 OK, so let's start. So we had been talking about non-singular points on curves. So we had so that the theorem if c is a curve, p a non-singular point, then the local ring, OCP, is a discrete valuation ring. And this meant two things that first, the maximum ideal at p is a principal ideal generated by some element t, which will be called the uniformizing parameter. And that, in fact, any element can be written. So for all f in OCP, I can write it as f equal to a times t to the n, where a is a unit in OCP, and n is some non-negative integer. So and we had proven this, this was an application of Nakayama's lemma. And then we had shown that, so we had defined the evaluation associated to that. So if f is an element in the local ring, we had defined then f, after all, can be written as a times t to the n, where a is a unit. And then we set the order of f at p, then we define the order of f at p to be this number n. And we can think of this of the order of vanishing. And we saw that this can be extended to the fraction, the field of rational functions on c by mu p of f divided by g equal to mu p of f minus mu p of g. And you can also then say that, in fact, if we have an element h in kc, which is not 0, then its order, then we can write it h as a times t to this number mu p of f, where now mu p, the multiplicity, can actually be any integer, not just a positive number. OK, so that's more or less where we had stayed. And in fact, we found that if an element, so f in kc, lies in the local ring, if and only if its valuation, mu p of f, is non-negative. OK, we wanted to first now use this to say something about morphisms from non-singular curves. This was this theorem which I had stated. So theorem, let c be a non-singular curve, say p be a point. And let phi from c to y be a morphism to a projective variety. So not from c without p, to a projective variety y. Then phi can be extended to this point. So then phi can be extended to c. So there's a morphism phi from c to y. From, say, I still write just phi. Maybe I call this phi 0. And then I put phi from c to y. So it's a restriction to c without point is the original phi 0. Obviously then, if we have this, we can also say if c non-singular curve and u subset c is open and non-empty, then we can also, in the same way, extend the morphism from u to c. This is basically trivial because u is an open subset in c. So the complement is closed. It's a finite set of points. In order to extend phi to c, we have to extend it onto a neighborhood of each of the missing points. We can do this one by one. So if we can extend it for one point, we can extend it for many points. So we only have to look at this. And as I said, somehow this has the idea that if you have a projective variety, then somehow no point is missing. So if we know where to map all points except one, then we also know where to map this point. So the intuition is that, that it somehow y is complete. I mean, we have proven that it's complete in the sense of how we defined it. But this condition is essentially equivalent. I mean, if, for instance, in Hartzohn's book, you find some proof that this is essentially the same statement as being complete or proper. So now let's prove this. So first we want to reduce that y is just equal to p to the n. So that's simple because assume we can extend phi 0 to phi from c so to p to the n. So assume that y is closed in a closed sub-variety of p to the n. Or y is a projective variety. It lies in some p to the n. So assume we can extend phi 0 not as a map from c to y, but just as a map to p to the n. So the missing point might in theory lie outside y. Then it actually will lie in y because then if I take phi to the minus 1 of y, this is obviously, y is closed in pn. So this is closed in c. And it contains c without this point p. Because I know that all the points in c without p are mapped to y. So only one point might map outside y. But this cannot be. I have a closed subset in c which consists of c without a point. So the closure of c without a point is obviously a whole of c. So thus, phi to the minus 1 of y is equal to c. So that means phi will be morphism from c to y. So thus, we can just assume that y is pn. So we just have to extend it as a map to pn. Well, and now we want to use a story with a discrete variation ring. So we only have to extend, we know the morphism. So on this open subset, on the subset c without p, we have given our morphism. And if you have a morphism, if you have that something is a morphism on two open subsets whose union is the whole set, then we have a morphism. So we just have to extend the morphism from a small neighborhood of p without p to y to a morphism on this neighborhood of p. So we can replace c by any neighborhood of p in c. Because if we then we have our morphism on c without p, and we have an extension of it on this other neighborhood. So we have it on two open subsets whose union is the whole thing. And so we have our morphism. So we can do that. And so making this neighborhood sufficiently small, we can assume that our morphism phi 0 is just given by some regular functions f0 to fn, where fi are regular functions on c without p, without common zeros. We had that every morphism to projective space is given by such a n plus 1 tuple of regular functions without common zeros. And so now the t, the uniformizing parameter. So we can write each fi as say fi equal to ai times t to the mi, where mi is some integer. And because the fi are, after all, fi are elements in kc, certainly, mi is in z, and ai is a unit. So if we want to, again, we can make our neighborhood possibly smaller. We know that ai is a unit, so it doesn't have a 0 at p. So there will be a neighborhood of p where it has no 0. And so we can assume ai has no 0. I haven't actually given a name on our neighborhood, on u. And well, actually, I had to replace c by a neighborhood, so I can also just call it c. So we can assume by replacing c by a smaller neighborhood of p that ai has no 0 of it, and that t has no 0 on c without p. Because we know that t of p is equal to 0, but it is an element in a maximal ideal, but it's not the 0 function, so it has only finitely many zeros. So by making the neighborhood smaller, we can assume that in our neighborhood, there's only 1 0 of t, which is here. And now we choose the j in such a way that it's the minimal of all the mi. So we chose the j for which this power is the lowest. And then we put for all i, we put g i for all i to be, well, I just say ai times t to the mi minus mj. So in