 This talk will be an informal survey of complex rational surfaces. So we will start by giving a few obvious examples. So the most obvious examples of rational surfaces are the projective plane and P1 cross P1. So we recall a rational surface is anything birational to the projective plane, and it's kind of obvious that these two both are. And next we can look at non-singular hypersurfaces in P3 and try and figure out which of them are rational. Well, first of all, degree one surfaces are trivially rational because they're all just hyperplanes, which are isomorphic to P2. If we look at degree two hypersurfaces, these are also rational, but you have to sort of think for a few seconds to see this. So they're given by the zeroes of some sort of quadric, which can be put in the form Wx equals Yz. And these are in fact isomorphic to P1 cross P1. And this is not difficult to see. For instance, we could take coordinates of P1 cross P1 and map this to, say, A, B, A, sorry, C, D, A, C, B, D, which is Wx, Y, Z. And you can see that this gives isomorphism from P1 cross P1 to this conic. So that's easy. Degree three is much more subtle. These are also rational. But it's much less easy to see it. These are the famous cubic surfaces which have 27 lines on. This is a famous result due to Cayley and Salmon and it is almost the first non-trivial result of high-dimensional algebraic geometry. Anyway, these turn out to be isomorphic to P2 blown up at six points. So I'm going to discuss these quite a lot more later on in the talk after finishing off a few examples of rational surfaces. You might ask what happens for degree four and above. Or degree four, or greater than or equal to four are not rational. Degree four turn out to be the famous K3 surfaces, or at least some of them. And degree five and above are surfaces of general type in the Federa-Enrique classification. Anyway, before discussing cubic surfaces in more detail, let's just give a few more examples of rational surfaces. These are the, what I mean to discuss are the hits of brook surfaces. And these surfaces are the surfaces that are projective P1 bundles over P1. So the surface maps to P1 and its fibers are all copies of P1. And you can get any projective P1 bundle by taking a two-dimensional vector bundle, and the two-dimensional vector bundles over P1, all of the form OM plus ON, where this is, stands for Sarah's usual twisted sheaf. And if we twist this by line bundle, then it doesn't change the P1 bundle we get, and we can swap M and N. So we may as well assume that it's of the form O0 plus ON for N greater than or equal to zero, because every hits of brook surfaces isomorphic to one of these. And they're named hits of brook surfaces because hits of brook had a paper where he carefully discussed them and worked out their topology and so on. And actually they were well known before hits of brook studied them, but mathematical objects are never named after the person who first studied them, so whatever. Anyway, so we get this family of hits of brook surfaces for N equals zero, one, two, three, four, five, and so on. And the one for N equals naught, you're just taking the trivial bundle, so that's just isomorphic to P1 cross P1. The surface for N equals one turns out to be P2 blown up at one point. So it's got a map to P2 with fiber, a copy of P1. And do you remember we said last lecture that any surface is obtained from a minimal surface by blowing up in a number of points? Well, the minimal rational surfaces are these ones there, the hits of brook surfaces apart from this hits of brook surface, which can be blown down to P2. So these are the minimal rational surfaces. The minimal rational surfaces, I think were first classified by Macaron about 1948 or so. So any surface can be obtained by blowing up one of these. However, it can usually be obtained by blowing up one of these in many different ways because these surfaces are all related. Each surface can be obtained from the previous one by blowing up a point and then blowing down a line. So each of these surfaces has a zero section. And if you blow up a point in the zero section, you kind of go one direction. And if you blow up a point that's not in the zero section, you go in the other direction. So all these hits of brook surfaces are all quite closely related. Incidentally, this shows that we can get from P2 to P1 cross P1 by blowing up two points and then blowing down a point. And that's the example we discussed in the previous lecture. I'm not going to go through the remaining examples because those are not too dissimilar from the example we did. Anyway, so now what we want to do is to look at the relation between cubic surfaces and P2 with six points blown up. Well, first of all, we're going to look at the problem. What happens if you blow up endpoints on P2? What do you get? Here I mean to take the endpoints to be in general position. So what does general position mean? Well, what general position means is that I'm being lazy and can't be bothered to write down the exact conditions that these points should satisfy. This was very common in algebraic geometry in the early part of the 20th century. And it's very annoying trying to read papers because you never actually know what the theorems are stating because they've always got this annoying general position condition in there. Anyway, what we're going to do is we look at the space of