 This talk is part of a series giving an introduction to complex projective surfaces and this one will be mostly about minimal surfaces. This is minimal surfaces in algebraic geometry. They have almost nothing to do with minimal surfaces in differential geometry. So we recall that every algebraic surface is birational to one that is projective and non-singular and minimal. And the classification of surfaces tends to be a classification of projective non-singular minimal surfaces. So I'll just briefly recall how you get each of these properties. First of all, for projective, this is easy. All you have to do is take an open affine subset, s naught of your surface which is birational to the whole surface, and then s naught is almost by definition contained in some affine space which is contained in some projective space. And you can take the closure of s naught in some projective space and this will give you a projective surface. This projective surface might have singularities, even if s naught doesn't have singularities. So we need to remove the singularities. So the second step is to make it non-singular. Well, making a surface non-singular is actually completely trivial. All you can do is discard the singularities, but of course that will no longer be projective. So the problem is to resolve the singularities while still making sure the surface is projective. This is a really hard problem called resolution of singularities and it was solved by Walker for characteristic zero and by Zariski for characteristic greater than naught. There were several claims of having done this before Walker, but none of them seemed to have been quite complete. So they did this for surfaces. Resolution of singularities and high dimensions is an extremely difficult problem that there's been a lot of work on. So Hironaka didn't characteristic zero and positive characteristic is still open. So I just quickly sketch how you resolve singularities and characteristic singularities of surfaces. What you can do is we can remove co-dimension one singularities by normalization. So normalization is a operation you can do which on affine schemes just corresponds to taking the integral closure of an integral domain in its quotient field and you can do this for all schemes and it turns out to be slightly stronger than resolving all co-dimension one singularities. So you can get rid of the co-dimension one singularities and now you can get rid of the co-dimension two singularities. So let's remove co-dimension two singularities. Well, co-dimension two singularities for surfaces are just points and we can just blow up the points. So we seem to have got rid of all singularities, right? Well, not quite because when you blow up a point, this can introduce new co-dimension one singularities. So you first do step A, then you do step B to get rid of the co-dimension two singularities. Well, it doesn't get rid of them, you blow them up. Then you've got some new co-dimension one singularities. Well, that's okay. You go back to step A and then back to step B and you keep on going and the problem of resolution of singularities is to show this process. So it's fairly easy to write down a method for resolving singularities. The problem is just to prove that it works. For example, you could try to measure the nastiness of a singularity by some suitable invariance and try and show that this process reduces the maximal nastiness of all singularities of the surface and show that you can't decrease the nastiness an infinite number of times. The trouble with this is it's rather difficult to find a good measure of just how nasty a singularity is. In fact, this is almost the essential problem in resolution of singularities. Anyway, we now come to the third step, which is what I really want to discuss today, which is to show that the surface is minimal, which means it's not the blow-up of some non-singular surface. It's quite often the blow-up of the singular surface. So let's just recall what the blow-up is. Well, let's just give the blow-up of the point, the origin of two-dimensional projective space for simplicity. So two-dimensional projective space is set of points with projective coordinates x, y, z. And for its blow-up, we first take the product of this with p1. So p1 has coordinates x, column y. I'm going to use capital letters here. And we take the subset of p2 times p1 of all points, such that x times y equals y times x. And this gives us the blow-up of p2 at a point. And the blow-up has a natural projection map to p2. And there's an isomorphism, except over the point 001. And the inverse image of 001 is, in fact, a copy of a projective line. So we've replaced a single point of p2 by an entire line. What happens is something like this. So here we've got a copy of p2, which I'm going to draw as a plane. And here's the point we blow up. And blowing it up kind of changes it like this. So we replace the point by an entire copy of p1, which I'll draw as a little circle, though p1 isn't really a circle, but I can't think of what else to draw it as. And what is going on is that each line through the origin corresponds to a single point of this p1. I mean, the way I've drawn it looks like it corresponds to two points, but that's because I'm not very good at drawing in four dimensions. So what we have here is a bi-rational map, sorry, is a regular map from the blow-up of p2 at a point to p2. And this map is actually bi-rational. So that's what blowing up a point is. And we want to show, find a minimal surface, which is one you can't obtain by blowing something up. Well, the obvious thing to do is to start with a non-minimal surface and blow down copies of p1, which is an obvious inverse of this operation, and keep blowing down points until we can't do so anymore. And the problem is, is this process finite? And the answer is it is finite. And one way of seeing this is that blow-up at a point increases the dimension of the second homology group of a surface