 This talk will be an informal survey of ruled surfaces. Here, surfaces will always be complex, projective surfaces, and a ruled surface is one that is birational to C times P1 for some curve C. So, some examples of these, so first of all, C times P1 is obviously birational to a curve times P1. Another example is got by taking a two dimensional vector bundle over some curve C, and then this gives an associated P1 bundle. So, we saw some examples of this last time where if we take the curve C to be P1, two dimensional vector bundles look like O, M, plus O, N for some M and N where this is says twisted chief, and the surfaces we get, we mention, we call the hits of brook surfaces. So, more generally, if the curve has genus greater than zero, then the two dimensional vector bundles are much more complicated to classify. There's usually a sort of modular space of them. We might have, for example, just to give one example of one that's not usually a product, we might take the chief of regular functions plus say the tangent bundle of the curve. And there are large numbers of other examples. In fact, the goal is to sort of show that complex projective surfaces with kadara dimension minus infinity are more or less the same as ruled surfaces. So, what I'm going to do is I'm going to sketch, give a very brief sketch of why this implication holds. And by the way, sometimes surfaces are called geometrically ruled. If the surface has a map to a curve C, such that the fibers are mostly copies of P1, so you can think of that as saying the surface is covered with copies of lines where a line is just P1. And it's not too difficult to show that a geometrically ruled surface is in fact ruled. In the early days of algebraic geometry, surfaces were embedded in projective space and ruled surfaces were supposed to be embedded in such a way that all the lines on the surface were in fact lines of projective space. Well, before discussing the relation between ruled surface and kadara dimension minus infinity, I want to review invariance of a surface. So, first of all, we have that there's really only one important invariant of a curve, which is the genus. So, the genus G is either dimension of H1 of omega zero or it's the dimension of H0 omega one where these are the same by ser duality. So, we recall that this is the dimension of the space of one forms, holomorphic one forms on the surface. And this is roughly the space of obstructions to the Mitterg Leffler problem. So, for surfaces, there's the geometric genus which is the analog of this. It's usually written as PG where the G stands for geometric and I've no idea why P stands for genus. So, this can be either written as the dimension of H2 omega zero which by ser duality is the dimension of H0 omega two. So, this is the dimension of the space of holomorphic two forms and this is some sort of cohomology group. Well, as well as the geometric genus, there's also an arithmetic genus, which is denoted by PA where the A stands for being arithmetic. The arithmetic genus was originally defined as some sort of complicated expression of a surface. For instance, you can write it as mu zero minus two thirds, mu one plus mu two over six plus mu two over 12 minus one and one notation. So, obviously you want to know what mu zero and mu one and mu two and mu two are. Well, mu naught is something like the degree of the intersection of the surface with a hyperplane and the others I haven't really managed to work out what they are. It's actually quite difficult deciphering old algebraic geometry texts to find out what various things are. If you want to try and find out more, you could try looking in the book by Semblon Roth but it's a bit tricky because it doesn't have an index. So, trying to track down where it defines these is a bit tricky. But anyway, so there was this complicated expression giving you an arithmetic genus and there was a geometric genus defined as the analog of the genus for curves. And an obvious question is where the arithmetic genus is equal to the geometric genus. And this is true for rational surfaces. And it's also true for non-singular and hyper surfaces. Sorry, hyper surface. I'm not sure what hyper surface. Non-singular surfaces in P three. And for some time, it was unclear whether these two invariants were actually the same. Anyway, Kaley pointed out that they're actually different. P A is less than P G for certain ruled surfaces, at least of ruled surfaces of genus greater than zero. And the algebraic geometries apparently got a little bit upset about these two not being the same because they called the difference P G minus P A. They called it irregularity. It's sometimes denoted by Q. And I think at the time it was thought that the surfaces for which the irregularity was none zero were kind of a bit exceptional but it turns out that they're actually very, very common. It was just a historical accident that the first surfaces people examined happened to have irregularity zero. Well, all these invariants are a little bit of a mess to keep track of. It's much easier to write them all in terms of hodge numbers which gives a systematic way of writing down all invariants. So the hodge numbers denoted by HP Q is just the dimension of HQ of omega P. So this is some sort of homology group of a piece power of the space of one forms. So this is just the sheaf of P forms. And it has the following properties. First of all, HP Q equals naught