other words, this is just fi divided by t to the mj. Now I divide everybody by the smallest power of the t that occurs in them. So then we find now, as this one was the smallest, all these powers are positive. So these actually lie all in the local ring. The g i are all in the local ring. And again, the way I've set it up here, I could assume that also t is regular on the whole of c. So I mean, by making the neighborhood smaller, we can assume that also ai is regular on the whole of c. And I can assume that t is regular on the whole of c. So therefore, I get that g i is regular on the whole of c for all i because here we have just a regular function times a positive power of t. So we are that. So that means that we have a morphism. I put phi equal to g0 to gn. And this is certainly a morphism from c. Well, we have to see whether it's a morphism. I claim it's a morphism from c to pn. So these are all regular functions on c, really on the open neighborhood of p that I have to place c by. These are all regular functions there. And they have no common zeros. So because the g0 and have no common zeros. Because one of them doesn't have any zeros. gj is just aj, and aj had no zeros. And so we have this thing is given by an n plus 1 tuple of regular functions with no common zeros. So it's a morphism. And clearly, it is an extension of phi0 on this open subset because it's just obtained by taking the phi0 is given by f0 to fn. And the new one is given by taking the same and just dividing it by t to the mj. So they are all multiplied by the same factor. So and furthermore, so on c without the point p, we have that phi is equal to g0 to gn, which is the same. The t is a function which has no zero. I mean, the regular functions has no zero on c without p. So if I multiply by it, they put the points, the image in the projective space does not change. I just multiply each factor by the same non-zero constant. So this is the same as, so this is equal to phi0. So it's rather simple. But so to just, the thing, if we are given a morphism, I mean, we can always replace our curve by a neighborhood of the point where the map is maybe not defined. And if the morphism is not defined at the point, then this comes from the fact that either the functions which are used to define it have a common zero at that point or that some of them has a pole. But at any rate, I can divide or multiply by the correct power of a local parameter so that this does not happen. So that if I have a pole, I multiply with the sufficient power of this thing so that it precisely, the pole goes away. If they have a zero, I divide by the corresponding power. If I have a common zero, I divide everybody by the smallest possible power so that the zero goes away. So it's quite simple. But only it's, and then, so as a corollary for instance c, that if c and d are two curves, projective curves, objective non-singular curves, so if they are non-singular, and if they are birational, then they are isomorphic. This is just, I mean, maybe I just said in words, if I have a birational map, if they are birational, I have an isomorphism from an open subset of one to an open subset of the other. There's a non-zero open subset of c and a non-zero open subset of d and an isomorphism between the two. And now I can just extend this isomorphism to a morphism from c to d. And the inverse map, I can extend to a morphism from d to c. And I can see that the composition in both directions is the identity. And that's it. So it's basically trivial. But it is a somewhat, in some sense, surprising fact. So we have this equivalence relation of being birational to each other, which is supposed to be different from isomorphic. It's supposed to be more coarse relation. But if we stick ourselves to projective non-singular curves in every equivalence class, there's only precisely one curve. And it's easy to see that the assumptions are precisely used. So we need all the assumptions. So we need that c and d are both projective, non-singular, and they are curves in order for this to be true. So if it's, for instance, always we have a map. So we need all the assumptions. So if, for instance, you have a1 to p1, is obviously birational map, but not an isomorphism, then if I have a, you can, if you look at the cuspital cubic, this is by itself a subset of a1. But you can complete it to a curve in p2. So if you take the cuspital cubic, this we have defined this as a closed sub-variety of a2. But we can close it up to a closed sub-variety of p2. And then we have a birational map from p1 to this thing. But it's not an isomorphism because the cuspital cubic is singular, and something singular cannot be isomorphic to something non-singular. And so we also see that the statement is not true. The curves have to be non-singular. And in higher dimension, we know that it's certainly true, not true, this statement because, for instance, p2 is birational to p1 times p1. But we have seen they are not isomorphic. Okay, there's something really about curves. Okay, yeah, now, I mean, I somehow, so that somehow more or less covers this chapter. I decided, I mean, in spite of some wishes that I would give some kind of overview about the last chapter. Obviously this is not part of the exam or something, but I just to see, give some kind of impression what the real use of this thing is that for a curve, the local ring is a discrete valuation ring, how you use this, and also in some sense, how it leads further. I mean, even much of the theory for higher dimensional varieties is somehow, the approach is somehow motivated and analogous to what one does for curves using this. So there's a certain way how, so this is somehow, I could say this is the theory of divisors on curves. I will not say so much about it, but I mean, I will not have very much time. But we want to, there's a divisor on curves. It's just a formal linear combination of points on a non-single curve. So I only talk about non-single curves. And in fact, non-singular projective curves. So a divisor is just, will be a formal linear combination of points with multiplicities. And we'll see that we can, I mean, if we have time, we can use them to describe morphisms. And one can use them to describe many other geometric properties of curves and study them. And in the higher dimensional case, one finds that the thing that instead one wants to use is not linear combinations of finitely many points, but of subvarieties of co-dimension 1. So a divisor on a higher dimensional non-single variety will be linear