by one. It's not too difficult to see that because blowing up a point is replacing the point by a copy of p2, sorry, a copy of p1. And the projective space p1 over the complex numbers has a second co-homology group of rank one, which is where this one comes from. So blowing down decreases the dimension of the second homology group by one. And since the rank of the second homology group is finite, you can't blow down more than a finite number of times before you come to a stop. So this shows that every non-singular projective surface is bi-rational to a minimal non-singular projective surface. And it's the minimal ones we really try to classify. And now I'll say a bit more about what the structure of bi-rational maps is. So here we want to talk about the structure of bi-rational maps. This was worked out by Zariski, although it was sort of implicitly known to the Italian algebraic geometers. I think they just didn't state it explicitly. So I suppose we've got a bi-rational map from the surface s0 to s. So we might take s0 being p2, for example, and s1 being p1 times p1. And these two surfaces are bi-rational but not bi-regular. So what Zariski said is that you can turn this into a regular map by blowing up s0 repeatedly. So we blow up a finite number of points. And these need not be points in s0. For instance, s1 might you might blow up a point in s0, but then s2 you might blow up a point in s1. And this might not be a point in s0. It might be a point in the in the extra copies of the projective line you get. So he said you can do that a finite number of times. So this map is now regular. So that factorizes any bi-rational map into a sort of inverse of a lot of blow-ups at points followed by a regular map. This isn't too difficult to prove. The idea is that if you've got a bi-rational map, it's sort of defined everywhere except at some points. And you can make it more defined by blowing up these points. And if you keep careful track of this, you'll find it can be factored like that. More generally, any bi-rational map can be written entirely in terms of blow-ups. So we can, in fact, write if we've got a bi-rational map from s0 to s. So this is bi-rational. We can find a sequence of blow-ups and then a sequence of blow-downs. Let's call this tn minus 1 down to s equals t0. Such that the bi-rational map is got by first of all blowing up a number of points and then blowing down several times. So in some sense, if you can understand blowing up at a point, then you've sort of understood every bi-rational map. So now let's look at this operation of blowing down. So what we want to do is ask, when can you blow down a surface? So we've asked, when can we blow down surface s? In other words, we want to give an s, can we find a map from s to s0 such that s is the same as blowing up s at a point? And so this is blow-up at, say, a point p in s0. And we call the inverse image of p. Let's call it e is called the exceptional curve, or the exceptional curve of the first kind, if you want to be really pedantic, but we won't worry about exceptional curves of the second kind. So when can, we can ask the question, when can we blow down a curve e? And the answer was given by Castelnuovo. He said, well, there's one obvious property. First of all, e must be rational. e is isomorphic to a projective line. It's kind of obvious. The second is a much more subtle property, that e equals minus one, where this is the self-intersection number. And the easy part of the theorem is to show that if you blow up a point, then the exceptional curve has self-intersection minus one. The hard part of Castelnuovo's theorem is to show that conversely, if you've got a curve like this, then you can blow it down. So what I need to do now is to show that, is to show what the intersection number of two curves is. So let's talk about intersection numbers of curves. So we have the following problem. Define the intersection number cd of curves, cd on a surface s, where s will be projective and non-singular. And I'm going to describe three ways of doing this, all of which have some disadvantages. The first is the naive definition, where we just define cd to be the sum of the, well, let's define it to be the number of intersection points. Well, there are several problems with this. So first of all, if the curves look like this, there's absolutely no problem that there's a single intersection point. On the other hand, if the curves look like this, then we kind of have to count that as a double intersection point. So this shouldn't quite be the number of intersection points. It should be the sum of intersection points with some sort of multiplicity. And the multiplicity can get quite complicated. For example, one of the curves might have a singularity. Well, obviously then you calculate two intersection points, but what happens if both curves have a singularity and they meet at the singularity? What on earth are you going to count this multiplicity as? And it's not entirely obvious. But there's an even worse problem. What happens if you take the intersection of this curve with this curve? In other words, if you take the intersection of a curve with itself, there are an infinite number of intersection points. So what are you going to do? Well, what we can do is deform one curve. So we now take the intersection multiplicity of this curve with a slightly different curve. We might sort of shift it very slightly, that might end up like that. Well, can we do that? Well, there's one problem. Can we deform the curve? And the second problem is the intersection multiplicity independent of the deformation. So you see we're actually getting quite a lot of problems in trying to define the intersection multiplicity. The answer to question one, can we deform the curve, turns out to be not always. So already we've run into yet another problem that we have to think of something else to do. If we can deform it, then you can at least show that this is independent of the