unless naught is less than or P is less than or N naught less than or Q is less than or N where N is the dimension of the manifold. And HP Q is equal to H of N minus P, N minus Q by sort of Poincare duality. That's not really Poincare duality, but whatever. And you notice that we switched P and Q here and it's very difficult remembering which were and P and Q go. But fortunately, this often doesn't matter because HP Q is equal to HQ P provided the surface is projective and complex. Should be warned that these numbers can actually be different for non-projective surfaces or non-complex surfaces. So for curves, you can write the hodge numbers as follows. There are four of them. There's H11, H01, H10 and H00. And you write them into a square like this for a reason that be apparent soon. And the numbers H00 and H11 are both one. So, and these two numbers are just the genus. So the hodge numbers for curves look like this. And for surfaces, we write them like this. H22, H12, H21, H02, H11, H20, H01, H10, H10, H11, H00, and these are equal to one. And then by these symmetries, these four numbers are all the same and they're just the irregularity. And these two numbers are the same and are just the arithmetic genus. And there's one more hodge number that we hadn't accounted for, which is H11, which doesn't really have a fashionable name other than H11. So you can ask, why do we write them in a sort of diamond form like this? Well, we write them in diamond form because there's a rather famous hodge decomposition for the ordinary cohomology of the surface with real coefficients. This is just the sum of, can we decompose as the sum of hq of omega p for p plus q equals i. So this means that each row of the hodge diamond corresponds to one of the betting numbers. For instance, for surfaces, we have H22, H12, H21, H02, H11, H20, H01, H10 and H00. And if we add up all the numbers in each row, these are just the betting numbers, B4, B3, B2, B1 and B0, where of course the betting number is just the dimension of the ordinary cohomology of the manifold with real coefficients. So instead of using all these historical terms like arithmetic genus and irregularity and so on, you can just write them in terms of the hodge numbers. So the geometric genus is just H20 equals H02, the irregularity is just H01 equals H10 and the arithmetic genus is just H20 minus H10. As you see, the arithmetic genus is really a bit of a silly invariant. It's just a historical accident as it was defined like that. There's actually a rather neat invariant called the holomorphic Euler characteristic which is one plus the arithmetic genus. Well, why is that neater? Well, it's because it really is an Euler characteristic. It's the dimension of H0 of the regular functions minus the dimension of H1 plus the dimension of H2. So we have an alternating sum of dimensions of cohomology groups just as for the usual Euler characteristic. This is rather nice properties. You can obviously define it for higher dimensional manifolds and it turns out that it's actually multiplicative, thus proving it's a more sensible invariant than the arithmetic genus. But people still use the arithmetic genus out of historical inertia. So next, you can ask about various other invariants. So there are several other invariants which can also be written in terms of hodge numbers. First of all, people sometimes like using churn classes or churn numbers. So there are two churn numbers you will often see and these can be written in terms of hodge numbers. Well, C2 is just the Euler characteristic of the surface which is B nought minus B1 plus B2 minus B3 plus B2. So that's a very good question to ask. So that's B1 plus B2 minus B3 plus B4 and we showed how to write these in terms of hodge numbers and the churn number C1 squared turns out to be 12 chi minus E, I think that's Nert's formula and there are other invariants. There's a signature which turns out to be four chi minus E. So it's also commutative in terms of hodge numbers. So that was proved by Hertz and Brooke. So historically there were large numbers of invariants, genuses, churn classes, signatures and so on and they all turn out to be essentially minor variations of the hodge numbers and there are really three independent hodge numbers, H nought one, H nought two, and H nought one. Under blowing up, H nought one increases by one and the others H nought one and H nought two are fixed. So what this means is that these two invariants here are birational invariants. So you see the old algebraic geometries really knew what they were doing. But what they managed to find was a sort of complete independent set of all the birational invariants you can write in terms of hodge numbers and all the other invariants people later found are just sort of variations of these ones. So for example, we can write out the hodge diamonds of various surfaces we've seen so far. So for the projective plane, the hodge diamond just looks like this, which is quite easy to work out because it's betting numbers are all just one. For P one times P one, it's quite similar. We get one, two, one. And you remember the projective plane can be turned into P one times P one by blowing up two points and blowing down one point. So we should add two to this middle hodge number and then subtract one. And indeed we get one plus two minus one is indeed two. If we look at rational surfaces, sorry, ruled surfaces P one times C, the