 This lecture is part of an online algebraic geometry course on schemes and we'll be about divisors on some Riemann surfaces, especially Riemann surfaces of degree zero or one. So what we want to do is to look at the Riemann surfaces and look at divisors on them. So what is a divisor? Well, a divisor on a Riemann surface is just a formal sum, sum of Ni Pi, where the Pi appoints on the Riemann surface. Now, if you've got a meromorphic function on a Riemann surface, what we can do is we can form a divisor on this meromorphic function. So we take the function to sum of Ni Pi, where Ni is the order of the zero of f at Pi. And Ni less than naught means there's a pole. And the problem is for meromorphic functions, the sum is in general infinite. So we usually assume C is compact, so sum is finite. I mean, you can do a certain amount of stuff if you allow infinite sums, but then you're really doing analysis, not algebra. So we have a map from meromorphic functions on a compact Riemann surface to divisors. And you can look at the divisor class group, which is the group of divisors modulo principle divisors. So the principle divisors are the ones that come from meromorphic functions. So what this group is really telling you is suppose you're given a divisor, in other words, someone gives you some points with multiplicities. You want to ask, can you find a meromorphic function with exactly those zeros or poles? So let's look at a few examples. So first of all, let's look at the case where the Riemann surface is just the complex numbers. Here, of course, meromorphic functions can have infinitely many poles and zeros. So we will just look at rational functions, which have only a finite number of poles and zeros. And there's a map from rational functions to divisors, which just takes any rational function to its zero. And then there's the Riemann surface and zeros and poles come with multiplicity. And now we can work out the group of divisors modulo principle divisors. And this group is rather boring. It's just the trivial group zero or one, if you're writing things multiplicatively. And the reason is that if we have any divisor, say P1 plus P2 plus P3 and one P1 plus N2 P2 plus N3 P3, just take Z minus P1 to the N1 times Z minus P2 to the N2 times Z minus P3 to the N3 and so on. So every divisor is a principal divisor for rather trivial reasons. You notice this is related to the fact that the ring of polynomials on C is a unique factorization domain. So the primes correspond to the basic divisors Pi and saying that given any finite collection of Pi, you can find the meromorphic function to it is very close related to saying that given any primes, you can find a polynomial with exact, whose zeros correspond exactly to these points or primes. So the complex numbers divisors are not terribly exciting. They're just all principal. Let's look at a slightly more interesting case. Let's take our Riemann surface to be the complex plane together with the pointed infinity. So this is the Riemann sphere. And now meromorphic functions on C are just the same as rational functions. And now it seems we're just in the same cases before, but it's slightly different because now we've got this extra possibility for divisors that we're also allowed the pointed infinity to be part of a divisor. And now if we look at divisors modulo principal divisors, this is no longer trivial. This is now isomorphic to Z. And that's because any divisor has a degree. So the degree of a divisor sum of Ni Pi is just sum of Ni in Z. And now you know that the degree of any principal divisor is zero because any rational function has the same number of poles and zeros. So the degree of a principal divisor principal always get the adjective in the noun mixed up. So the degree of a principal divisor is zero. So this is slightly different from the case of C because now the divisors modulo principal divisors become Z because there's this non-trivial obstruction to finding a rational function with given poles and zeros. And you can also do the same for any compact Riemann surface. So C is a compact Riemann surface. We again find that the degree of a principal divisor is equal to zero. So we get a homomorphism from divisors modulo principal divisors to Z where this map is given by the degree. And this will be on to, but well, we're going to have to investigate the kernel. So we've shown the kernel is trivial for the genus zero Riemann surface of the Riemann sphere. And what happens for others? Well, we're going to look at elliptic curves. Before we go into them, let's just recall why the degree of a principal divisor is zero. The point is, you can find the number of poles and zeros of any meromorphic function inside a region by integrating round a curve C. And taking 1 over 2 pi i of f prime of Z over f of Z dZ. So this is for functions in the plane, but you can do something locally on any Riemann surface. This is 1 over 2 pi i times the integral of d log of fZ. And this log of f of Z sort of increases by 2 pi i whenever you go round a zero. And now locally you can do the same thing on a Riemann surface. And if the Riemann surface is compact, then if you think about it integrating along a curve containing the whole Riemann surface is the same as integration round a curve containing just nothing at all. So this integral is zero if you try and use it to calculate the number of poles minus zeros on the whole Riemann surface. So the number of poles is equal to the number of zeros. Now we're just going to look at elliptic curves, which we're being analysts rather than algebraists this lecture. So an elliptic curve is going to be the complex numbers