 This is a mathematics lecture on the Riemann-Roch theorem for the special case of genus zero surfaces. So genus zero surfaces are particularly easy and the Riemann-Roch theorem for these is very elementary and we can just check it explicitly. So what we're going to do, we're going to verify the Riemann-Roch theorem for a particular Riemann surface, the projective line, and then we'll use the Riemann-Roch theorem to classify genus zero surfaces and show that they're all isomorphic to the projective line at least over the complex numbers. So the genus zero surface, if we're talking about compact surfaces, it means the surface is just a sphere, so you can either take the surface to be the projective line if you're an algebraist or if you're an analyst, you can take it to be the Riemann sphere which is the complex numbers together with a pointed infinity. So the general Riemann-Roch theorem says that L of D is degree of D plus 1 minus G plus L of K minus D where you just remember that D is a divisor, the degree is the number of points in the divisor, G is the genus, K is the canonical divisor and L of a divisor tells you the number of the dimension of the space of functions whose poles lie only on the divisor D, roughly speaking. So let's look at the special case of this for genus zero. Well, first of all, the genus is zero, so we can omit that term. Next, we have to evaluate the canonical divisor. So the canonical divisor is just the set of zeros of a one form, so we could take a one form on C and it has no zeros or poles as you can see and we need to know what happens at infinity. Well, we can see what happens at infinity by making the change of variables y equals 1 over z in which case D z becomes minus 1 over y squared dy and you see this as a pole of order two at y equals zero which corresponds to z being the point at infinity. So the canonical divisor which is the set of zeros of D z is minus two times the point infinity because a pole of order two counts as a zero of order minus two. In particular, we see the degree of k is equal to minus two and L of k is equal to zero. There are no holomorphic one forms on the surface. So the Riemann-Roch theorem for genus zero now looks like this. It just says L of D is equal to the degree of D plus 1 plus L of minus D minus two times the point infinity where L of D is the dimension of the functions f such that f plus D is greater than or equal to naught. You remember this means the zeros of f and this essentially just means that all the poles of f have to lie on D and in fact this formula here can be written down more explicitly as follows. First of all, if degree of D is less than zero then L of D equals zero. This is true for any compact Riemann surface whether or not it is genus zero because the function can't have more zeros than poles. On the other hand, if the degree of D is greater than or equal to zero then L of D is the degree of D plus one and this is because the degree of minus D minus two times infinity is now less than zero so L of minus D minus two times infinity is equal to zero so this term up here cancels out. So we have an explicit formula for the number of functions whose zeros are on D that depends only on the degree of D and we should point out that this depends only on degree of D and this is a simplification that occurs only for genus zero Riemann surfaces. As soon as the genus is bigger than zero then working out this number here becomes rather more complicated but this is one of the reasons I said that genus zero is just a sort of easy warming up exercise. So what we're going to do is just verify this directly and the point is that suppose we're given a divisor D is sum of n i p i and let's take all p i to be on the complex plane and we're not going to take p i to be as infinity for the moment and now let's find a function on C let's say it's rational such that f is equal to D so we're not worrying about what happens at f at the point infinity we're just looking at its values on the finite plane and it's obvious how to do this we can just take the product of Z minus p i to the n i and this obviously has a zero of order n at p i and what happens at infinity well it has the order of the pole at infinity is just sum of all the n i's so on C union infinity zeros of f are just D minus sum of n i times the point infinity so you see this this now has degree equal to zero and conversely you can see that any degree zero divisor can be written in this form for some divisor on C and now it's easy to check that what L of D is first of all for any Riemann surface and if p is a point then L of D plus p is at most L of D plus one so it's equal to L of D if there's no function in this in this space with a pole of maximum possible order at p and if there is a function in the space of the pole of maximum possible order of p then that then it gives you an extra plus one to the space of functions with with poles on on on this divisor on the other hand if the function is genus zero and the degree of D is greater than or equal to zero then L of D plus p is now going to be greater than or equal to L of D plus one because we can always find a function with the pole of maximum possible order for this divisor which will not be in L of D so L of D plus B has to be bigger than L of D so we can find this function just by writing down it explicitly like this so we find that if the degree of D is at least zero then L of D is exactly equal to the degree of D plus one so now that we've worked out L of D in all cases we can just check the Riemann Rock theorem explicitly so suppose the degree of D is one of these numbers minus three minus two minus one zero one two three and so on then what is L of D well it's zero zero zero one two three four and so on what's L of K minus D well this is degree minus two minus the degree of D so so L of this goes like this it goes um it's a zero zero zero one two three and so on so now we can look at what is L of D minus L of K minus D but it looks like this it goes zero one two three four minus one minus two minus three and so on and you can see that this is just the degree of D plus one and what you should also notice is that this line here is a polynomial whereas this line here is not a polynomial and this line here is not a polynomial either and what's going on here is that more generally um L of D turns out to be the dimension of a zero co-homology group and L of K minus D turns out to be the zero to be the dimension of a first co-homology group and in general the dimensions of co-homology groups don't vary like polynomials I mean that it's a polynomial here but then it changes on the other hand if