combination of subvarieties of co-dimension 1. And then many of the things, and again, one can describe morphisms in terms of this and all kinds of geometric properties of the higher dimensional varieties, one describes in terms of these divisors. But the kind of model and where everything works at this best is in the case of the curves. And it's already difficult enough. And then in order to study these divisors, one needs then, at least in the higher dimensional case, to develop the theory of homology and so on, which we certainly are not going to study. Now, this thing with the divisors will remind you a little bit, maybe, of the beginning of our algebraic topology course. We start with some kind of fear-bearing group generated by something. So I start so in this last hour or so that I have, so a curve will always be a non-singular projective curve. So I only deal with this because what's the divisor? So let C be a curve. So a divisor on C is a formal divisor. It's just a formal sum. So sum over all points in C, A p times A t times p, where p is a point in C. So we have for all points in p, we associate the multiplicity. A p is an integer. And all but finally many are 0. So it's actually a finite sum. And it looks very infinite here because I sum over all of them. But actually only finally many are 0, are non-zero. And if I want, I can also write it just, if I know which one of them are non-zero, I can just write the sum over those which are non-zero. So this can also be written as sum i equals 1 to l A i times p i if the only ones for which, so if I know that there are precisely l points for which the A p is non-zero and I call them p i, so this would actually be like this. I can also write it directly as a finite sum. But it will be more practical to write it like this. But remember always that the sums are always finite. This is like when you basically a divisor is just a zero cycle in the sense of algebraic topology. What? Yeah, it's a zero chain. Actually, yeah, it's not a cycle. I mean, we're not talking about, we don't have homology. And obviously they form a group in the obvious way. So the group and the B in group, the divisors on C which I call divisors on C. Namely, it's kind of clear that if D is equal to sum A p times p and E is equal to sum B p times p, then I say D plus E is equal to the sum A p plus B p times p. And if this was a finite sum, if both of them are finite sums, then this will still be a finite sum afterwards. So this is over all p and C, only finite in many non-zero. And you can also take the difference obviously by putting minus B p. And the zero element in the group is just for every point the multiplicity is zero. And so they form a group. And I will also write, so if D is a divisor, if D is equal to sum p and C A p times p, I will also write the multiplicity of D at p is defined to be the number A p. So I call D is called effective if all these multiplicities are non-negative. If nu p of D is bigger equal to zero for all p in C. And I also write this just as D is bigger equal to zero. I can also, anyway, that's maybe good enough. So this is just some words. And if we have a function, a rational function, I can associate it to the divisor, namely just given by the multiplicities of the function. So remember, so if H is a rational function on C and not zero, we had defined for every p in C, we had defined the multiplicity nu p of H, which is some integer. Remember, if I write H as A times T to the k, where A is a unit and T is a uniformizing parameter, and then the multiplicity is this. And I always have to think of this multiplicity as the order of pole or vanishing of the function at that point. And before, I'd always fixed the point, but I can do it for every point, as C is a non-singular curve. And so for every p, I can associate this multiplicity. And so therefore, the divisor of H, which I write as div of H, this is equal to the sum of all points in C of the multiplicity of H times p. So for each point, I remember the multiplicity. And it's easy to see that this is a divisor, so that only finitely many of these numbers are non-zero. So it has only finitely many poles and finitely many zeros. That would be an exercise. What is a little bit less? So I forgot one thing. So another word which I have to introduce here is also the degree of a divisor. So the degree of a divisor d equal to sum p, Ap times p, is just the sum of the coefficients. We know that only finitely many are non-zero, and they are integers, so I can form their sum. And so is dig of d equal to the sum Ap. So this is some integer. So if I have, for instance, if the divisor is p1 plus p2, then the degree is 2. And one thing which is not so easy to prove with the things that we know about which is quite fundamental is that, so first I should say, a divisor of this form is called principal divisor. So if it's a divisor of a function, these are special divisors, they are called principal divisors. And one thing that I will not prove is that if I have the degree of a principal divisor is 0. So every principal divisor has degree 0. I mean, somehow you can imagine that if you think back, then a rational function can be viewed as a quotient of two homogeneous polynomials of the same degree. And then you would somehow think that if you have a homogeneous polynomial of a certain degree, then it always has the same number of zeros counted with multiplicity. And so the number of poles counted with multiplicity is the number of 0 of the denominator. And then somehow it should cancel and be 0. But you have to think how to give a real proof. Anyway, but I will not do it. So now it's again a little bit like, the next step is a little bit like what we have in algebraic topology. So we make an equivalence class on these devices. Which I could view as 0 cycles. And if I divide by this, we get a group which is called the picar group, which is actually an interesting group. So we have the picar group. So we have, again, c, our curve. So divisors d and e on c are called linear equivalent. If the difference is just here you put minus, no? If the difference is a principal divisor. So for instance, if d and e, so that there is a, if I take the poles minus the zeros minus the poles of a rational function, then I can find such a rational function that the zeros minus poles with multiplicity is equal to this, I mean, according to this definition. And now this is obviously, it's easy to see that this is a subgroup, no? The principal divisors form a subgroup. And then just