 The other concept I wanted to tell you is about the type of a polarization, so that will come into our picture later, but now we have all these definitions. If you have E, the an alternating form representing a polarization, L on A, so A is a quotient, so there is a, how's the name, there is a theorem that tells you that there exists, there is exist a basis of elements. Yes, someone has the answer. Okay. There exists a basis of elements on the lattice. So it's a basis of B as a real vector space, with respect to which you write this E is given by the matrix. So you can always find a basis such that E looks like that, T minus T zero, so this is a 2G matrix, 2G matrix, so this is a basis. A basis of B as our vector space. Such that D is a diagonal matrix, diagonal matrix with elements in diagonal D1, DG, and every DI divides the next one, the E plus, the I plus or E, we're starting one, two, G minus one. Okay, this is the condition. And okay, since, okay, and from now on I will just work and concentrate online bundles that they are ample and so then, so in particular this is non-degenerated, yes? So all the DI has to be positive. Non-degenerated. Okay, they are all positive. And that's right, so there is a full theory of looking at all line bundles, even all the line bundles, right? But for me, I will concentrate on those ones. And these numbers is a discrete invariant on our ability, right? So this is the vector D1, DG, is called the type of the polarization. The polarization. This is another important concept. And we say that the polarization is principal, if the type is, if it is of type, it is of type 1, 1, 1, 1. So you have as many ones as the dimension, yeah? So remember this is, this is a real dimension and a complex dimension is G, so we have G is the dimension of, okay, a billion bright. So as usual, we have also a Riemann-Roch theorem. Let me see how it, there are several versions. So in our case, Riemann-Roch, let's concentrate L, non-degenerated, and positive definite, yeah, as you say, positive definite, positive definite, and the trap class. The Euler characteristic of this line bundle is just to the fafian. The fafian of E is the square root of the determinant. So in this case, it's just the product of the elements of the DI's, okay? So the determinant, the determinant of E is the Fafian of E square. Okay. So yes, so actually that, that corresponds. So you know, you know the Euler-Polkare, the Euler-Polkare characteristic of this is defined as the alternating sum of the dimensions of the homology groups, A0 of minus 1 nu from A0 of A H, sorry, no, AL. But in that case, yeah, all of them, all except the A0 is going to be 0. So this in this case is just A0 of A. But just to mention that there's a more general, more general theory. But in our case when L is non-degenerated and the channel class is positive definite, I, yeah, all my eigenvalues is positive. So I just to get, I just to get all the, all the, all the homology groups Spanish is and only the A0 survives. And let's, let's keep it simple. Moreover, you, we can also compute the degree of EL. So this is also the isogenic EL is given by the determinant of E. So this is the, the Fafian. Yeah, the Fafian square. So, so the, the Euler characteristic square is coincides with the degree of this. Okay, so there is a relation also between the, the Euler-Ponkari characteristic and the degree of EL. Okay. I won't say more. Of course, this is all included in a course, a billion varieties. But you see is what is, what is nice is that you can read the Riemann-Roch A0 of a polarized a billion variety from the, from its type of polarization. Okay. It's very simple. Now, let's go to the examples. So let's, let's, let's start from something the dimension one. And I'm sure I've seen this picture, maybe. So you are in the complex plane. In the complex plane. See, we have a lattice. So every lattice is given by, can give a basis of a lattice. We can fix that one, one basis is one. And the other is an imaginary number tau. Or imaginary complex numbers such that, so that is very, very important. You want the imaginary part of tau is different from zero because you wanted the linear independent with, with one. Okay. So that's a basis like this. And so, oops, oops, oops, oops, that, that. Okay. Oh, God. It was not safe. Okay. Sorry about that. So let's start again. Examples. So, so he has the complex. So let's do G1s. So, you take, it's not so straight, but you have nice. Sorry, you have one. It's an element of the basis. And the other is tau. The imaginary part of tau is positive. And then you copy, you translate these vectors. So it's a little bit. It's supposed to be a little bit shift, but so you copy, you copy