 Okay, so it is a pleasure. We should have done this yesterday, but we were so freaking out about the interface to introduce Dr. Amhela Ortega from the Humboldt University in Berlin for her second lecture. Thank you so much. Thank you. Well, let's continue with a little bit of our background over about the billion varieties. It was a little thing that I didn't say yesterday before going into the primarities, and I think Babel mentioned today in the exercise sessions. So, I think I defined it yesterday's map, beta, from the symmetric product of N-symmetric product of a curve. So the parameterizes the effective divisors on a curve. So of course the genus G curve into the peak N group of C. So it is the group of line bundles of degree N on the C. So C is a curve, smooth curve, yeah, so far everything is smooth, of genus G. And so essentially just to take the divisor and consider the associated line bundles, okay. So, let's define the image of this map. Omega VN, the image of this map is a subset in peak N. So essentially it's all line bundles of degree N with non-empty L as well. So let's put a line bundles of line bundles of degree N with non-empty linear system. Yes. So there is effective divisors that define this line. Okay. So you know by Riemann-Roch, Riemann-Roch theorem, for curves, Riemann-Roch theorem for curves, you know that when a G is, when N is bigger than G, this is the whole space. So essentially all the line bundles of degree N have effective divisors for any, at least G. This is Riemann-Roch. And then for a general, and for a general divisor in the image, for general, let's write it L as line bundle, for the general line bundle in peak N of C, for G between one and, okay. And, sorry, in between one and G, we have only one section. That is what you expect. Okay. This is the general. Of course, you have the real letter, low size, when this, and this letter, the low size will come into the picture later. Okay. So that says that beta is birational in this range. So beta is birational with this image, but that's not onto its image in this way. And so, so WN is birational to C. So in general. Okay. So since beta is proper, you can show that it's a proper map. But actually, the image is a closed subarid. So it's a, it's a, it's an irreducible subarid. So irreducible, in particular, in the case, we are very interested in the case where a, when N is G minus one, WG minus one is a divisor in peak G minus one. So this degree plays a bigger role in the whole theory. And this divisor is canonical associated to the core. Okay. And this is what we call the canonical device. We will see why canonical, canonical data divisor, because it's essentially and it's going to be a translation of the core data divisor. So one, one can show that there is a limo though. I'm going to call it. Maybe not a good idea when I use it later. Doesn't matter. In peak G minus one. Yeah, of the of this degree, such that the pullback of data divisor is. So this, this alpha eta is the map that goes to from pick NC into is a translation essentially you can also translate from one degree to the other. Like tensoring, we have L and you tensor with the, the inverse of it is like, okay, something I didn't mention yesterday, but remember pick zero of C is a group. So the trivial line model is the zero element. And the sum is denserization of, of line bones. And this is it corresponds to, to the, to the sum into the Jacobian we have established this is more business. So the sum, the natural addition in the Jacobians translate into tensor of line books. So I'm going from the Jacobians as a sum of points. Okay, this is, this is, maybe I sometimes I just write it and I don't see it. Okay. So it exists and it is you have a data device on the Jacobian you can always translate it from this canonical data device on degree G minus one. But we like very much in minus one because of the following theorem. And this last thing I'm going to say is the Riemann singularity theorem, very famous one. So you have for every line bundle of degree G minus one. Yes, you want to compute the multiplicity of this canonical divisor at the point L is given by the number of sections. Yes. As I mentioned, the general element here will have only one section. Yes. Only one section. And so it means the element is smooth. But as soon you have two or more sections, you have a singularity of the canonical divisor. And it's just a very nice theorem. So you really can read the singularities from, from, from the space of sections. So many things gonna come from this theorem. Now, that was from yesterday. So I want to go. So the second lecture should start here. So let me start to define the prim varieties. So, so today you will see prim varieties. All the way you have already saw them in the morning with the lectures. Actually, I'm very happy that the lecture happens today because you already saw the motivation of the primary so motivations appears in other context, hitching vibration or, or this Shimura varieties and so on. So let's, let's do it slowly. So where the motivation comes from. Okay, let me, let me give you some sort of motivation of prim varieties. The original motivation. Okay, so yesterday we saw that we have, we have a for every, every curve we can associate it up in a principal price of in variety namely Chacovia. So let's, I haven't done it, let's, let's give names to all the modular spaces I'm going to work with. So MG is going to be the modular space of parametrizing all the curves, see a smooth projective curve of Geno C. So do isomorphism. So it's a class of isomorphism of, of course, and I'm going to work also with AG AG is the modular space parametrizing pairs of a billion varieties such that a is really polarized a billion variety and data is its data divisor. So, traditionally, every time that the divisor is principally principle we call it data divisor, although it's not necessarily coming from from a Jacobian or from a chorus just a principally polarized data device. And it has to be done also up to isomorphism. Okay, so isomorphism classes of these pairs. This, okay, I'm, I, I'm not in the position of talking about how to, how to construct MG is so it's a more long, it's a longer story but there is the GIT construction. On the other hand, it's easier to tell just basically what is AG. So, AG is going to be the quotient of the single