taking the divisors and divide by the subgroup, so the picar group. And I write this as d is equivalent to e. And I take d, I write the equivalence class of d. And then I say the picar group of c is the group of divisors on c divided by the linear equivalence. And so this is, picar group is called pick of c. So if one goes to more general setting, so like to higher dimensional varieties or to schemes, then one can see, one can identify these equivalence classes of divisors with something which then in scheme theory would be called invertible sheaves, whatever that might be. And then so the picar group would also just be the group of invertible sheaves. And on a smooth variety, this would be the same, would be also, again, isomorphic to the group of line bundles. So there are different things that you can identify on a variety, which in this case can just be, which are always in some sense, the simplest interpretation is to just say them as divisors, modular linear equivalence. And so, yeah, I mean, I should maybe have said here, so this is really, the principal divisors are subgroup of the divisors. Because obviously, if I take the divisor of f times g, f and gr, then this is, by definition, I mean, almost by definition, the divisor of f plus divisor of g. Because if you remember how the multiplicity was defined, if I take the product of two function, it is the sum of this. And so the group of principal divisors is the image of the multiplicative group of this and the multiplication in the divisors. And so in particular, we see that, so we can also look at the peak, so write, say, div 0 of c to be the divisors on c of degree 0. And I can write peak 0 of c to be div 0 of c divided by the equivalence class. So this is the group of equivalence class of divisors of degree 0. You see, as a principal divisor has degree 0, this will be a subgroup of this. I can also take this portion. And so one thing that one can prove, which is actually very difficult. So maybe I can first make a simple proposition. So what does group b, small remark, so if our curve c is equal to p1, then this peak 0 of c is the trivial group. And actually, I should say differently, c is isomorphic to p1 if and only if. I just want to sketch one direction. So what does it mean that this is the case? It just means that maybe I just say it in words, you can easily see that this statement is equivalent to the fact that if I take any two points, so a divisor which just consists of one point, any two points are linearly equivalent. So you have to find, so easy to see, peak 0 of c is equal to 0, means that for all p and q in c, if I take the divisor p times 1, so just the one point with multiplicity 1 as divisor, is linear equivalent to q. And this is very simple. So if p is the point, say, AB and q is the point CD in p1, then I can just take as a rational function f equal to ax0 minus bx1, I hope it's correct, divide by cx0 minus dx1. And I claim that then the divisor of f, so this means where this thing has 0, I mean, I claim the divisor of f will just be equal to p minus q. So the numerator only vanishes at the point p. The nominator only vanishes at the point q. And it vanishes with multiplicity 1. So you have that, if you put a local parameter at this point, it will be just this times a constant. So this would, I mean, this is just a sketch of the fact that for p1 you get this, a much more difficult theorem, which I cannot possibly prove. Actually, it's quite difficult to prove is that px0 of c is actually the points of some non-singular variety. So it can be given the structure of a non-singular variety, and in fact, of dimension, the genus of c, g of c. So g of c I will define later. So where g of c is the genus of c to be defined later, c later. But at any rate, whatever the genus is, for a non-singular curve, the genus is 0 if and only if the curve is isomorphic to p1. And what I'm saying is that this group is always a non-singular variety of some given dimension. But that's a rather subtle theorem. We are not going to use it. So this px0 of c sometimes called the Jacobian of the curve c, and it has been studied quite a lot. Also, classical. It's a genus of c, yes. So just to have a picture, I mean, this we will not say. But anyway, if our field k is the complex numbers, then c will be a Riemann surface. So it will be a complex manifold of complex dimension 1. So it is a differentiable surface. So it's a two-dimensional manifold, a compact two-dimensional manifold. And so it is a sphere with so many handles. So it looks like this. So it's a sphere with so many holes. So in the case g equal to 0, it's a sphere. In case g equal to 1, it's a total sense on where g is the number of holes. But this is not obvious from anything I say. We are completely in the algebraic setting, so we cannot see why this would be the case. And now one thing that one wants to study when one studies these things is how many sections a divisor has. So we have d, a divisor on c. So the space of sections of d is the following, is l of d. This is a set of all rational functions, such that if I take the divisor of f plus our divisor, this is effective. And well, in kc, we always have excluded the 0. We want this ld to be a vector space, so we add back the 0. So this is kc without 0, and then we add back the 0. So again, you could think of it, if you think, so you could also say it like that, if again the divisor of f is the divisor of 0s minus the divisors of poles, then you would say that this is all the f in kc, which have to have 0s of sufficient order whenever d has a negative multiplicity and is allowed to have poles up to the order of the multiplicity here. So you kind of describe by this divisor what kind of poles and 0s f is allowed to have. So you could say that, so for instance, if d is equal to some AP times p, then we must have that for all p, new p of f must be bigger equal to minus AP. Is this correct? Yes. OK. And you can easily see that this is a vector space, because if this is 2 for f and it's 2 for g, then it's also 2 for f plus g by this additivity property of the new p, the multiplicity at the point of a rational function is bigger equal to the minimum of the multiplicities of both of them, and from this it follows that this is actually a vector space. So this is a vector space. And what is not so trivial is that, and I in the moment can also not explain to you why, I mean, ld is a finite dimensional sub-vector space, sub-k vector space of kc. So this will always be, for every divisor, this will