises in three ways. Well, let's see if that's possible. So suppose we had a point here that lied on, that lay on two different lines. So let's suppose there was a line here that also intersected these two points. Well, this would give us a contradiction because you see these two lines here form a plane. And this plane would have to contain the blue line because there are two points in common and it would have to contain the red line because there are two points in common. So this red line and this blue line would line the same plane. So they would have to intersect, but we assume they didn't intersect which is a contradiction. So this line here can't exist. For any point on the cubic surface there's at most one pair of points such that it's in their image. So this strongly suggests this map is generically one to one and probably has a rational inverse. So this strongly suggests that the cubic surface is birational to p one times p one. And if you push in a certain amount of work checking all the details you can find this really is a birational map. Well, this shows that any cubic surface with two non-intersecting lines is birational to p one times p one. And in fact, any non-singular cubic surface contains two non-intersecting lines and in fact it contains quite a few lines several of which don't intersect. This isn't completely straightforward to see. Anyway, I'm going to give a different argument now for why most cubic surfaces are rational. What we're going to do this time is to pick six points in the plane. So I'm going to pick six points here. One, two, three, four, five, six. And I'm going to pick these in general position. So general position is an extremely annoying phrase if you try and read older books on papers in algebraic geometry. What saying points in general position means is that these points must be points that don't satisfy a number of special conditions but you're not going to be told what these special conditions are. So in the older days in algebraic geometry there was a really big tradition of not actually specifying your theorems precisely or just make this vague comment about general position and it would just be rather tough on the reader because the reader wouldn't actually know what this means. I'm actually going to tell you what this means. It means no three on a line and not all on a conic. So any two points lie on a line and any five points lie on a conic. So I'm saying you can't have three points on a line or six points on a conic and this turns out to be enough. But in the spirit of old style algebraic geometry I'm not going to worry too much about what general position means. Now what I'm going to do is I'm going to look at all cubic functions. So the space of all cubics, all cubic polynomials in three variables is 10 dimensional because we have what we have x cubed, x squared, y, x, y squared, y cubed. And then we've got x squared, y, z, x. Can't add up x squared, z, x, y, z. And y squared, z. And then we've got x, we've got z cubed and here we've got x, z squared and here we've got y, z squared. So if I'd drawn this a bit, you'd be able to see these monomials lie on a triangle with exactly 10 points. So we've got a 10 dimensional vector space of cubics and I'm going to look at the cubics vanishing on all six points. Well, each condition about the cubics vanishing is a linear relation on these. So we've got six linear relations on a 10 dimensional space. So we end up with a four dimensional space. This requires the points being in general position if we chose all the points to be the same, for example, and obviously this space would be more than four dimensional. And let's suppose this is spanned by F1, F2, F3 and F4. And now I can try and define a map from P2 to P4 by mapping a point x, y, z to F1, F2, F3, F4. So x, y and z are only determined up to multiplication by constant, but if you multiply them by constant, all of these get multiplied by the cube of that constant. So this almost gives you a well-defined map from P2 to P4. The problem is this is not defined at six points because at the six original points, all these four functions vanish, so this isn't well-defined. So we've got a map from P2 to P4, but it's not a regular map, it's just a rational map because there are some points where it's not really defined. And now we can ask what is the image that this map from P2 to P4? Well, so not this isn't P4, it's P3. So it's got four coordinates, so it's actually three-dimensional, not four-dimensional. So what's the image of this map from P2 to P3? Well, it's very plausible to guess it's a hypersurface. So let's assume it's a hypersurface. And we can ask what is the degree of this hypersurface? Well, you can work out the degree of a hypersurface by counting the number of intersection points it has with the line. So let's pick a line in P3. And we might have this line being say F1 equals F2 equals zero. So these are both qubits and Bezut's theorem, which we haven't quite proved, but we'll talk about later. So if you've got two, a curve of degree M and a curve of degree N in the plane, then they generally have M times N points. So these have three times three equals nine intersection points. Well, six of these are the six points we started with, which don't really count because their images aren't in P3. This leaves three points. So generally, if we take two lines like F1 equals naught and F2 equals naught, there are going to be three points of