cubics through these endpoints. And the first question is, why cubics rather than degree four curves or whatever? Well, it turns out this is related to the fact that cubics are essentially sections, related to sections of the anti-canonical bundle. So you remember any non-singular variety as a canonical bundle, which is the highest exterior power of its cotangent bundle, and the anti-canonical bundle is the dual of that. And normally, you embed varieties by using the canonical bundle, but that doesn't work so well for rational surfaces. So instead, people use the anti-canonical bundle. This is high dimensions. These are things where you embed using the anti-canonical bundle, a close relation to things called far-no varieties, which is a measure to del petso surfaces. Anyway, we need to work out what is the dimension of the space of cubics? Well, a typical cubic is going to look like somewhere A300x cubed plus A210x squared y plus all the way down to plus A003z cubed. And you see there are 10 coefficients. And this gives you a nine-dimensional projective space of cubics. And you remember, if you've got a space of cubics, then we get a birational map from P2 to some projective space. Except it may not be defined at a few points because our space of sections might all vanish at some given point. But generally, if we get an n-dimensional space of cubics, we'll get a map from the projective plane to n-dimensional projective space, except it might not be defined at a few points. But you can make it defined by blowing up these points. So generally speaking, what will happen if we blow up n points where we can take n to be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and so on? Now, if we have more than nine points in general position, then there will be no cubics at all through those nine points. That's one of the conditions that general position means. Of course, there are some sets of 10 points that do line a cubic, and most of them don't. So if we look at the cubics through these n points, then we will get a map, but a rational map from P2 to P9, P8, P7, P6, P5, P4, P3, P2, or P1, or P0, which is a point. So we get a map to this. And its degree will be 9, 8, 7, 6, 5, 4, 3, 2, 1, or 0. That's one of the conditions for del petso surfaces, as it should have a degree n embedding into n-dimensional projective space, except these ones at the bottom are a bit funny because they don't actually give you embeddings, but anyway. And we can say what these are. Well, first of all, this one is the Veronaise embedding. So the Veronaise surface usually means a degree, an embedding of P2 into P5, defined by looking at quadrics, but instead of looking at quadrics, you can use cubics, and then you get an embedding into P9, and you could use quartics or high-degree curves if you wanted, but this is sort of just analogous to the usual Veronaise surface. So it's well understood. This one turns out to be the Hitzerbrook surface, or its image is the Hitzerbrook surface H1 that we mentioned before. This is the one we're going to discuss in more detail. It's the famous degree 3 embedding into P3. As you can see, it's really part of a family of degree n embeddings into Pn. And so this is the cubic surface. This one also has a name. It's called a Segre surface, and turns out to be an intersection of two quadrics in P4. So these things here are the famous Del Petzso surfaces, if I pronounce the Italian right, except the definition of Del Petzso surface is a bit fuzzy about these ones here. Not everyone agrees about whether or not these cases here are the ones as Del Petzso surfaces. So now let's look at the case of cubic surfaces a bit more and ask why does a cubic surface have 27 lines on it? Well, I'm going to cheat a bit here because I'm not going to answer why a cubic surface has 27 lines on it. I'm going to answer why a projected plane blown up in six points has 27 lines on it. So showing that's a cubic surface takes a little bit of work. So first of all, what are these lines? Well, these lines turn out to be exceptional curves. So you remember these are exceptional curves. They're curves E with self-intersection number minus 1 and E rational. So let's look at P2 blown up at six points. And let's find exceptional curves on it. Well, it obviously has six exceptional curves, at least six exceptional curves, given by the blow-ups of these points. Well, six is rather less than 27. So there must be some other exceptional curves on this. Well, we saw some of them in the previous lecture. Suppose we take two of these points that we blew up and we take a line through them. And if we blow these up, these exceptional points become little copies of P1 that I'll draw like that. And the inverse image of this curve consists of a line together with these two exceptional curves. And if we call this line L, then we saw last time that the intersection number of this line with itself is 1, which is the intersection of this line with itself, minus 1, minus 1. So these are the self-intersection numbers of these two exceptional curves E1 and E2. So we can get another exceptional curve by taking two of the six points we blew up and drawing a line through them. And let's see what happens. Well, we've got six points. So we just draw a line through two of them. And there are how many ways to do that? Well, the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. So this gives us an