be a finite dimensional space of rational functions. And we call ld, small ld, is defined to be the dimension of this vector space. It's easy to see that in a suitable sense, this ld depends only on the linear equivalence class of d. So if d is linear equivalent to e, so that means the difference of d and e is a principal divisor, then we find that ld is isomorphic to le as a k vector space. Because if in fact it is true that d, so that e is equal to d plus the divisor of h, that means for h a rational function, then we have a map from an isomorphism from ld to le, which sends a rational function f to f times h will be an isomorphism. Because again, by the additivity of these multiplicities, this is what it is. I mean, unless I made some sign error, but anyway. OK. And it's also easy to see that if, so another remark, first, if the divisor is just 0, so no multiplicities, then I'm looking for rational functions. ld, then it follows that ld, is equal to a set of all f in kc, which are regular f regular on c. But we know that on a projective variety, the functions that are regular on the whole of the projective variety are the constants. So it follows that ld is just equal to k. So just a constant, so it's one dimensional, ld is equal to 1. And if the degree of d is smaller than 0, then it follows, I'm asking for the set of all rational functions such that if I take the divisor of the rational function plus our given divisor, this is effective. But if the divisor is effective, then certainly its degree is bigger equal to 0, because all the coefficients are bigger equal to 0. So if the degree here is 0, then whenever, if I add a smaller than 0, if I add the divisor of a rational function which has degree 0, the degree is still smaller than 0, so it can never happen that this is true. So in particular, it means that then l of d just consists of the function 0. So in some simple cases, we can do that. Now I come to some big theorem, which again I will not prove, and I will maybe just try to roughly explain, which is we have the Riemann-Roch theorem, which is one of the kind of important theorems of the 19th century, the beginning of the 19th century about algebraic curves. The course version was proved by Riemann, whose name you know, and the complete version by his student, Roch. Is a student of Riemann. But this is all some, this would be in the 30s of 1830 or something, I don't know. So it's a long time ago. And so I should maybe say, so this is for curves. Then there are versions of it for varieties of any dimension. The most general version, so for smooth varieties of any dimension, this is called the Hirze-Buchemann-Roch theorem, which says if you have a, it's a bit more general, but if you have a line, a divisor on any non-singular right, projective non-singular variety, how many sections it has, and well, not quite, you will see. I mean, something about how many sections it has in terms of topology. So this is due to my advisor, Hirze-Buchemann-Roch. It was what made him famous. And then there is an even more general version by Grotendieg, the Grotendieg-Riemann-Roch theorem, for which Grotendieg got the Fields Medal, which is some kind of relative version of this. So it's somehow a very, it's the beginning of a very important story, which is also, you know, some parts of it are quite deep. So first, I have to talk about something which is a bit difficult to understand. Maybe so I have to talk about differentials. So it's related to what we said about differentials before, but not in a completely obvious way. So I will not try to make this connection. So I do this now in a rather formal way. So we have again C is our curve. So a differential form is always actually one form, but anyway differential form on C is an expression omega equal to sum i, so some final sum, fi times dgi, where fi and gi are rational functions on C. And this is, so we look at all sorts of expressions, but we have the following relations. So first, for any rational function, we can write d this, which is in the moment just the symbol. But we have the relation that if you take d of the sum of two rational functions, this is the sum, for the product, we have the Leibniz rule. And say if the function is constant, so dA for A element in K. And so basically, if one wants to say it in a more abstract way, we take kind of the K vector space generated by all these symbols dgi, and we divide it by these relations we get there. And now, so the remarkable fact is, so in this way, these differential forms form a vector space over Kc. We have this coefficient is an element in Kc. And what is not so obvious, but what is true is that this is actually a one-dimensional vector space. If you look at here, it looks very strongly infinite dimensional. If you look at this, you have an usually uncountable set of generators. But then we have also extremely many relations. And the statement is that it's one-dimensional. And in fact, what you find is, so I mean, I'm just saying it. So if P is a point in C, and omega is a differential form, we can write using only these relations, we can write omega equal to F times dt, where t is a uniformizing parameter at the point P, and F is a rational function. So we can always, for any point in C, we can bring the whole omega into this form. So whenever this differential form was, we can always write it in this way. And then we say the divisor of F of this differential form at the multiplicity of the differential form at P is supposed to be that of F. So the dt, which somehow doesn't contribute. So we say, and we put, but I mean, I should say that this statement here, that we can write it like this is a theorem. It's not obvious at all. But I claim we can, every differential form can be written in this form as F times uniformizing parameter at any point that I want. So we put the multiplicity of omega at P. We define this to be the multiplicity of F at P. So if I'm given a differential form, I can do that. So for instance, and then we can define, so the divisor of omega is defined to be just as usual the sum over P in C of the multiplicity of omega times P. Just for every point, we say what the multiplicity is. And so what we find is that in this way, we get another divisor, which somehow is something which is given naturally. If you know what the curve is, we can say what a differential form on it is. And so then we know what a divisor on it is. So there is, one can prove, so theorem. So for all differential forms, the omega 1 and omega 2 on C, we have that