the, the, so you translate these two vectors in with set so plus, minus and, and then you, you, you, you cover, you, you, you generate the latest to generate the lattice in and in C. Right. So you are looking for the quotient of this is the lattice. The lattice is generated by tau times set. Those are the points of the lattice. One, you want one. Okay. And so you ask what is, what is the quotient. So the quotient of C, all of these lattice. So it has to be compact first of all. So you copy. So every element in the quotient has a representative in this fundamental parallelogram. And moreover, you can decide is, is identified to decide because it's a translation by an element of the lattice and decide is identified with this side. So you glue both sides. So first of all, you, you get a cylinder where you have, you have glue here. This side, and then you glue these two sides together to obtain a toss. So this is the doors. That's right. Identify. So this is this, this is the picture of CA. This picture comes with the, with the element of the basis. So on one side, let me call the image of these sites of the parallelogram one color. Let's say I have names for that. Oh, actually it's one and tau. So for instance, this year. It was the image of one. And this year is the image. Okay. So actually they are also elements on on the first homology of, of the, of the doors. Okay, so in the basis. In the basis. One. Of C. This is V. We have a, we have a billionaire form into a given by given by this matrix, given by the matrix. Which is actually the intersection for. This is the intersection for intersection form. And in the first homology group. On the first. From the quotient. Let's call it C. Yeah. E for elliptic. Mm hmm. Intensive. And that is well known that this is isomorphic. And the basis is these two, two, two loops get. So every, you can. Yeah, you can define an intersection form on the, on the closer paths defining the baby. And this is. And this is in this matrix. So. Mm hmm. So that's what we call an elliptic. This is. An abelian. This is a principal. Okay, we have here. We have here. Thanks to this form, we have defined a principal. Will arise. A billion variety. Of dimension one. Which is the same thing as an elliptic. So elliptic. So elliptic curve. Meaning. Genus of the curve. The associated code is one. So this is given by. One whole topological topological invariant here. Chief was one. And so the difference essentially between elliptic curve and a principal price and then variety is that the. Of dimension one isn't this doesn't decide we have a privilege element called zero. There is a zero over there. And the elliptic curve. Well, we don't have a privilege point. See, but. Yeah. But once you fix the zero, every elliptic curve is an abelian. But I don't dimension one. Okay. So more. So in this, in this baby case, this is the baby case. The line bundle. What is the line bond? What is the polarization? I give you an incarnation polarization via the billionaire form, given by the intersection for. But they can give you also the line bundle. The line bundle. Is given by this point zero. Is the. The shift. Such that the divisor is to see. Yeah. That is. So zero as, as divisor, right. My device around. On, on, on E. Okay. Moreover. And you take. Three times L. So there's a three times. This is. The shift three times this point zero. This is a. This is a. Align bundle of degree three. Because that is a degree one. Yeah. Of degree three on a curve. And by Raymond rock. So. These have exactly. Three sections. So actually. A zero of L. Has one section. And a zero L three times has three sections. And this is, this is, as I mentioned, that is already very ample, very ample. That means that with this line bond, you can embed your elliptic curve. In P two. So in the A zero of CL. That's for three. This is a P two. And actually the embedding given by this, this is an embedding of degree three. That means that the, the elliptic curve is always a cubic, a plane. You can see. Yes. It's a plane cubic. Okay. Which is also very nice. So. When you're right. See, you mean. Right. Yes. I am so in local. You for elliptic. Thank you. Thank you. Okay. So I also want to remark. I forgot. I forgot to say that. That by Riemann Rock. So the Riemann Rock, the number of section is giving them a multiplication. By the, by the multiplication of, of, of the type of the, of the numbers on the type of the polarizations. So in particular. So if the line bundle is principal. That means that it's only one section up to multiplication by scar. And it's if only if, if only if. Okay. If you have only one section and that means that the polarization is of this type one, one, one, one, one. Okay. Only one