extra 15 exceptional curves. So how many have we got? We've got 6 plus 15 equals 21 exceptional curves so far. And there ought to be 27. So we're still missing six of them. So where can we find these other six exceptional curves? Well, we can do it like this. What we do is we take five points that we blow up. 1, 2, 3, 4, 5. And let's take a conic through them. And this is easy to do because if we've got any five points in the plane, there's a, well, any five points in general position, there's a unique conic through them. So let's see what happens if we blow this up. Well, we get five exceptional curves. And we get the inverse image of the conic, which turns out to be a curve with self-intersection number. Well, the conic has self-intersection number four because any two conics intersect in two points if they're in general position. So the self-intersection number is going to be four minus one, minus one, minus one, minus one, minus one, which is equal to minus one. So we found another exceptional curve, given by taking a conic through five of the points we blow up and looking at its inverse image. So let's count the number of points, exceptional curves. We've got six coming from the points. And we've got 15 coming from lines through two points. And we've got six which come from conics through five because there are six ways of choosing five points of our original six points. And if we add these up, we find there are indeed now 27 exceptional curves that we found. Well, at least 27. In fact, there aren't any more, but I'm feeling too late to show that. Well, we can look at what happens for other numbers of blown up points. So let's blow up P2 in endpoints and ask how many exceptional curves are there. So here we can take n to be zero, one, two, three, four, five, six, seven, eight, nine, 10. And let's count the number of exceptional curves we get. Well, first of all, we can get some exceptional curves just from these points. So these come from the blown up points. And then we've got some exceptional curves by taking two of these points and drawing a line through them. So we need to count the number of ways of picking two points out of this. And the number is 0, 0, 1, 3, 6, 10, 15, 21, 28, 36, and so on. So these come from a line through two points. And I'm going to indicate this as 2, 1, 1, meaning it's 1, 1, 1, meaning it's a line of degree 1 passing through two points. We'll see what this notation means a little bit later. Next, I can take a conic through five points. And the number of ways of doing that is the number of ways of choosing five of these points. And obviously that's 0 until we have at least five points. And then it's the number of ways of choosing five out of this number of points, which looks like 1, 6, 21, 56, 84, I think, and so on. So then we can get the total number of blown up points. So we get 0. Sorry, the total number of exceptional curves goes 0, 1, 3, 6, 10, 16, 27. But I said there were no extra exceptional curves when we blow up six points. But in fact, there are some other ways of getting exceptional curves when we blow up seven or more. So let's mention some of the ways you can get more points by blowing if you've got seven points blown up. Well, what we can do is we can take a cubic through seven points, one of which it passes through doubly. So I'm going to indicate this as 3, 2, 1, 1, 1, 1, 1, 1. Sorry, I forgot to indicate I was drawing a conic through five points as 2, 1, 1, 1, 1, 1. So if you check, this is self-intersection 9 minus the sum of these squares, which is 9 minus 4 minus 6, which is minus 1. And you can also check that it's a rational curve by using something called the adjunction formula. Anyway, you see there are seven ways of doing this. So we should add another seven here. And there are 56 ways of doing it for degree 8. And there are 252 ways of doing it for degree 9. Now, that turns out to be all the exceptional curves if you blow up seven points. So we get 56 here, which is 7 plus 21 plus 21 plus 7. Degree 8, things get even more complicated. So for degree 8, we also get degree 4 curves passing through eight points. Three of them passing through as a double point. So I'm going to indicate that by this notation. And if you count up, we get 56 of these. We can also get degree 5 curves, which pass through five points with degree 2 and three with degree 1 if I've got it right. And you can check there are 28 of these. And we can also get degree 6 curves passing through one point with degree 3 and the others with degree 2. One, two, three, five, six, seven. And one of these seems to be wrong. I think maybe that should be six, two, because I'm not quite sure about it. Put a question mark there. And there are eight of these. So if you add them up, you get 8 plus 28 plus 56 plus 56 plus 56 plus 28 plus 8, which is equal to 240. Now, for degree 9, this process goes on forever and you actually get an infinite number of exceptional curves. So we've got this rather funny sequence here. 1, 3, 6, 10, 16, 27, 50, 6, 240, infinity. Well, if you've studied Lie groups in the algebras, you may recognize this because these are dimensions. 240 is the number of roots of the exceptionally algebra E8. 