the divisor of this are linearly equivalent. So if you have two, so this is this, in some sense, if you have another differential form, it can be written as F times dt for some possible other F. And so the difference in the function in the divisor only is multiplying by a rational function. But this is also not obvious. And so a divisor, so we call Kc in pick of C, the divisor class of a differential form. Either this Kc, so the class in the Picard group or one of these divisors in the class are called the canonical divisor on C. But I should say that I have not explained this very properly. It actually takes quite a lot of effort, maybe a whole lecture, to explain why this should make sense. So I've just given some rather rough sketch. Anyway, given this, I can try to state the Manroch theorem. Maybe if I can find this. First I can give an algebraic definition of the genus. I mean, I've said over the complex numbers the genus is something. But in general, we have the definition. So the genus of C is G of C equal to now the dimension of the space of sections of Kc. So in other words, this is the same as the dimension, basically it's just the dimension of the space of canonical divisors. Anyway, this is the genus of C, how many sections this thing has. So we can, for instance, see that if C, if x is equal to, so if our curve is equal to P1, I had claimed that the genus of C is equal to 0. And why is that? I cannot explain it very properly, but just as a sketch. So I can take, for instance, so on, I can view P1 as A1 union infinity or something. So if I take here the coordinate z, then so I can also say, so I can view it like this, but I can also view P1 as, say, U0 union U1. So if here I take the coordinate z, then this is the local where z, so maybe I can write it out, so the coordinates on P1 are, say, T0 and T1, the homogeneous coordinates. The coordinate here would be T1 divided by T0. And the coordinate here would be actually 1 over z equal to T0 by T1. So we see that the relation on these two open subsets, so here we have a coordinate, which is also essentially a local. So if this is coordinate here, we have this coordinate here. So now we can ask ourselves, what is the divisor of dz? Z is a function, a rational function on it. And so if you look at it, if I take, so dz, obviously, so if I have z minus the value of z at the point A in P1, this is a local parameter at the point A. So for A, this is a local parameter, a uniformizing parameter. And d of z minus zA is just dz, because this is a constant. Actually, z of A is just A, after all. It's just dz, because A is a constant. So we see that over A1, the divisor of dz is 0. Everywhere, it is just dz times nothing. But now here, we have, at near infinity, we have the local coordinate, the uniformizing parameter at infinity is 1 over z. So I put, say, Bt equal to 1 over z is the parameter here. And it's a uniformizing at infinity. So at infinity, we have that dz is equal to d of 1 over t. Now, where t is the uniformizing parameter at this. And now, from the rules that I have given, I have the product rule, the Leibniz rule for the derivative, we find that it also works in the usual way with the quotient as if it was the derivative. So we find that it's equal to minus 1 over t squared times dt. And so we see this has a pole of order 2 at infinity. So we find that the divisor of dz is equal to minus 2 times infinity. So minus 2 times the point infinity. Maybe I might make a bracket around it so that this infinity is not a number, but it's a point in P1, and this is how it is a number. And so obviously, the degree of this divisor is minus 2. And therefore, we find that L of dz of, say, of the divisor of dz is equal to 0. So the genus is 0. And now, the Riemann-Rhoff theorem says how it works in general. It's a little bit more complicated than one might want to think. So what we are interested in, the Riemann-Rhoff theorem will tell us about, we want to know if you have a divisor, how big is the space of sections. And we don't quite get an answer to that because there's no simple answer to that. But we get some kind of partial answer. But it's the best possible thing that you can get if you ask the question like this. It's the following theorem, Riemann-Rhoff. So d is a divisor on our non-singular curve. C, non-singular projective curve, which has a genus g of c. And then we have that if we take L of d, so the dimension of the space of sections of d, well, we would want to know what it is, but we don't quite know. We know if we take this, kc minus d. So k is a canonical divisor. This is equal to the degree of d. You somehow would think the bigger the degree is, the more sections there should be. But then if the curve is very complicated, maybe they are less. And in fact, it's 1 plus 1 minus the genus of c. So this is the answer. So this is the Riemann-Rhoff theorem because there's also Riemann's theorem, which was that Riemann didn't know about this thing with the canonical divisor. So he has this without this. And this is a non-negative number, and he had bigger equal here. And Roch found the missing term. I don't know, maybe that was it. So Riemann found the previous version of the theorem, and then maybe he gave as a thesis project to his student, can you tell me what the missing term is? And he actually could. So as a corollary, we, for instance, find what is the degree of a canonical divisor, if you look at this. I claim the degree of a canonical divisor is equal to 2 times the genus of the curve minus 2. And I mean, how does that work? This we can actually do. We apply Riemann-Rhoff to Kc. So we get the L of Kc minus the L of Kc minus Kz. Kc, so L of the zero divisor, is equal to the degree of Kc, which we want to know, plus 1 minus the genus of C. Now, the genus was defined as this. This is the genus of C. I have told you that the degree of the zero divisor, that the L of the zero divisor is 1. Because for the zero divisor, the L of it are the constants. And now I think you would be able to bring this to the other side to deduce the statement. Well, yeah, that is actually remarkable. It is minus the Euler characteristic. Minus the Euler characteristic, yes. But that is true. This is because this is by some theorem. I mean, the Euler characteristic, there's a more general statement, which is sometimes called Hopf-Index theorem, which tells you that if you have a complex manifold, say, then the Euler number is equal to whatever, the top churn class evaluated on the manifold. Now, in this