deformation. So although you can push this through, it's really rather difficult and it's less effort to use a different, less intuitive definition. So here are some easier definitions. First of all, you can map any curve to a cohomology class in the second cohomology of your surface S with real coefficients. And the second cohomology class has a cut product from the second cohomology group times the second cohomology group to the fourth cohomology group, which is isomorphic to the reals. So this gives you some sort of bilinear map on curves. And this does give you a reasonable intersection multiplicity. One obvious problem is it involves working with cohomology over with real cohomology, which is rather hard to make sense out of over fields of characteristic greater than zero. You can do it using etal cohomology, but it's a lot of work. Fortunately, there's a more algebraic definition that's easier to work with, where you define the intersection multiplicity of two curves on the surface. So here's C and D are curves on the surface, and we define it to be the Euler characteristic of O minus the Euler characteristic of O of minus C minus the Euler characteristic of O of minus D plus the Euler characteristic of O of minus C tensor with O of minus D. So what on earth does this mean? Well, this is the sheaf of regular functions on the curve. This is a line bundle associated to a divisor, and this is the tensor product of line bundles. And chi of a line bundle or invertible sheaf is the Euler characteristic, which is the dimension of the zeroth cohomology group minus the dimension of the first cohomology group plus the dimension of the second cohomology group. This is coherent sheaf cohomology, which is in chapter three of Hartron's book, and which I strictly speaking, we haven't yet covered in the course. Well, this definition actually works great. There's one obvious problem with it. The main problem with it is that looks weird. It's completely unintuitive and not at all clear how you would ever think of this. The advantage of it is it's actually really easy to work with, because cohomology groups, although they're a bit unintuitive, are actually very easy to work with, and the Euler characteristic is very nice properties. So if you're willing to put up with the fact this is not at all intuitive, this is actually the easiest definition of the intersection multiplicity, and then you can then go off and prove a theorem saying that it's the same as the sum of the multiplicities of the intersection points and so on. So next I want to discuss how is it possible for a curve to have negative self-intersection multiplicity? So how come we can have cc being negative? There's certainly no problem having it equal to zero. If you look at p1 cross p1, we see we can think of this as a lot of horizontal lines times a lot of vertical lines, and the any two vertical lines obviously of intersection multiplicity zero and any two horizontal lines of intersection multiplicity zero. So intersection multiplicity zero is no problem, but what about less than zero? Well, let's prove that we can't have two curves with intersection, a curve with self-intersection multiplicity less than zero. So suppose we've got a curve c. What we do is we deform c slightly to get a curve c prime, and now intersection multiplicities is invariant under nice deformations of curves. So we see now that cc is equal to cc prime, which is greater than or equal to zero because it's just a sum of intersection multiplicities of a finite number of points. So this seems to prove that curves cannot have negative self multiplicity. Well, there's a problem here. What this actually proves is that if cc is less than zero, we cannot deform c to have finite intersection with with itself. In other words, for instance, c might just be rigid. There's no way to deform it, or c might have two components, and you can deform one of these components, but can't deform the other one or something like that. Well, when I say we can't deform c, what I mean is we can't deform c algebraically. We can deform c topologically. So the difference is we can't deform a curve c to be an algebraic curve inside the surface. But if we think of the surface as being a four-dimensional topological manifold, we can deform c as a topological manifold. Well, doesn't this give us the same contradiction? Well, no, because the topological intersection multiplicity of two surfaces in some four manifold can be less than zero. Because topological intersection multiplicities, you have to sort of worry about orientations and things like that. And intersection multiplicities can be less than zero. For instance, if we're just talking about curves in a, sorry, real one dimensional curves in a real one-dimensional surface, if you've got a curve like this, these have intersection multiplicity one, but if you deform the curve to look like this, it now seems to have intersection multiplicity three, but you have to give this intersection number multiplicity one, and this minus one, and this plus one, and so on. So so algebraic intersection multiplicities of, at a point always finite, but you can't always deform curves, topological intersection multiplicities can be negative and you can deform curves. Anyway, so that's why the intersection multiplicity of a curve with itself can actually be negative. Now I want to show informally why is the intersection multiplicity of an exceptional curve equal to minus one. So remember, exceptional means it's a blow up of a point. Well, I'm going to explain this by look at the case of blow up the origin of two-dimensional projective space. So let's draw a picture of the blow up of two-dimensional projective space. So what you think of this is two-dimensional plane, except the origin has kind of been replaced by a copy of p1. And now I'm going to take two curves two straight lines C and D in p1. And you see that these two lines are generally going to have intersection multiplicity one