 This algebraic geometry lecture will be a rather informal review of classification of algebraic curves. So we're going to look at algebraic curves and we're going to do it over the complex numbers just to simplify things a little bit. So there are three ways of viewing algebraic curves over the complex numbers. First of all, we can view them as algebraic curves. If you've got an algebraic curve, this is birational to a complete curve because you can just embed it in projective space and take its closure. And as we will see later, if you've got any algebraic curve, you can actually desingularize it. So it's birational to a non-singular complete curve. So these are all birational to non-singular projective curves. And there are two other ways of looking at non-singular projective curves over the complex numbers. They're more or less the same as compact Riemann surfaces. So if you've got any projective curve over C, you can look at it underline complex manifold and that's a compact Riemann surface. Conversely, if you've got a compact Riemann surface, it's a fairly deep theorem that it does actually come from a non-singular projective algebraic curve. The difficult part of this theorem is showing that a compact Riemann surface has at least one non-trivial meromorphic function on it, which requires some quite serious analysis. The third way of looking at curves is they correspond to finitely generated fields over the complex numbers of transcendence degree equal to one. So to go from a non-singular projective curve to a field, you just look at all meromorphic functions on it. So this just takes meromorphic functions. If you want to go back from a finitely generated function field to a non-singular algebraic curve, this is discussed in detail in Hawke-Schuln's book on algebraic geometry in chapter six. So I won't be discussing that in too much detail because it's already done. And of course, you can go backwards and forwards from compact Riemann surface to finitely generated fields. Sorry, that shouldn't be meromorphic functions. That should be rational functions. To go from here to here, you take the meromorphic functions. So there are three ways of looking at the same object. An analyst would talk about compact Riemann surface. An algebraist would talk about finitely generated fields over the complex numbers of transcendence degree one. And a geometer would talk about non-singular projective curves over the complex numbers. And they're all more or less the same thing. So let's look at how they're classified. The main invariant is the genus. Defining the genus for an algebraic curve and for a field is a little bit tricky, but defining the genus for compact Riemann surface is really easy because a compact Riemann surface is a topological surface and it's orientable. So it has a genus, which is the number of roughly speaking, the number of tori you have to stick together to get the surface. So here we have a surface of genus zero, which is just a sphere or one or two or three and so on. Defining the genus over for algebraic curves can be done, but it involves a little bit more work which we've maybe doing later. So the genus is the basic invariant of a curve. For each genus, there's a family of curves of that genus and there's something called a modular space whose points correspond to isomorphism classes of curves of that genus. So it's a modular space of curves of genus one which classifies curves of genus one and so on. And it's a very fundamental problem in the theory of algebraic curves to try and understand this modular space which is more or less equivalent to classifying all curves of a given genus. So genus naught is easy. The only curve of genus naught, we're talking about non-singular projective curves is the projective line. So the modular space is just a point. Genus one is an extremely interesting and important case. These curves are called elliptic curves. So we discussed these briefly in an earlier lecture. If you're thinking of them as Riemann surfaces, any elliptic curve can be obtained as C modulo elatis L. So L is elatis consisting of all the points M plus N tau for some non-real complex number tau. And as we saw, if you've got a complex number, if you've got C modulo elatis, you can use the Weierstrass function which satisfies this differential equation. And this maps C modulo L to an algebraic curve Y squared equals four X cubed plus B X plus C. So over the complex numbers, the elliptic curves of C modulo elatis. Writing a curve of C modulo elatis is a very non-algebraic operation. Lattices often can't really be defined in algebraic geometry in some sense. And this is certainly not an algebraic function. It's a very transcendental function. So over fields other than the complex numbers, it's rather harder to handle elliptic curves. Anyway, if you look at this, the elliptic curve maps to the X plane or rather the X line. Unfortunately, there's always a problem whether algebraic curves are called curves or surfaces because an algebraic curve is one dimensional when considered algebraically. But if you consider as a Riemann surface, it's two dimensional as a topological manifold. So there's this constant confusion about whether these things should be called curves or surfaces. So this is the X plane or the X line depending whatever. And this is a two to one map from the elliptic curve to the X plane. And it's ramified at four points. So if we draw the X plane, there are going to be four points where it ramifies. That means it sort of branches out a bit. So there's going to be one point at infinity and three points that are the roots of this equation. So what you can do is you can sort of join these up by lines. So this goes off to the point at infinity. And you can construct this surface by taking two copies of the plane, cutting along