hodge numbers turn out to look like this, where G is the genus of the curve C. And what you notice in all of these is that this number here, which is the geometric genus is equal to zero. In fact, these are all special cases of ruled surfaces. So ruled implies the geometric genus is zero. We can ask, does if the geometric genus is zero, does this imply the surface is ruled? Well, the answer is often, but not always. So now what I want to do is to discuss the surfaces of geometric genus zero. And how do we classify them? So let's try and classify surfaces with geometric genus zero. And first we do the ones with irregularity greater than zero because this is actually the easy case. The point is for any surface, or for that matter, any variety, there's a map to something called as Albany's variety. And this is an analog of the Jacobian for curves. In fact, for curves, the Albany's variety is just the Jacobian. And the dimension of the algebraic variety of the Albany's variety is equal to the irregularity. This map isn't onto, of course, in general, but... But so if Q is equal to naught, this gives no information because the Albany's variety is just a point. However, if Q is greater than zero, then we have a non-trivial canonical vibration of the surface. And this makes it much easier to identify. If Q is greater than one, then the surface has image, a curve in the Albany's variety. And the fibers are all P1, so the surface is ruled. If Q is equal to one, then it's either ruled or you can analyze this case and you find it's a product of two curves, modulo of finite group. And it turns out if these are not ruled, then they all have cadera dimension, at least zero. So you can classify them. I'm not actually going to bother. So this leaves the case when Q is equal to zero, which is actually the hard case. It's a little bit odd. You'd expect the case of Q equals zero to be the easy case, but in fact, it's the difficult case. Well, one obvious question is, are these ruled? Are these rational? So this was actually an early question, an algebraic geometry for none of the early surfaces people looked at with PG was not, and QG was not, were not rational. But the answer is actually no. Enriquez found some counter examples which are now called Enriquez surfaces, which we might talk about briefly later. Turns out there are quite a few examples, although they're quite hard to find. One of the more interesting ones, the fake projective planes, which have the same hodge numbers as the projective plane. And for a long time, it was an open question whether any of these exist. The first examples were found by Mumford and are rather difficult to construct. They're not things you would come across by accident. Anyway, so PG equals zero and Q equals zero is not enough to show the surface is rational. If you strengthen these conditions, we come across a very famous theorem by Castell-Novo, which says that if Q equals zero, the irrationality is zero and P two equals zero, this implies the surface is rational. This is Castell-Novo's rationality criterion. I might just remind you what this is. P n is just the dimension of the space of sections of the nth power of the canonical bundle. So P one is just the geometric genus and P two is a sort of generalization of this. If P two equals zero, then this implies there are no sections of the square of the canonical bundle other than zero. So there are obviously no sections of the canonical bundle. So this implies PG equals zero. So this condition is strictly stronger than this condition here. Anyway, with Castell's criterion, we can now construct a sort of flowchart of how you classify the surfaces with cadara dimension minus infinity. So what you do is you first of all look at the geometric genus and if the geometric genus is greater than zero, then the cadara dimension is at least zero and the classification of these is extremely complicated. And we'll say a little bit more about it later. So the other case is if the geometric genus is equal to zero. Well, now let's look at the irregularity. So we've seen that if Q is greater than one, then we have a ruled surface. And to prove this, we use the Albanese variety. We have them back from the surface the Albanese variety that more or less gives us our surface as a ruled surface. If Q is equal to one, then what we do is we look at the invariance P4 and P6. So you remember these, the plurigener where you take the fourth and sixth powers of the canonical bundle. And if they're zero, then the surface is now ruled. And if some of them are non-zero, then it's a little bit complicated. We get some hyper elliptic and various other surfaces. But these all have cadaura dimension greater than or equal to zero. So if we're doing the cadaura dimension minus infinity, we don't need to worry about it. So now we've got the case where the irregularity is zero. And now we look at the size of the second plurigenus. And if it's equal to zero, then the surface is rational. And this follows by Castle-Novo's theorem. And if P2 is greater than zero, then this implies the cadaura dimension is greater than equal to zero. And as I said, the complication of these is complicated and rather difficult. So that's a sort of very brief summary of how the classification of surfaces works with surfaces of cadaura dimension minus infinity. So next lecture, we'll be taking a look at the surfaces, some of the surfaces of cadaura dimension zero.