section. And, and this is very important. Well, very important. It's nice for, for geometry. Because one section means only one effective device. If you have a single object, then you can have two objects on the ability as a right. So the green, the principal polarizer being right this calm. Canonically with a super ID. One unique super ID. Inside there when we, you can do geometry. If it's not principally polarized. You get a linear system and you get many other objects. So this is why we like principally polarized, as we will see, right? So this example generalized to higher dimension, higher dimension. So you can do the same thing is historically is the first example of a billion variety, the Jacobian of a curve. Actually, this is how Abel and Jacobi discovered, so to say, the billion varieties because they first discovered the Jacobian. So they tried to understand how to integrate over this. Yeah, this is a long story, but yeah, the way up to integrate hyperliptic functions is to consider in the quotient. Okay, so let's let's be see, let's see, be a projective smooth projective, smooth projective algebraic curve. So if you prefer, people prefer to call them Freeman surface. And such as smooth projective algebraic curve has as first homology group. It's always an abelian group, if you have to say to G, where G is the genus of the curve. So, okay, let's have a record, you have a curve. So the complex version of a curve, you can have more, but it looks like this. You have some holes in the curve. So for instance, you have three holes. So this is the genus of a curve, the number of holes. And so for each hole, you have two elements on the homology, on the homology. So two loops, one here, yeah, and one on the other direction. Like we did for, okay. So just to make sure what we are talking about, this is the group of, this is the group of closed paths, paths in C. So how do you construct it? Well, you take a point, P0, and you go all the closed paths. So up to, yeah, modular homology, yeah. So two curves, one of, yeah, this curve, this path is essentially different to the other, but this, because this captures a hole and this doesn't. Okay, this is conductive. And so, okay. So the point is, this homology group can be seen as a lattice, as a lattice inside of the holomorphic, in the dual of holomorphic differential. So we have a map from omega C into H0 of C, omega C to what? So this is the vector space of holomorphic differentials on the curve, let's see. So this is naturally, this is isomorphic to Cg. So it has dimension, the genus of the curve. And then you go, how do you take? Can I notation this lambda or gamma? Yeah, you take a close path gamma. And yes, you see it into that, the linear functional that it takes an holomorphic differential omega. And what do you do with the holomorphic differential? What do you integrate them along this path? And that's the fine unelement in the dual space. Okay. And it turns out that this map is injected. All right. So we can see this lattice, it's a full rank lattice inside of this vector space. So they define the Jacobian of a curve because of C is the complex torus. So this is the quotient of the differentials modulo these lattice. So the image of these lattice inside. So this is isomorphic too. Okay. This is not so important. The important is this definition. Moreover, the dimension of this abelian variety is the dimension of the holomorphic differential is the genus of the curve. Okay. And the intersection product that I mentioned before defines the polarization intersection product on H1. So H1CZ. Times H1CZ, so it takes integer values. So you take two paths and you associate the intersection of these two paths. Right. So for instance, this is zero if they don't intersect. This is one intersect in one point in the right directions. And this induces an alternating form. So you can always choose the basis like here. How I put it. So this is called gamma 1, gamma 2, gamma 3. And this is mu 1, mu 2, mu 3. For example, if you choose it like this, the alternating form in these spaces looks like you have a matrix. This actually is the identity of size g minus identity. Okay. So this one means that the loop's lambda intersects the mu in one point. In this order, only with one. And this means that if you change the order, this is minus 1. And the order in the multiplication. So that gives you a principal polarization. And actually, we work very much with the divisor version of the polarization. Denoted by theta. So there is one unique divisor in the Jacobian up to translation, but this is canonically associated to one divisor, giving this principally polarized. Giving the principal polarization. Okay. So the chair in class of L is