these lines, and then you sort of get some scissors and sticky tape and join up your two copies of planes by copying tab A in one plane to tab A in the other plane and tab B in this plane to tab B in the second plane and you do the same thing here. So you sort of join up, you cut along this line and then if you've got two surfaces, you join part of this surface to part of the other surface. And you end up constructing the surface as two sheets of paper with two cuts in each two sheets and then you stick them together in the right way. So in general, if you take any four points in the projective line, you can do this trick of joining up the points and pairs and forming an elliptic curve out of them by gluing them. Alternatively, you can look at the equation Y squared equals X minus A, X minus B, X minus C, X minus D, where A, B, C and D are the four points that are ramified. So the question is, when are two elliptic curves the same? Well, if you've got any three points on the projective line, there's a linear fractional transformation, tau goes to A tau plus B over C tau plus D, taking any points, lambda one, lambda two, lambda three to zero, one infinity. So if you've got four points, lambda four is going to go to some point lambda. So any elliptic curve over the complex numbers can be defined by giving a point lambda and you're then looking at the equation Y squared equals X, X minus one, X minus lambda. However, two different values of lambda can sometimes give the same algebraic curve. For example, there's a group of order six permuting zero one infinity, which takes lambda two, lambda or lambda one minus lambda or one over lambda or one over one minus lambda or there's one more that I can never remember, lambda over lambda minus one, I think, one, two, three, four, five, let me start one other, one minus one over lambda. So these form a little group of order six isomorphic to the symmetric group on three points. And this acts on lambda and any two lambdas that are equivalent under this group of order six give the same elliptic curve because you're just permuting these four points. And it turns out the modular space is the affine line parameterized by lambda modulo, this group of order six. So in order to take the quotient of the affine line by this group of order six, we just want to find the invariance of this action. And as I think I mentioned earlier, there's an invariant called the J invariant, which is 256 times lambda squared minus lambda plus one over lambda squared times lambda minus one squared. The factor of 256 makes things work well in characteristic two in case you're wondering where it comes from. So this is the famous J invariant of an elliptic curve. If you've got elliptic curve given like this, is J invariant as just this? Well, we also said elliptic curves could be given as C, the complex numbers modulo lattice one tau. So given tau, there should also be a corresponding J invariant. And this is actually quite difficult to write down. Q is equal to e to the two pi i tau. Then the J invariant turns out to be Q to minus one plus 744 plus one nine six eight eight four Q plus 21493760 Q squared and so on. And there's a whole theory of elliptic functions about these sorts of functions. This is the basic example of an elliptic modular function, as I think I mentioned in an earlier lecture. The coefficients of all sorts of rather bizarre properties, for instance, one nine six eight eight four is one more than the smallest dimension of the smallest representation of the monster simple group. And there are all sorts of other strange properties. So elliptic curves are more or less classified. Two elliptic curves turn out to be isomorphic, if and only if they've got the same J invariant. So the modular space, which is this lambda line minus S three is more or less isomorphic to the affine line A one. Actually, the modular space for elliptic curves isn't really the affine line A one because it's not really a variety at all. Strictly speaking, the modular space is something called an algebraic stack. And there is a problem caused by the fact that elliptic curves have automorphisms. And when things have automorphisms, it always turns out to be a bit iffy writing down a modular space for them. So there's a generalization of varieties called stacks. And I'm not going to give you the definition of a stack because although I've read the definition half of dozen times, I've probably forgotten it immediately afterwards each time that the definition is notoriously difficult to remember or work with. So that's genus one curves. They're pretty well understood. Now let's take a look at genus two curves. Well, you can write down genus two curves in a similar way to the way you write down genus one curves. You can just look at the equation y squared equals x minus a one, x minus a two, x minus a three, x minus a four, x minus a five, x minus a six. So what we're doing is we're taking six points in the complex plane and taking a curve that is ramified to order two over all of these six points. In general, if you've got y squared equals x minus a one up to x minus a two n, it's not very difficult to work out what the genus of the corresponding curve is. You can take this surface and chop it up into cells and you find there are two two cells and you can have two n zero cells and number of lines you need is going to be two n minus one times two. And if you work this all out, you find this Euler characteristic is two plus two n minus four n plus two, which is four minus two n. That's the Euler characteristic. So the genus is related to the Euler characteristic because the Euler characteristic is two minus two g. So the genus is just n minus one. So in particular, if there are six points, the genus is two. Incidentally, this shows there are curves of any