particular case, you can prove that you don't know what the churn class is, a certain homology class. But it turns out it is, in some sense, easy to see that the first churn class of the curve is equal to the canonical class, is equal to this thing. So it follows from the Hopf-Index theorem, whatever. That means that the degree of the canonical divisor is minus the Euler number. So the top churn class, you can think of it like this. If you have a section of a vector field on your manifold, so that to every point you associate a tangent vector, then the Hopf-Index theorem says that the Euler number of the manifold is equal to the number of zeros of that vector field. So how many zeros it has? This is a, sometimes it's called, what is it called? Anyway, I mean, it has also comes with other names, sometimes it's called Gauss-Bonnet formula. And now, it turns out that an element, so this thing, can be viewed as zeros of a section, not for a vector field, but a co-vector field. So you take the dual vector space of the tangent bundle and you take a section of that. And then you get the number of zeros of the section of a vector field will become the number of poles of this thing and vice versa. And so you get the negative. Anyway, so by some rather deep results, what you say is true, and it's part of some very general story. Yeah, now I'm not. And so here we had found this. And so just to see, so that you see that this genus is not just something random, that you don't know what this is. For instance, it's a theorem which is actually not difficult to prove, only a lot of work. If C is a non-singer curve of degree, say, D in P2, so it's a zero of a homogeneous polynomial of degree D, and it's done in such a way that it has no poles, then the genus of C can be computed. And it is equal to 1 half times D minus 1 times D minus 2. So we find, for instance, if the degree is 1, then this will give 0. So in P2, if the degree is 2, this will also give 0. If the degree is 3, this gives 1. So a curve, a hypersurface of degree, so a cubic in P2, the zero of a homogeneous polynomial of degree 3, will have genus 1 and will therefore be an elliptic curve. And then when the degree becomes larger, the genus becomes larger and larger. So all kinds of genera occur here. And this is proven by just explicitly computing in coordinates. You can compute what the genus, what the degree of a canonical device is. You just write down a differential form, just Df for some function, and you see what the multiplicities are at the point of the curve, and you find that you get this number. It's just elementary, but it's a bit of work. So now let me see what I want to say. Maybe I can say, yeah. So I can say what this has to do with morphisms, how to make morphisms out of this. So morphisms. So again, let D be divisor. So we can define a morphism, say phi D. So there are some assumptions. Anyway, from C to the P of L of D minus 1. So the protective space of dimension L of D minus 1. L of D is the space of sections here. And this is as follows. Phi, I define it phi D. It's called like that. That's its name. So it depends on the linears. I could also maybe, I like to call it like this because it depends on something which I haven't defined, so maybe I can just write phi D. That's maybe simpler. And so how does it work? It's just very simple. So let H0 until H L of D minus 1, so like this, be a basis of our vector space of sections of D. We know that dimension is L of D. So maybe I write it down here. So obviously, the number of elements of there is like this. And then we have phi D. I just define this to be the morphism defined by H0 to H L of D minus 1. So this is a power i irrational map, because it's only defined where they have no common zeros and so on. But we know that this can be extended to a morphism on the whole of C. I should maybe say that what makes it nice to describe morphisms in this way is that if I take a hyperplane, so if I take the image of this map under the hyperplane, and I take the intersection of the curve with that hyperplane, so I take the curve maps into the space, then if I have any given hyperplane, I can look at the intersection of the curve with the hyperplane. Then for every, so let me see, so then if I start like this, I have to see. So if I take the, I can write D to be the set of all D, E, which with D is linear equivalent, with E is leverally equivalent to D. All the divisors on C, which are linear equivalent to D, and E is effective. So these are by definition precisely equal to the set of all D plus the divisor of F, where F is in L of D. Because L of D was precisely defined as the set of all rational functions, such that if I add its divisor to D, I get something which is effective. And so in this way, it is the same thing. So this, and then I just say it in words. So the divisors, the elements, say E in D, are precisely the inverse images of the intersections with phi D of C with hyperplanes in P L of D minus 1. So what I want to say is you have, so what I mean by intersection is, this is a divisor, so these are points with multiplicities. But at least as a first approximation, if I just take the set of points where the multiplicity is non-zero, this is precisely the inverse image of the intersection of the image with the hyperplanes. And so for instance, if you take just the curve itself, it means if I want to describe, if I have a divisor like this and look at this linear system, so just if I'm given a divisor and I look at this set, which is the same as looking at the space L of D, then what it tells me is if I embed my curve via this map into some projective space, what the intersection of the curve with every hyperplane is. So it's a little bit as if I want to compute a tomography. If I am given a divisor and I use it to describe my thing, I have kind of all the cuts. I know all the cuts of my curve for all the hyperplanes. So this divisor has given me a way to describe somehow how the curve looks like. And if I'm given the curve in a somewhat abstract way, I might not know this before. I know it only when I find such a divisor. And one can use this to, for instance, describe something I mean, maybe I will close there. But I want to at least close with that. So one thing one can prove is, in this way, is the following theorem that if D is a divisor on a curve C of G is G, so if the degree of the divisor is bigger equal to 2G minus 1 plus 1, then we have that this map phi D from C to this projective space. So in this case, we find that L of D is the