E. And this, the polarization is O. Okay. In particular, it has only one section. Okay. So what happened with the elliptic curve that we saw at the beginning? Well, in that case, the Jacobian. You don't have to do anything because the Jacobian of the elliptic curve is the curve itself. Elliptic curve is isomorphic to itself. Okay. To the curve itself. Okay. Let me see. Yes. We are very much behind on my program. Okay. Is there any question? No. Okay. So let me tell you about Abel Jacobimac. So as I said, pick zero, let's, I talk about pick zero of the abelian varieties, but we can also take pick zero of a curve. So again, is is the group of line bundles of line bundles of chain class zero. But this is in the level of the curve. This Ne-ronse very group is isomorphic to set actually. And so this number is just to the degree of the line bundle of degree zero on C. And as you might know, that it can be also constructed as the quotient of all the divisors, the group of all divisors on the curve of degree zero, modulo, principal, divisors. So divisors that define it from functions. Okay. Okay. So we define the Abel Jacobimac. The Abel Jacobimac. Alpha. So they are very ways of defining it. Let me keep you this one. You take a divisor of degree zero on C and you can go to the Jacobian the following way. So take a divisor. So divisor is what? Oh, sorry. Sorry about that. So usually nobody calls me. Ah, yes. Ah, so you take the sum. Okay. You can you can you can write it as something like this. So difference of points. So to be sure that it's a degree zero. And then so what is the Jacobian? So it's a linear forms on the differentials, holomorphic differentials, modulo, homology. So I keep you a linear form that takes every differential. The natural thing to do is to integrate between the point penulti, this differential and then make the sum. Okay. This is a linear, that's a linear form on the differentials. Okay. And then to do that modulo homology. Yes. Okay. This is this is actually how Abel Jacobi started to study this by studying this integrals. And so the theorem is that this map induces, induces an isomorphism between pick zero. So if you make the quotient and the Jacobian, pick zero C. So in other words, the kernel of this map, it's precisely the principal devices. And yeah, that also makes sense because you integrate. It has to do with the fundamental theorem of the calculus. I'm not going to do the details, but okay, that is the result. Then alternatively, you can also define alpha dn from Cn into the Jacobian. So what the Cn is, is the Cn is a symmetric group. So you take Cn times modulo, the symmetric group Sn. So on order, on order points or the n-topos. Or you can also say effective divisors. This is the defective divisors. Oh, let's see. Effective, I need to forget, defective divisors. So you take a sum, a formal sum of points, n points, which could be repeated, right, with some multiplicity. And then you go to this functional. So what do you do? You integrate again, but you have a fixed point C. C, P, no, it don't make up. On modulo, homology. So dn, dn is a divisor of degree n, which is n times a point. So the point is fixed. Okay, this is another way of defining it. Okay, let's see if I can arrive. We have more maps. You can also go to, from the effective divisor of degree n into Pn. So what is Pn? Pn is the line bundles of degree n, of degree n on C. And you take every effective divisor, just to take the associated line model. Okay, so you know that the inverse image of a line bundle, it's just going to be its linear series or linear system. We probably call it linear system of L, is the set of all effective divisors, linear equivalent to the effective divisors, linear equivalent to 2, 2, 2, 2, 2, L. So that they have, yeah, okay. Yes, I wanted to finish in a more precise note. There is something I'm going to use later, is the following proposition, that the project device, the project device differential of the Aveljako we map. So the Aveljako we map that goes, so you can, this is what I described here, alpha dn for n equals 1. So for n equals 1, you have an embedding of the curve into the Jacobian when you fix a point of the curve. Okay, so you fix a point C and you take the line bundle associated to the difference of these two points for a fixed, C is fixed. Okay, so this is Aveljako we map. And the differential, the project device of the differential, so what is it? So you have to put the differentials of alpha C, you have to project device, correct device, the tangent to C, the tangent bundle to C. So it's essentially, you can identify to the curve because in every point of the curve you have a tangent line, and you project device this line, it's just