or per half plane on space in the plane, H, G, modulo the action of the group, the symplectic group G. So let me tell you briefly how it works. Okay, just to do the single or per half plane of dimension G is simply the space of all G time G matrices, it's a complex entries, such that the matrix is a symmetric so the transposition is equals to T, and the imaginary part defines, yes, a billionaire form positive definite. So maybe that reminds you this style that we described yesterday, the case of G equals one, G equals one is just to the upper half planes of the complex numbers. So, yeah, so for each time that we took a complex numbers on this half plane, we could construct a particular. Okay, and the question by the group. It gives you the isometry them isomorphism of the classes. So here acts. Let me put names because it's going to be a sense. See go over half space. And then SPG is the symplectic group of all the matrices, invertible ones size to G to G with rational entries with the property that they preserve the polarizations you want to preserve the polarization so the polarization with so it can be represented by a matrix. So in this case is precept is our principally polarized for the matrix would say this. So this is the identity. G m transpose. So is that should not change the polarization. Okay. So this, this group acts as a symplectic group. And this symplectic group acts on on the signal or perhaps play space by so. So you take, take a matrix of the form a B CT. So each one is a small matrix of the size G times G here. And we act in in a matrix G times G. So you see this is this is complex by by the following formula. So just now it makes sense because a B our matrices G times G, and then do some this blood multiply them with C plus detail and you take the inverse of this matrix. Okay. So it's such a way that every time. Okay, we know every you choose an element tau in HG, you can define naturally a complex source by taking as a basis for for the latest. Sorry, that would be a basis for the latest. Just like we did yesterday for the liquid curve but now in more dimension higher dimension. And, and, and we say to to to such complex terms are isomorphic as principally polarized a billion varieties, if only if they are related by the action of the simple. So it exists a matrix SPG to G cool. Such that you can go from one towel to the other. Okay. That's very explicit. You can work with that. And it's the only thing that I'm going to say. So, so, so every point in the quotient represents an isomorphism class of principally polarized a billion variety of dimension. Okay, so the same construction also works for other polarizations. So but in this case you have to change, you have to change here the polarization. Okay. And then you get another group. But it's a similar construction. So in any case this signal or perhaps space is the universal core of this modular space. And in particular, you can compute compute the dimensions. So the dimension of, of a G, it will be also the dimension of the, the signal space G, and this is just parametrized by symmetric matrices so the symmetric matrices. G time G has a dimension of G plus G times G plus one over two. And this comes from this construction. Okay. See, now, our Jacobian construction allows you to define a map between these two modular spaces, namely, the Dorelli map. So what is the Dorelli map. It's a map that's associated to every curve of G and G goes to HG, it's Jacobian. So you take a class of isomorphism of course, and you take the class of Jacobian together with its data device. Okay. And the Dorelli theorem, and this is, this is the first term, the Dorelli theorem. So all the Dorelli terms you have here, Dorelli type theorem, it comes from this one. Yes. From this Dorelli. So the Dorelli map is injected. It's injected. So with, with other words, that means you can recover the curve from the Jacobian and its polarisation. You need the polarisation also. So with this pair, you can recover the curve. So, so people then start to study them, the modular space HG through curves, and let's see how far can we go. So you make a little table on the dimensions. So the dimension of MG is 3G minus 3, and that's works only for G at least two. For G1, you are in the case of elliptic curves, and we saw elliptic curves is isomorphic to Jacobian. So in this case is an isomorphism of the Dorelli. So here you have G, G plus one over two. G plus two. Well, you can get one, one, one, this is already one case. But for G plus two, this is dimension three, dimension three, three and six. And this is also six. For G plus four, this is dimension nine, and this is dimension 10. For dimension five, this is 12. For dimension, this is 15. And then, and then you go 21. So this grows linearly and this grows quadratically in G. So, my point is 28. My point is when the, with the, with the Dorelli map, you go from, from here to here, right. So in those two cases, they have the same dimension. So you can recover every, so with another words, the Dorelli map says that the general, the general, the principal, the polarizability and variety of dimension and this dimensions three and three and two are, are Jacobians of some curve. Okay. And this is actually generically fine. You cannot say all because because you have things on the boundary for instance the product of elliptic curves. Well, you are actually there is a Jacobian of the product of two curves. So it's not, it's not a smooth curve, right? You have to do something else on the boundary and then we will see what's happening about, but people were very happy with this because you can study AG through curves. So this is, this is, let's say the philosophy. The philosophy is study AG through the geometry. Of course. Okay. So people ask, okay, can, can I get, and this is very successful because you can study the singularities of the data device source and so on. So the question is that now is, can we get older principally polarize a billion varieties using curves. And the answer is, yes. Some more. So for that we are going to introduce them. Bring varieties, bring varieties. Consider first, double covering. Consider a double covering between curves, between smooth curves. Let's call it P or Pi, Pi, C, C, C, T, C, C, such as the genus of C is G and the genus of C, T, T is G, Tila. Okay, so you can compute who