degree of D. So I just say is an embedding. So if we have a divisor of at least this degree, then this defines me an embedding into some projective space of that dimension. And with this embedding, I can then, in this sense, I know what the intersection with all the hyperplanes is if I can describe all the classes, the divisor's linear equivalent to it. And this one usually can. So I just maybe say in words two things about the higher dimensional case. So if you have a higher dimensional variety, this still works, as I said, if you don't talk about points on a curve, but about subvarieties of co-dimension 1. So if x is a non-singular, I always need non-singular to have this kind of subscription, projective variety of bigger dimension, a divisor on x is a linear combination, is a formal sum, say d equal to sum a z times z. So overall z, where these elements z are closed subvarieties. So irreducible closed subsets of x of co-dimension 1. So if this dimension is d, this is of dimension d minus 1. And so again, these are finite sums. So for every closed subvariety of x of dimension d minus 1, we associate a number az, which is an integer. And this gives us a divisor. So now if f is a rational function on x, we can associate to it a divisor, a principal divisor. We have d of f is a corresponding divisor, is a corresponding principal divisor. So what one has is that if you have a rational function and you have a subvariety of co-dimension 1, you can say what its order of vanishing or what the order of vanishing of f along the subvariety is, or what the order of pole of the subvariety along this thing is. It's not obvious how to define it for us now, but you can. And you will find that if you have such a function, there are only finitely many subvarieties for which this multiplicity is non-zero. This is also not obvious, but it's true. And so we have again a divisor like this. And then we can again define the spaces L of d for divisor. So first we define linear equivalents. So two divisors are linear equivalents if the difference is a principal divisor. And then we define the spaces L of d. So this is, again, the same thing. This is the set of all for a given divisor. This is the same formula as before. This set of all f in kx such that the divisor of f plus d is effective. The effective means, again, that for all subvarieties of co-dimension 1, this number is non-negative, as in the other case. And then we can again associate there's some extra condition. We can again associate to such a divisor a morphism from x to some projective space. It's a little bit more complicated, because I will not precisely say it's a prora, a rational map. And then you have to see whether it's a morphism or not. But anyway, so to, but I just say it to a divisor d, we associate a morphism, say, phi d from x to some projective space of dimension of the L of d minus 1. And again, we have the property that the image of a hyperplane for every hyperplane. So the divisors which are here, which are obtained here, are precisely inverse images of the hyperplanes in this thing. So we again have some kind of tomographic description of x in terms of if we know these divisors. And there's also a Riemann-Roch theorem, which I cannot possibly state. No, no, it's always 0 never defined. What? Yeah, it's the same definition. I mean, except that we are in a different setting. Was there some more thing? I mean, it's certainly over time, one question, whether I want to have one more sentence. I somehow wanted to say something very important. Now I lost it. OK. So we also have this Riemann-Roch theorem. And, ah, yeah. And so, but as we already saw in the previous thing, it's, you know, you know, you can somehow, these morphisms, phi d, are somehow the main tool, one of the main tools to study the varieties x. You know, because you also have this description in terms of the divisors of how they actually look like. So in order to study them, you have to have some way of understanding these spaces L of d. And this you do via cohomology. So, you know, this is then just to say the words. So you have, there is some kind of, you know, you have something, so if d is a divisor, you have something which is called a coherent chief, whatever that might be, which is called O of d. OK. It's just some work. And then you find that you can, there is a so-called chief cohomology. So you can associate to a chief f, whatever a chief is, you can associate this cohomology group H i of x f. So now we can do this here. And we find that the zero-thomology of, this thing is set up in such a way. I take the zero-thomology of x O d. This is just L of d. OK. It's set up in such a way. And the higher comb, there's also H 1, H 2, and so on. And these are much more complicated to understand. OK. And the thing which makes things, but in order to study these bases, you have to study this cohomology so that you can finally understand these bases, L of d. And it's a very big machine, this cohomology. I mean, if you look at Hartjohn's book, he takes about 100 pages to introduce and study the cohomology. And anyway, the Riemann-Roch, so we are interested in, we would want to know what this is the dimension of L of d, for instance, which is the same as the dimension of H 0, according to what I said. But this is not something for which there exists a formula. The Riemann-Roch formula is one. You look at something like the sum i equals 0 to the dimension of x, the minus 1 to the i, the dimension of the i-th homology group. And this is equal to some formula, Riemann-Roch formula. So this is how the Riemann-Roch formula goes. You have some formula here in terms of some topology. It's equal to this. In the case of curves, you precisely see that for a curve, we find that H 1 of x or of d is precisely equal to this thing that we had L of ks minus d, the thing that we had in the Riemann-Roch formula. Actually, it's not quite true. It's a dual vector space, but that's not. Anyway, so somehow what we have for curves is a special case of all this general setting. Anyway, this was just an overview of some of the things. It's not kind of modern. I am not going to the research problem. I mean, the last things are the last result. The Riemann-Roch formula is from 1950. So we are still a little bit away. And most of what I said about curves is from the beginning of the 19th century. But algebraic geometry is a very old subject. These are the last things which are only hinted are very powerful and very difficult results. Anyway, OK, maybe it's enough.