given genus because you can just take a sufficiently large number of these curves of this form are called hyper elliptic curves. And in fact, in genus two, it turns out that all curves are hyper elliptic. So this construction gives all of them. Then you want to classify them. And the classification turns out to be equivalent to the problem of taking six points in the projective plane, so in the projective line. But then you've got the group PSL2C of all matrices A, B, C, D, acting on the projective line by mapping tau to A tau plus B over C tau plus D. So what you want to do is to have the problem of finding six distinct points of the projective line, modulo the action for group PSL2C. Well, taking six distinct points of the projective line is more or less the same as classifying binary forms of degree six. So we'd be looking at forms A naught x to the six plus A1 x to the five y plus A to the six y to the six. Modulo the action of the group SL2C. And this is the old problem of classifying invariance of binary quantics. And this problem has been solved, although it's quite difficult. The final answer you get is that this looks like A3, the affine plane, modulo the action of a cyclic group of order five over five z with the group of order five acting as x, y, z goes to zeta of x, zeta squared of y, zeta cubed of z, where zeta is a fifth root of one. So in order to find this space, you need to find the set of invariance of this and the set of generators for the invariance is given by x to the five x cubed y, xy squared, y to the five x squared z, xz cubed, z to the five yz. So you see there's an explicit description of the modular space of hyperliptic codes of genus two, but it's a bit messy. I mean, in higher genus, as far as I know, there's no similar explicit algebraic description of the modular space. It just starts getting too complicated. Well, what do genus three curves look like? Well, genus three, there are two sorts of genus three curves. There are the hyperliptic ones, which are given by y squared equals x minus a one up to x minus a eight. But not all curves are hyperliptic. Another sort of degree three curve are just degree four non-singular curves in the plane. In general, you can work out the genus of a degree D non-singular curve and single plane curve. And the generic degree D plane curve, you can think of as a default cover of the line and generically it will only have two-fold branch points and it will have D times D minus one, two-fold branch points from which it's not too difficult to work out that the Euler characteristic is two D minus D times D minus one, which is equal to two minus two G. So we find the genus is equal to D minus one, D minus two over two. For example, when the degree D is two, we get zero if it's degree three, we get one if it's degree four, we get three if it's degree five, we get six and so on. So the only possible genus of a non-singular plane curve, these triangular numbers zero, one, three, six and so on. So it's actually quite unusual for an algebraic curve to be a non-singular plane curve. There's in fact a six-dimensional family of degree four non-singular curves in the plane. The reason is the space of polynomials has dimension 15. You subtract one because you really want the corresponding projective space that gives you 14. And then you subtract another eight because there's a group PSL2, 3C acting on this. On the other hand, the hyper elliptic curves form a family of dimension five. So in some sense, most genus three curves are in the six-dimensional family and the hyper elliptic curves form a five-dimensional family. We can give you just one example of a genus three curve. A typical example is the trot curve, which is 144 X to the four plus Y to the four minus 225 X squared Y plus Y squared plus 350 X squared Y squared plus 81 equals zero. So what's the point of this rather bizarre curve? Well, a degree four non-singular curve in the plane generally has 28 bi-tangents, which are lines meeting the curve twice. And the question is, can you find an example of a curve such that you can see all these 28 bi-tangents at once? And this particular curve you can, if you draw a picture of it, kind of looks like four beans. I'm not very good at drawing beans, but it ends up looking something roughly like this. And then you can see the 28 bi-tangents because first of all, each of these four beans is a bi-tangent that looks like that. And secondly, there are six ways to choose a pair of these. And if you've got any pair of them, it turns out you can find four bi-tangents, two going between them and two going like that. So this is a particular example of a genus four, so a genus three plane curve where you can draw all 28 bi-tangents explicitly. If we go to genus four, these can't be obtained as singular plane curves. It turns out genus four turn out to be the intersection of a cubic and a quadric in three-dimensional projected space. Genus five can all be given by intersection of three quadrics in P four. As the genus gets bigger and bigger, it gets more and more and more complicated to describe what the curves look like. I think people have pushed this sort of a script not to about genus 10 or 12 or something. I'm not sure what the current record is, but it really gets very difficult to describe these curves. Representing curves of high genus is quite difficult. You can embed them into a projective space using things called canonical embeddings. The trouble is that embeds them into a rather high-dimensional projective space. You can embed it into a plane with singularities such as double points. You can represent those branch covers of the plane. You can represent them as quotients to the upper half plane by discrete groups. Next lecture, I want you to describe some special sorts of curves called herwitz curves.