 So this lecture will continue the overview of complex projective surfaces by talking about the surfaces with cadera dimension equal to one or two. So in previous talks we talked about the cadera dimension minus infinity, which we saw with the ruled surfaces, and cadera dimension zero, where we found there were four different classes of surfaces. And cadera dimension one surfaces all turn out to be elliptic surfaces. Here an elliptic surface is a surface S with a map to a curve C such that the fibres are mostly elliptic curves. I will explain what mostly means later, some of the fibres aren't elliptic curves. A typical example of an elliptic surface is just a map from E times C to a curve C, and obviously the fibres are all copies of this elliptic curve E. Actually not all elliptic surfaces have cadera dimension one. We can have elliptic surfaces with cadera dimension minus infinity or zero. Some examples of these are E times P1 for E elliptic, which maps to P1. So this is cadera dimension minus infinity. Or we could just take a product of two elliptic curves mapping to an elliptic curve and this would have cadera dimension zero. So the reason why cadera dimension one give you elliptic surfaces is that for any surface we have a map to some projective space, which is the projective space corresponding to the graded ring, where you take some of, sorry that should be a gamma, global sections of nth powers of the chronicle bundle. So this is always some projective variety or empty. And if the cadera dimension is one, this space is a curve and you can then check that the fibres are mostly elliptic curves and this gives you the vibration making it into an elliptic surface. So the examples where you just take a product E times C to C are the sort of trivial examples. There are more complicated examples as follows. So first of all, it may be a non-trivial bundle. So we see that this map here is a trivial bundle over C because it's just a product but you could put in some sort of twisting in where E is sort of bent round a bit. In particular, the fibres can vary. So instead of just having one elliptic curve as the same fibre, the elliptic curve could vary from one point of C to another. Moreover, the fibres may be singular. So they need not just be elliptic curves but they can sometimes be sort of degenerate elliptic curves with singularities. In fact, you can sort of classify elliptic surfaces by using this data. So to specify an elliptic surface, you have to specify a curve C. You have to specify what the singular fibres are and which points their fibres over. And then you sort of have to describe the bundle over the other points of C. For example, you get a map from the fundamental group of C minus the singular points to the first homology of an elliptic curve and so on. So that gives you some sort of invariant for getting hold of the surface. I should remind you that if you've got a map from a surface to a curve, even if the surface and the curve are non-singular, some of the fibres can be singular. So a very simple example of this is if you just take the surface z equals x, y and you map it to the affine line by mapping x, y, z to z, then the fibre at naught is x, y equals naught which is singular, whereas the fibre at z not equals zero is some sort of hyperbola. So it's non-singular. So the hyperbola, you've got a family of hyperbolas which sort of degenerate to a singular curve at one particular point. You can do the same thing with the fibres being elliptic curves. For example, suppose we take y squared equals x cubed plus ax and map this to the a line by... So this is a surface in three-dimensional space and we're looking at the fibres for various values of A and for A not equals zero, this gives you an elliptic curve, but for A equals zero, we get y squared equals x cubed, which is a cusp. So here we've got a family of elliptic curves except that one of them degenerates into a cusp and this is a fairly typical example of what a singular fibre looks like. Of course, these should really be projective varieties and here we've just done the affine case for simplicity but should give you the idea of what goes on. So the next case you can ask is, what are the possible singular fibres? So these were actually classified by Cadiara. First of all, the fibre can be irreducible and if it's irreducible, it can be an elliptic curve, obviously. Or it can be a cusp and we saw an example of this with the family y squared equals x squared plus Ax at A equals naught, it becomes a cusp. Or it can be a node looking something like that and here we can, for example, take y squared equals x cubed plus x squared plus A and as A goes to zero, we get an elliptic curve with a node. And the other ones, I can just list the other ones that Cadiara found. If we just describe the ones without multiple fibres which I won't worry about we can also get ones that look like that. So these can be two or three copies of P1 intersecting the following way. Here they touch in a double point and here they