modulo lattice L. So we take a lattice L and we quotient out by the lattice and we look at meromorphic functions on this quotient. So we're looking at doubly periodic functions on the complex plane. And here we might call the periods omega 1 and omega 2 because why not? So as before, we can find, we saw that the number of poles minus the number of zeros of a doubly periodic meromorphic function is zero, where of course we just count the zeros inside some fundamental region. And let's just do that explicitly for this case and see why it cancels out. So what we're doing is the number of zeros minus the number of poles will be the integral around this curve of 1 over 2 pi i times the integral of f prime of z over f of z d z, where f of z plus omega 1 equals f of z and f of z plus omega 2 equals f of z. And now if you look at this integral, you can see that this is equal to zero because this integral just counts as out with this integral because of this periodic condition and this integral just counts as out with this integral because of this periodic condition. So we can see very explicitly that the number of poles and the number of zeros of an elliptic function are the same in a fundamental domain. So, now we ask if a divisor has degree zero, is it divisor of an elliptic function? Of course we're taking the divisor points as divisor to be not not in the complex plane, but in the complex plane modular lattice and an elliptic function as doubly periodic function is doubly periodic so it is a well defined divisor on this quotient. And it turns out that there's an extra obstruction, not only must the degree be zero, but also the sum of the poles minus the sum of the zeros must be zero, so let's see why that is true. What we do is we look at the following integral, we look at the integral of z times f prime of z over f of z d z. And here what we're integrating over the boundary of a fundamental domain of the elliptic function. If the elliptic function happens to have poles or zeros on this red path here we have to shift it slightly but we won't worry about that. It's just a minor technical point. And let's think about what this is. Well, this function here has residue n z, so let's call this n pi at z equals pi if f has zero of order n i at pi, where of course poles counters zeros of negative order. That's because this thing here has a has a residue of one at a zero of order one and so you just have to multiply it by that. So the integral of this over a fundamental domain will be sum of n i p i at least one over two pi i times that will be some of n i p i. So, however, if we try looking at this integral, the intervals over the lower and upper parts no longer cancel out because z is not doubly periodic. So, so let's see what actually happens. So, so we've got these points not omega one, omega two and some of them. And if we look at the integral along here minus the integral along here. What we get is the Z the Z is almost cancel out so here. Z is omega two more than the Z here so what we do is we get the interval from naught omega one of omega two times. d log of Z. Sorry, the log of F of Z times one over two pi i. And now this thing here just increases so long of F of Z increases by a multiple of two pi i. That's because F of Z has period omega one. So, F of Z is the same here as it is here so if you look at what log of F of Z does it must increase by multiple of two pi i. So, this integral here is an integer multiple of omega two. In exactly the same way if you take this integral minus this integral. This gives you an integral multiple of omega one. So, we see that one over two pi i times this integral Z F prime Z over F Z d Z is equal to some integer times omega one plus some integer times omega two, which is an element of the lattice L we started with. So, we now have this extra condition that if some of N i P i is the divisor of some doubly periodic function F, then some of N i P i equals naught in C over L. And this notation is a bit confusing because this means a divisor. Here I'm taking the P i to be the basis of a free Abelian group. And here I'm actually adding them up in C. So, we think of P i as being an element of C. So, we're adding them up as complex numbers. And of course, P i is a zero of an elliptic function and that's only defined modulo L. So, this sum is only defined modulo L. So, we can't really say what it is in C. It's only an element of C modulo L, but it has to be zero. So, we get a map from the degree zero divisors, which is actually the principle divisors maps on to C over L. So, there's a non-trivial obstruction to finding an elliptic function with given poles and zeros, even if we say that total number of poles is the same as the number of zeros. In fact, this isn't isomorphism. And I just sketched the proof of this using some classical facts about elliptic functions. So, what we want to do is to show that given a degree zero divisor, we want to find an elliptic function which has those as its poles and zeros. So, let's just have a quick review of elliptic functions. This is going to be a very quick review with most details missing. You can easily spend several hours giving the details of this. First of all, we have the famous Weierstrass elliptic function, defined by rho of z is sum of lambda in L of one over z minus lambda or squared, which is obviously double periodic, except it isn't because unfortunately the sum doesn't actually converge. If it did converge, it would obviously be doubly periodic because you're summing over all elements of L. So, what you do is you twiddle it a bit. So, this is one over z squared plus sum of lambda in L of one over z minus