you take the Euler characteristic which is an alternating sum of dimensions of co-homology groups that often turns out to be a polynomial so so this number here is really a sort of Euler characteristic and Euler characteristics often behave really nicely and turn out to be polynomials um well so that verifies the Riemann-Roch theorem at least in the case of the Riemann sphere now what we want to do is to classify Riemann surfaces with with genus G equal to zero so let's write down the Riemann-Roch theorem here we get L of D is equal to the degree of D plus one minus G which is zero plus L of K minus D now as before in a previous lecture we find that the degree of K is minus two and L of K is zero and as usual L of D is equal to zero if the degree of D is less than zero so the Riemann-Roch theorem becomes L of D is equal to the degree of D plus one if the degree of D is greater than or equal to zero and not if the degree of D is less than zero so in particular we see that L of zero is equal to one in fact this is always true this is just the space of constants which is one dimensional and L of P is equal to two if P is a point so this is just to devise a consisting of one point and in particular you notice that L of P is greater than L of zero so there is a function f with pole of order one at P and no other poles so for Riemann surfaces of genus greater than zero we don't get functions with just a single pole and no other zeros so this is a sort of characteristic of genus zero Riemann surfaces well and we can use this function it's now a function from our Riemann surface or algebraic curve to the complex numbers and we notice that f is injective because f minus c has exactly one zero for any number c in c because it's only got one pole so we've got an injective function from x to c this doesn't sorry c union infinity this doesn't actually imply that x is isomorphic to c union infinity because it might x might have singularities if it's an algebraic curve however if x is non-singular this easily implies that f is an isomorphism if you just want to see an example of an algebraic curve with a singularity such that there's an injective map that's not an isomorphism we can just take the curve y squared equals x cubed which looks has a sort of cusp like this and if you take the function f to be y over x then it's an injective map from this curve to the Riemann sphere you can add a pointed infinity if you like but it's not an isomorphism of algebraic curves because of the singular point here it's actually a homeomorphism of topological spaces so it doesn't in some sense have an inverse however this inverse isn't a isn't a regular map of Riemann of algebraic curves so this shows that any genus zero curve any non-singular complete genus zero curve is isomorphic to the projective line over the complex numbers another thing we can do with genus zero curves is they're very closely related to unique factorization domains so genus zero sort of is more or less equivalent to being a unique factorization domain what we can do is is we can define a curve y to be say p1 minus a finite number of points and we can let r be the ring of rational functions which are regular on y for example if if we take y to be just c then r is just the ring of polynomials in one variable over over the complex numbers now the point is that r is a unique factorization domain and you can see this because the Riemann rock theorem implies that for each p in r we can find a function with a pole at this point p and no other zeros and poles so we can find a function gp which is zero at p no other zeros and then we we get unique factorization because any other function f on r is just a product over p of gp to the np where np is the order of the zero of f at p i guess this should be times a unit where u is a unit so we've got a factorization of f into primes where the primes are these numbers here so you see unique factorization to primes has this following very strong geometric meaning it says roughly that given any collection of points and multiplicities you can find an essentially unique function with with those zeros or opto unit or whatever so for genus greater than zero it's very unusual for rings like this to be unique factorization domains that I think there are a handful of examples over small finite fields but usually being a unique factorization domain in the case of curves is almost equivalent to its completion being genus zero so genus zero curves are actually really quite different from higher genus curves really you should you should divide curves into three classes genus zero which we've covered today genus one which i'll be doing a bit later and genus greater than one and the differences are well here the degree of the canonical device is less than zero here the degree of the canonical divisors equal to zero and here the degree of the canonical device is greater than zero if you do differential geometry you know these correspond to surfaces with curvature greater than zero here the curvature equals zero and here the curvature is less than zero in fact in differential geometry the Gauss-Bernet theorem says that you can relate the genus or rather the Euler characteristic to the curvature the Euler characteristic which is 2 minus 2g is just the integral of the curvature times some bunch factor which I can't remember there's also something called the Kadera dimension which is indicated by Greek letter kappa the Kadera dimension is related to the number of one forms which in turn is related to the degree of the canonical divisor so the Kadera dimension is by convention given as zero what minus infinity zero or one and the Kadera dimension really controls the behavior of curves or more generally algebraic varieties and higher dimensions genus zero also has many automorphisms so if you've got the projective line you know it's acted on by pgl2 of the complex numbers genus one also has met quite a lot of automorphisms because this can be written as c over l which is a group so this group acts on itself here there are few automorphisms in fact the automorphism group of a typical curve of genus greater than one just as one element and in general it's always finite um genus zero as we've just seen tend to be unique factorization domains genus one and genus greater than one are usually not unique factorization domains so you see the genus zero surfaces we've discussed genus zero Riemann surfaces we've discussed today are really quite unlike the curves of higher genus we're going to discuss later on okay the next lecture on this we'll be talking about uh the Riemann rock theorem in the case of curves of genus one