all intersect in a single point. You can also get copies of P1 arranged in a sort of cycle like that. So here I've drawn a cycle with one, two, three, four, five, seven of them. You can also get them all arranged along the line like this except that you get the sort of double ones on the end. And you can also get them arranged like this. Sorry, that should be a... This should both be separate. Or you can get them arranged like this. Two, three, four. Or they can be arranged like this. It's very difficult getting the number right. And that's all. Well, where does this funny collection of diagrams come from? Well, it may look much easier if you swap lines with points. So here I've drawn a line for each copy of P1. It's a point where any two copies of P1 intersect. But if instead I draw a point for each copy of P1 and draw lines between them when they intersect, the diagrams end up looking like this. So first of all, we've got cycles with any number of points. Secondly, we've got something that's looking like this with any number of points. And then these three become these diagrams. Not one. It should be there. And now you recognize these. These are just the Dink and Diagrams A, N, D, N, E6, E7 and E8. That turn up in the classification of compactly groups and also turn up in the classification of lots and lots of other things. I should say if you think I put too many points on these Dink and Diagrams, these are actually the affine Dink and Diagrams, which are the usual Dink and Diagram together with one extra point. So if you delete these points, you get the usual non affine Dink and Diagram. So in other words, the classification of singular fibers of an elliptic surface turns out to be more or less the same as the classification of compactly groups. Well, there's another classification that's very similar to the classification of elliptic surfaces, which is the classification of elliptic curves over Z and the integers. Now, you remember, a speck of Z has dimension equal to 1, so it's a sort of curve. And an elliptic curve over Z is really a sort of surface. I mean, its spectrum is actually two-dimensional. You know, we have a map from the spectrum of the elliptic curve over Z to the spectrum of Z. And you can think of this as being not a curve but a surface, and of this as being a curve. So this is really a sort of very similar to an elliptic surface. So the fibers are elliptic curves over points of Z, which are really over the spectrum of various finite fields, except that sometimes they are degenerate elliptic curves. And again, some of the fibers may be singular. For example, let's just look at the following elliptic curve over the integers y squared equals x cubed plus x plus 1. And so you have a map from the spectrum of this elliptic curve to the spectrum of Z. And if we try and draw it, well, the spectrum of Z looks like this. And we've got a generic point nought. And if we look at the fibers, the fibers you should mostly draw them as some sort of elliptic curve. So here we've got elliptic curves. And there's one point where it's not an elliptic curve because if you work out the discriminant of this, it's minus 31. So x cubed plus x plus 1 becomes x minus 3 by x plus 17 squared if I've worked it out right. So this is no longer an elliptic curve, but a curve with a cusp. So over the point 31 of spectrum of Z, you should actually picture it as looking something like this. So you can picture this elliptic curve over Z as being some sort of family of elliptic curves that degenerate into a curve with a node over the point 31. Well, the degenerate fibers for elliptic curves over Z were in fact classified by NERON independently of Kadara. And I think it was noticed a little bit later that both of them had really... Kadara and NERON's classifications were both really the same. Unfortunately, this left us with two completely incompatible sorts of notation for the singular fibers. Okay, well now I'll say a little bit about the surfaces with Kadara dimension 2, which are those of general type. So examples where we might have a product of two curves with genus of C1 is greater than 2 and the genus of C2 is greater than 2. Greater than 1, I guess, is enough. Or a non-singular hypersurface in P3 of degree greater than 4. So there are masses of these. And there are huge numbers of other examples. For instance, instead of hypersurfaces, you could take complete intersections in some high-dimensional projective space that are surfaces. Nearly all of these are of general type. And it turns out there are, you know, dozens or hundreds of constructions of surfaces of general type that people have come up with. I'll just give one example of the many, many constructions people have come up with. So typical example, this is one of the easiest ones, might be the Goddow surfaces. And for this, what you do is you take the surface S given by w to the 5 plus x to the 5 plus y to the 5 plus c to the 5 equals 0 in three-dimensional projective space. And this isn't the Goddow surface. I haven't