the point itself. Goes to the project device of the tangent of the Jacobian, so the tangent of the Jacobian. So the tangent of the Jacobian we know because the tangent to the Jacobian in one point, let's say zero point or any point, is isomorphic to the vector space itself. Right? What do you call it? The C, space of the morphic differential is DUI. So you project device the dual of this, so yes. And actually the tangent is because we have these translations, is this trivial bundle, so you have the Jacobian times, yes, I wanted to write it properly. Okay. So what I mean, okay, of course you have, so maybe it's not so clear, but what I'm saying, this map is actually the canonical map. So every curve comes with a canonical divisor, a canonical shift, and that's give you a canonical map between, define it by the last one though. Usually it's embedding, but not always into P of HCO, mega C, which is PG minus one. This is, this is a very nice proposition we'll want to use later. I'm sorry, I'm over time, I think a little bit, not so much. But maybe I will finish here. It's any questions? I have a stupid question. Yeah, please. Can you remind us what is the projectivization, projectivization of TC? Do you mean the projectivation of vector bundle on other stuff? Yeah, of the vector bundle, yes. So the differential, the differential of any map goes from the tangent bundle from one to the tangent one to the other. Yes. Okay. So you put yourself in one point. So what they want to projectivize, I want to projectivize that bundle. And by that, I mean I projectivize every fiber, every fiber of the bundle. For, yes. So and on the left side, on the left side, well, there is nothing to, yeah, when you projectivize a tangent to the point, well, it's the point itself, there is nothing to do. And on the other side, what you projectivize is this part. So this is, this gives you the base point. So you have a base point, so you get, okay, maybe you go from C, this point, and you take, well, point P. This is a point in the Jacobian, and then you projectivize this one here. So what I mean is this should be the given by the sections of this bundle. So what's the section? Yeah, so did I give names? Yeah, some section of homogenize, yes, because I projectivize this part. Yes. So what is nice of the Jacobian is the tangent to any point is isomorphic to the differentials, because you get all the by translating, by translating. And you projectivize that, you projectivize this part here. Okay, so, and the canonical map, I mean, the canonical map is the second part. The point is fixed is this part. This is, this is the canonical map. Yeah, so you take a point and you go to, you get a section, a basis of sections, basis of differentials. Oh, is this cross, this P is not good. How many you have? So you have, you start in one, you have G. That's right. Yes. Omega zero. Omega zero. Yeah, I shouldn't start to omega zero. I just say, I start in one, and G is speech. Yeah, it's homogenize, it's homogenous. No, these are correct. No, no, no. Now it's all clear. Thank you. Is it clear? Okay. Yes, so this projectivize differential one has to understand it on each fiber extension, on each fiber. Okay. Other questions? Thank you. Sorry, I have a question. So on the right hand side, it's a projectivization of the tangent bundle. So is it the product of the Jacobian with the PG minus one? Yes. Yes. So the canonical map is actually the production to the PG minus one. Yes, the canonical map is defined by sections of these differentials. Sorry, Angela, maybe. That's a question. Yeah. Yes, maybe here you have to compose with the second projection, right, from the canonical projective bundle to PG minus one. Yes. From the trivial. Yes, from the trivial. Yes. Yes, I put it here, but yes, I think to make it clear. So essentially, I should, sorry for the mess, but here you have to project into the second one, yes, to the second factor. So the projectivization of each fiber. This, this composition is the canonical map. That's right. I hope it's clear. Is it clear? Yes. Thank you. Okay. Yeah. Yeah. That's why I don't have written notes, because all my notes will be much better than here. So I should, good. So I am into the behind my plan. I just want to talk a little bit about the data divisor, which is the next step, not so much more. And the next step tomorrow, come the prime varieties. So I will tell you what the prime variety is and the prime map. Today feels like a little bit in a hurry because there are a lot of background, but I hope it will slow down and be more concrete. More questions? Okay. Good. Then done. I think you have the exercise session with Pablo tomorrow. Feel free to ask him all the questions. Okay.