lambda squared. And now you sort of subtract this fudge factor minus one over lambda squared. Now that makes it converge. Unfortunately, that's a slight problem. It's no longer quite so obvious that this is doubly periodic, but it's not too difficult to show it's doubly periodic. You can go and look at a book on elliptic functions. So, the Weierstrass function, that's the property that rho of z equals, so it's not a rho, it's p of z equals p of z plus lambda whenever lambda is in L. And now we need another not quite elliptic function. We need the zeta function. And the zeta function is a sort of integral of the Weierstrass function, except it isn't quite because this is a tyson pole at zero. However, it's still true that the derivative of the zeta function is minus the Weierstrass function. And the zeta function is a pole of order one at zero. And before you get too excited about the Riemann zeta function, this is not the Riemann zeta function and has nothing to do with the Riemann zeta function. It's a completely different zeta function. And the Weierstrass function is doubly periodic, but when you integrate it, it's no longer doubly periodic because you pick up this constants of integration. And we find that the zeta of omega one plus z is zeta of z plus an integration constant, which is sometimes called eta of one and the same for zeta of omega two. And that's still not enough. What we need is the sigma function. And the sigma function is sort of given by e to the integral of zeta of z d z. Well, except it's not quite because this integral doesn't really converge if you start integrating at zero and sigma function actually has zero, so it can't be e to the anything, but it still has the property that the derivative of the log of sigma of z is equal to zeta of z. And sigma is even less periodic than zeta. If you unwind it to find that sigma of z plus omega one is equal to minus e to the eta one of z plus omega one over two times sigma of z. So it's almost periodic except you've got this funny extra more complicated fudge factor of z plus omega two is equal to something similar. Well, so these are used to be standard elliptic function formulas. And back in the 19th century, every undergraduate, every mathematics undergraduate British universities was thoroughly trained in all this. And you may think this was a bit of a silly thing to train people to do, but I will just point out that at that time Britain had the world's largest empire and the decline of the British Empire coincides exactly with the time that elliptic functions were removed from the undergraduate syllabus. Anyway, having done all these, we can now find an elliptic function with given poles and zeros. So suppose we've got a divisor sum of n i p i. What we do is we look at the product of sigma of z minus p i the n i. And let's call this f of z. And it certainly has zeros at p i because I forgot to say that sigma of naught is equal to naught and the question is, is this doubly periodic? Well, let's work out f of zeta plus omega one. Well, this is equal to f of zeta times a product of all these fudge factors we got from sigma, which is a product of minus x of a to one z minus a p one times n i. And this factor is equal to one if sum of the n i equals naught and sum of n i p i equals naught in C. So if you pick the p i, so their sum is actually naught in C and not just naught in lambda, then this is a doubly periodic function. So f is an elliptic function with divisor sum of n i p i. So what we've done is we've shown that we get an exact sequence. We get an exact sequence, naught goes to C over L, goes to divisors, modulo principle divisors, goes to z, goes to zero. And an awful lot of the classical theory of elliptic functions is sort of follows from this exact sequence. What this does is it tells you exactly when you can find an elliptic function with given poles and zeros, the obstructions of the degree and this funny group here. And this is called the Jacobian of the elliptic curve. And you may think it's a bit silly to call it the Jacobian of an elliptic curve because it actually is the elliptic curve. The point is if you replace the elliptic curve by some more complicated curve, then what you get here is not the curve, which isn't even a group in general, but something that looks like this only in higher dimensions and it's called the Jacobian of the curve. And what we want to do is to generalize this to higher dimensional varieties. So we want to figure out what we mean by divisor in higher dimensions. In fact, there are there are two plausible versions of what a divisor might be. You can either think of a divisor as being something like the zero locally like the zeros of a function. These are the analogues are called Cartier divisors, which look like zeros of functions. Or we can get things called very divisors. You might think of the divisors as being a sort of union of co-dimension one irreducible subsets. So these correspond to co-dimension one irreducible subsets. So for non-singular curves, Cartier divisors and very divisors are more or less the same, but in general they're slightly different. So in high dimensions, this bit here will sometimes at least turn out to be something called the Picard variety. And this bit here will turn out to be something called the Neuron-Severi group. So there's a sort of discrete group sitting here and a connected group sitting here. And the connected group quite often turns out to be an algebraic variety in its own right. So next lecture, we're going to start discussing how you generalize the notion of divisor in high dimensions and also what it has to do with invertible sheaves.