rather obviously. Now I'm going to move these two lines so that in the plane they both pass through a point. Well in the in the blow up something a bit funny is going to happen because if I move this line D so it passes through the origin then in the blow up of p2 it becomes two lines. First of all we have the entire exceptional curve and secondly we have a line D zero. So what we're really getting is a line D zero plus a copy of the exceptional curve. And similarly if I take C and move it along then it suddenly splits as a copy of some line C zero plus the exceptional curve. So here what I'm doing is I've got a map from the blow up to p2 and I'm moving lines around in p2 and looking at the inverse images in the blow and the intersection multiplicity turns out to be invariant if you move curves around like this. So so C D should be equal to C naught plus E intersected with D naught plus E. And as this should be bilinear it's equal to C naught D naught plus C naught E plus E D naught plus E E. And now we can work out what these are. So C naught D naught is just zero assuming C and D have different tangent directions at the origin because they intersect the exceptional curve in different points so this is equal to zero. And C naught E is obviously just one because C naught intersects the exceptional curve in one point corresponding to its tangent direction at the origin. So this is equal to one and this is equal to one and this sum is equal to one because it's C naught D naught. So if we have one is equal to zero plus one plus one plus this so this must be equal to minus one in order to make everything add up. So that's a sort of informal explanation of why exceptional curves have intersection multiplicity minus one. Now let's look at an example. So let's blow up p2 in two points and see what happens. So here's a copy of two-dimensional projectors space and I'm now going to pick two points p and q. And what I'm going to do is I'm going to look at a line L. So here I've got a line L in p2 and I'm now going to move this line so that it passes through both of these two points. And now let's look at what happens in the blow up. So in the blow up we've got p2 blown up at these two points so let's try and draw a picture of it. So we replace these two points by little copies of p1. So here we've got a copy of p1 and here we've got another copy of p1. And these are the exceptional curves. I'm going to call this exceptional curve e1 and this exceptional curve e2. And now when we move the line L so it passes through both points we see that the inverse image of L in the blow up looks like a line L0 plus e1 plus e2. And now let's work out some intersection multiplicity. So we have one is equal to the intersection multiplicity of L with itself and this should also be equal to L0 plus e1 plus e2 intersect with L0 plus e1 plus e2. And if we work this out it's equal to L0 L0 plus 2 L0 p plus 2 L0 q plus 2 pq plus p sorry that should be e1 that should be e2 and should be plus 2 e1 e2 plus e1 e1 plus e2 e2. And if we work out what these numbers are this L0 e1 is one you can see L0 is intersecting this exceptional curve in one point so this is 2 times 1 this is 2 times 1. e1 and e2 obviously don't intersect so we get 0 then we get minus 1 then we get minus 1 there so we get 1 is equal to L0 L0 plus 2 plus 2 minus 1 plus minus 1. So the conclusion is that this is equal to minus 1 so L0 is rational and it has intersection number minus 1 with itself so L0 is an exceptional curve what this means is we can now blow down along L0. So let's draw a picture of what's happening here so what we've got is we've started with a copy of the two-dimensional projector space we blew it up at one point to get projector space with one point blown up and then we blew it up at a second point to get projector space with two points blown up and this now contains a line L so we now blow down along this line L0 and we ask what do we get here and the answer is what we get here is a copy of p1 cross p1 and let's see why this is true well if we look at p2 and look at these two points we can see through each of these points there's we can draw a sort of p1 worth of lines because all the lines going through that point form a copy of p1 and similarly through this point we can draw lines forming a copy of p1 so we've got a second copy of p1 here and now we can map p2 to the product of these two ones so we get maps to p1 times p1 and this is this map is very easy what we do is if we take any point it lies on one of these lines through the first point and one of these lines through the second point so it's giving us a point from the first p1 and a point from the second p1 so we've got a map from p1 blown up at two points to p1 cross p1 because of course this map isn't quite defined at this point of p1 because it lies in all these lines you have to blow it up to make it well defined well now let's look at the line that goes through these two points and now you see this yellow line this entire yellow line maps to a point in p1 cross p1 because any point on this line lies on the same two lines through the point here so this is of course the line we blow down so this sort of shows informally that if we contract this line to a point then we're probably getting a copy of p1 times p1 so this is an example of a bi-rational morphism from p2 to p1 cross p1 so this is mapping x colon y colon z to x colon 1 y colon 1 in p1 times p1 and of course this isn't always defined it's not defined at all it should be a z so this shows that a bi-rational map from p2 to p1 cross p1 can be factored into some blowing ops followed by some blowing downs which is the riski's result about how bi-rational morphisms can be factored into blow ops and blow downs okay that's all I want to say for the moment about minimal surfaces so the next lecture I will probably be talking about rational surfaces which are the simplest class of surfaces and historically the first ones to be investigated