intersection of this line with the hypersurface. So the hypersurface has degree three. So this suggests we've got a birational map from the projected plane to a hypersurface of degree three in P cubed, where of course, as I said before, this map won't actually be defined at six points. And we can try and guess how many cubic hypersurfaces we've found. On the one hand, the dimension of the space of cubic surfaces in P3 is 20. Sorry, the dimension of the space of cubics in P3 as dimension 20. So we can work this out in the same way that we did for P2. Only there are going to be more variables. So we get 10 plus six plus three plus one variables. So the dimension of the space of cubic surfaces is 19. What do I mean by this? Well, what I mean is you can think of the space of each cubic surface as being a point in some sort of parameter space. And this parameter space is just 19 dimensional projected space because we should subtract one because if you multiply a cubic by a constant, you don't change the surface. So we've got a 19 dimensional space of cubic surfaces. On the other hand, if you look at the dimension of the space of six points, that the dimension of the parameter space of six points is just six times two because there are two dimensions for choosing each point. So there's a sort of 12 dimensional space of choosing six points. On the other hand, we should take a look at the group of, dimension of the group of automorphisms of P. And this has dimension equal to eight because you can see the automorphisms is the projective general linear group on three in a three dimensional space, which has dimension eight. And similarly, the automorphism group of P three has dimension equal to 15. So this is three squared minus one and this is four squared minus one and we've got PGL four of K. So if you put this together and sort of think about it a bit, you can see that you seem to be getting a subspace, the dimension of the space of cubic surfaces you get by taking six points in P two is probably going to be 12 plus 15 minus eight, which is, that's gonna be 27 minus eight, which is 19, which is the same dimensions of space of all cubic surfaces. So this strongly suggests that you can get all cubic or released all non-singular cubic surfaces from this construction because at least the dimensions of the space is add up. You may notice that this argument is nowhere near being approved. It's got more holes in than a piece of Swiss cheese. This is fairly typical for all style algebraic geometry arguments. And the problem was the arguments are quite short and generally give the right answer, but the problem is every now and then they give a hopelessly wrong answer. And this is one of the problems that algebraic geometry suffered from until about 1950 or so, that it was full of informal arguments like this, which mostly worked and every now and then went hopelessly wrong. So this isn't a proof, it's just a informal suggestion that all cubic surfaces can be obtained by choosing six points in P two and doing this construction. This is actually correct, at least for non-singular cubic surfaces, but actually proving that takes considerably more work. However, in spite of the fact that these arguments aren't valid, it's still useful to be able to do them because it gives you a quick and dirty way of guessing whether results are true or not before you sit down and do a lot of hard work of trying to prove them. I should just mention earlier that I said there are 27 lines on a cubic surface and I just want to say very briefly how you might guess it from this construction. So here's 27 lines on a cubic surface. So what we do is we take P two and we take our six points in it and I'm now going to informly describe what the 27 lines are. Well, first of all, there are six points and I said you can't really define the map from P two to the cubic surface on these six points. In fact, that the map turns out to be indeterminate and for each of these points, there's an entire line in the cubic surface that might be the image of the point. So six points give rise to six lines in the cubic surface. If you like, you can think there's actually a regular map from the cubic surface back to P two and there are six lines in the cubic surface whose image are just these six points here. Next, there are 15 ways to take a line going through two of these points. So we get 15 lines in P two through two of the six points and the image of each of these lines turns out to be a line in the cubic surface. So here we get 15 lines. There's one final trick we can do. What we can do is we can take a conic going through five of these points. So any five points in the plane, there's a unique conic going through them and there are six ways. There are six conics through five points and if you sit down and work out what the image of a conic is the cubic surface, it turns out to give you another line. So this gives you six more lines and if you add this up, you get 27 lines. So that's a very informal explanation of why you get 27 lines on the cubic surface with as you see quite a lot of details missed out. This construction of going from P two to a cubic surface and having six points where the function is indeterminate turns out to be a special case of something called blowing up. More precisely, what we really do to P two to get a cubic surface is you blow up these six points. So in the next lecture, we're going to discuss what blowing up is.