56 and 27 are the dimensions of the minuscule representations of E7 and E6. This is the dimension of the spin representation of D5. This is the, and this sort of corresponds to, what's the next one, A4, and these correspond to A2, A1, and so on. So this coincidence is not actually coincidental at all, as I will now explain. And I also want to explain what this funny notation here means. So what's going on? Well, what we do is we look at the Picard group of the surface blown up in n points. So the Picard group is the group of degree 0 divisors, modulo linear equivalents. And for a surface blown up in n points, it's not very difficult to work out what it is. It's just z to the n plus 1. And the Picard group also has a bilinear form and it's given by intersections. And the intersection actually makes this intolerance in space. So if we've got a point x0, x1 up to xn in z to the n plus 1, it does norm x0 squared minus x1 squared minus x2 squared and so on. With these sort of a relation to the fact that an exceptional curve is self-intersection number minus 1. So now the canonical divisor on this turns out to be divisor 3, 1, 1, and so on, 1. So it is norm is 3 squared minus n. And you notice that when n hits 9, which was where things went wrong here, the norm actually becomes 0. So this is greater than 0 for n less than 9, equal to 0 for n equals 9, and less than 0 for n greater than 9, which is, as we'll see in a moment, it's why the number of exceptional curves suddenly becomes infinite here. And now an exceptional curve comes from an element of the Picard group. So it comes from an element x0, x1 up to xn of the Picard group, and r, r is equal to minus 1. So x0 squared minus x1 squared, and so on, minus xn squared is equal to minus 1. And because the curve is rational, if you use an adjunction formula, this turns out to imply that it has in a product 1 with the canonical divisor. So what we're doing is we're looking at norm minus 1 vectors that are in a product 1 with the canonical divisor. Now, if we draw this as Lorentzian space. Now, if the canonical divisor has positive norm, it sort of lives in here somewhere, and its orthogonal complement is positive definite, and therefore only has a finite number of vectors of norm 1. So this is why we only get a finite number of exceptional curves if the canonical divisor has positive norm. And we can write out these vectors r, which have norm minus 1 and inner product 1 with the canonical divisor. So they can look like naught here and then minus 1 somewhere, naughts elsewhere. So these were the ones corresponding to the exceptional curves, or it can have inner product 1 with k and have 2 1s somewhere, or it can have inner product 2 with k and have 5 1s somewhere and 0s elsewhere, or it can have inner product 3 with k and have 2 and 6 1s and so on. So this is where all these funny vectors come from. They really represent elements of the Picard group that have norm minus 1 and inner product 1 with the canonical divisor. So we only get a finite number of them if the canonical divisor has positive norm, but we get an infinite number as soon as the canonical divisor has negative norm. Now for the relation with exceptional the algebras, it turns out that if we take the space z, the 1 plus n with norm x naught squared minus x1 squared minus xn squared, and we take the canonical divisor 3, 1, 1 and so on and take its orthogonal complement. So if we take n less than or equal to 8, then this is positive definite. And when I say positive definite, what I mean is negative definite because I'm just getting muddled up about the sign of this. And this turns out to be a certain lattice. It's the lattice E8, E7, E6, D5, A4, A2, A1, and something else when n is equal to 8, 7, 6, 5, 4, 3. So these are the root lattices of various the algebras. So what is the root lattice E8? Well, in fact, about the easiest way to construct it is simply to define E8 as the orthogonal complement of this vector in Laurentian space. You might want to change sign if you want E8 to be positive definite rather than negative definite. And the vectors r with r squared equals minus 1 and rk equals 1 turn out to be usually weight vectors of the weights of a certain representation of E8, E7, E6, and so on. So for E7 and E6, we get 56 or 27 vectors like this, which correspond to the 56 and 27 dimensional vectors of the Lie algebra E6. So the 27 lines on a cubic surface actually correspond to a basis for a certain representation of E6. And this gives you other relations. So the vial group of E6 acts on the 27 weight spaces of this representation. It also acts as an automorphism group of the configuration of 27 lines on a cubic surface. Similarly, here we get 240 vectors, which are not quite the weight spaces of our representation. Instead, they're the roots of E8. And you can see this because if r squared is minus 1 and rk equals 1, then r plus k has norm 1. So r plus k is going to have norm minus 2. That wasn't quite right. It should take r minus k as norm minus 2. So these correspond to the minus 2 vectors in the E8 lattice, which are just the roots of E8. So I'll just finish by saying that if you want to find out more about this, there's a very nice book by Manin about this on cubic forms, which is mostly about cubic surfaces. And he's got lots of chapters on the, and he's got a very weird chapter on Mu Fang loops to start off. I don't know why, but he's got several chapters on the, he's got a chapter on the 27 lines where he explains the relation with exceptional curves and viral groups in much more detail. OK, I think I'll leave it at that for today.