finished the construction yet. What you do is you take this surface, you take an automorphism sigma of order 5, which takes w to w, x to zeta times x, y to zeta squared times y, and z to zeta cubed z, where zeta to the 5 equals 1 as a fifth root of unity. And S modulo, the action of sigma, is then a Goddow surface. And these are kind of a bit interesting because they turn out to have geometric genus and irregularity both equal to 0. So you remember Castle Novo at one point asked whether this condition implied the surface was rational. Enriquez gave the Enriquez surface as a counter-example. And it turns out that Goddow surfaces are also counter-examples to this question. So in the last slide, I'll try and draw a picture of a sort of ball surfaces. So this is a sort of map of surfaces. And what I'm going to do is I'm going to classify surfaces Well, it's sort of traditional to use the churn numbers, which is the self-interception number of the canonical class. And the second churn number is just the Euler class, which is the usual topological Euler characteristic of the surface. And you can just draw all the surfaces in general type by plotting them on this diagram where you use c1 squared and c2 as the two axes. So for surfaces of general type, c1 squared and c2 satisfy the following inequalities. First of all, c1 is greater than 0 and c2 is greater than 0. Next, c1 squared plus c2 equals 12 times the holomorphic Euler characteristic. This is a Notar's formula. And Notar also found an inequality, which says 5c1 squared minus c2 plus 36 is at least 0. And there's a much more subtle inequality that was only found much later, found by Bogomolov, Mier, Ochre and Yao, which says that c1 squared is less than or equal to 3c2. So Bogomolov proved this with a 10 here. He reduced it to 4 and Yao found the best possible number of 3. So if you add all these conditions into the graph, you find all surfaces must lie on these lines where the sum of c1 squared and c2 is a multiple of 12. So you get various lines like this. And if we put in these inequalities, well, the Bogomolov, Mier, Ochre, Yao inequality sort of looks like this. I'll mark this as bm y, not bm w. And notice inequality sort of looks like this. So we find that the surfaces of general type must lie inside this funny shaped region here and they must lie on these lines here and most of the points on these lines inside this orange region are known to be churn numbers of a surface of general type. So you can also look at this in a bit more detail. For instance, we can look at the points on the boundary. So there should be 10 points here. This one isn't actually a churn number of a surface of general type because c1 squared has to be positive but we have all these other numbers here. And the Godot surfaces actually live here. So they're almost the simplest possible churn numbers of a surface. The ones here are called fake projective planes and you can understand this if I mark on the churn numbers of surfaces not of general type. So you find that the ruled surfaces all lie on this line here and most of them are kind of down there off this diagram and the projective plane has these churn numbers. And the projective plane has the same churn numbers as certain surfaces of general type called fake projective planes. These are quite hard to find and the first one was only found by Mumford and quite late on. For example, if you take a surface of degree 5 in p3 it's somewhere just off to the right here. I can't quite leave myself enough room to draw it. You can also ask where do things like Enrique's and K3 surfaces live? Well, these are the Enrique surfaces, these churn numbers. These ones turn out to be K3 surfaces and these ones are where you find the abelian and hyper elliptic that they're all at the origin. The surfaces of Kaderid Dimension 0 just live down here. The surfaces of Kaderid Dimension 1 all lie on this line here and more generally all elliptic surfaces lie on this axis here. So there's been a certain amount of work trying to classify the surfaces of general type and the answer is they're incredibly complicated to classify. Gazika showed that for any fixed values of C1 squared and C2 there's a scheme, a quasi-projective scheme whose points classify the surfaces of general type with given churn numbers. So for any given values of the churn numbers you can in principle classify all surfaces of general type with those numbers. There's been a lot of work studying these Gazika schemes. Mostly they aren't known except for a very few cases around the edge where it's just about possible to find them and all indications are in that in the middle these schemes are incredibly complicated. They have wild singularities and can be reduced at every point and things like that. Okay, so that's a summary of what we know about surfaces. Surfaces of Kaderid Dimension less than two, we sort of more or less know most of them at least if they're projective. Surfaces of Kaderid Dimension two, it's wide open for further exploration.