it's a formula, you can compute G, Tila in terms of G, in terms of the ramification as well. But for this map induces, induces a map between the Jacobian and Homo, in fact, is a homomorphism of groups called the norm map. The norm map of P between the Jacobians. And taking a bunch of the, I want my description of the Jacobians that they parametrized line bonds of degree zero. Well, I can take a divisor of degree zero upstairs, linear equivalence like the class of the linear equivalence so it defines a line bond of degree zero and take the push forward of this device. So push forward just take the image of these points and make the sum. Right, so the only condition is the coefficients, the sum of the coefficients is zero, they are integers. So in terms of divisors that is very, very easy to define. And it's also clearly an homomorphism of the groups who respect the sum and everything else. It's pi, right? Yes. I mean it was clear, I just wanted to. Thank you. No, no, it's good that you pointed out because yes, I mix up my notation sometimes. Yes, I mix up in my notes too. Okay, it's pi. That's right. So that means the connected component. And they can, okay, they can look ahead. Several, several components, and one component contains the neutral element of the zero, and this is going to be my brain variety. Okay, so let's define the prevariety. We define the prevariety associated to the cover of pi. By taking the kernel of the normal map. And take the connected component containing the zero. So this is not really a subgroup of the Jacobian C tilde. Yeah, which is, this is a subvariety also, which is a subvariety subgroup also. So, of dimension. Well, just the dimension. Let's call this P. And you want to buy here because it's associated to the morphism. The dimension is the difference of the dimensions. Right. Moreover, it comes with a polarization. We can restrict the data divisor, the principal polarization to this this variety. So it carries a natural polarization, namely just take principal polarization C tilde and restrict it to me. Okay. Now, the question is, when did I get a principal polarization. And actually this is one of the of the exercises, I think it's the last exercise of the list. I will tell you the answer, but you can, you can look at the, how do you prove that. So there is a nice hearing by, by mom for. So the paper of mom for I am referring to is a social deposed it you can have a look as well, which says that the restriction. Okay. The restriction data tilde restricted to P to the pream is algebraic equivalent to twice a principal polarity principal polarization with principal. Let me put here. algebraic equivalence. If only it is. Let's put it here. Two cases happen. P is a tad or. Or P is ramified in exactly two points. Okay. You see, let's do some observations. So first of all, this definition I gave, I started by simplicity just by simplicity. I started with a double cover. But there is nothing that doesn't mean that the cover has to be degree to it could be any degree. And you can still define the normal map. So I should put the name, the normal map. And you can still get the kernel of the normal map as a sub variety ability as a variety. The, the thing is that. So okay, first of all, the definition also holds for D bigger than two to the degree, the degree of the map, bigger than two. Okay, not only the coverings. And if you consider all the possible degree, the only cases where you obtain a principal polarization are these two. So for the bigger than three, there are, there are only two other cases, two other cases where the restriction induces a principal polarization. And those are the following one is. Yes, this is P is the degree three. So piece is triple, I sorry, triple, cyclically covering over a genius to curve. So in this case, the covering upstairs is a genius for. And that's associated bring varieties of dimension to, and there is another little case. Be is any, any, any, any final map, any degree, any final map. So any degree, but yet now the curve of stairs is genius to and downstairs is one. Yeah. So again for Riemann, Riemann, always formula. You can it allows any, any D, any D is allowed. But it's not so interesting because, because of the dimension the plane variety will have in an electric curve. So this case is less interesting. But this is possible. Yes. It's actually not psyche, the first. Non-cyclic, yes. Non-cyclic. For cyclic, it's not the case. That's right. Thank you. I am thinking of the other. Yes, I was working on cyclic and then, okay, it's non-cyclic. That's right. Okay, so, and that's all. So you cannot get anything principal, apart from what we move forward. Let me pick, alternatively, another way of looking at the plane, just to you can start with with a curve with an evolution and define, define the curve C as the quotient of the evolution and define. So this what they said the normal map is also equivalent to define the pre-map as the image. Okay, we have an evolution. So the evolution, every time you have an automorphism of the curve, the covering, that induces an endomorphism of the Jacobian. And you can, you can take the pullback of line moments through this, this element. So then you take the identity map in the Jacobian minus the induced evolution on the normal system. And this is, this is also the primary. Okay. And one can check that these two definitions are. So, so we can think of the prime variety is the Utah, Utah and the invariant part, part of the Jacobian upstairs. Okay, the invariant part is the pullback of, if this is spy, you can, you always have the pullback map between, so now it is from downstairs to upstairs. Yeah, so just take any line bond of degree zero, the pullback is a line bond of degree zero on C tilde. And in the case of etal coverings, so when pi is etal, this, this has a kernel. So the kernel of pi is, is, is generated by, I call it the game eta, eta is a two torsion point. So line bond is just the square is three. Okay. In any case, in any case, the pullback of the Jacobian is the Utah invariant part of this and invariant part, because actually you can see it as the as it gets the image from one plus Utah, so the tax, Utah, Utah acts here, trivially. And this, the, the, the, the, the, the, the, the orthogonal to the, to the pullback of the Jacobian downstairs. Actually, it's more is complimentary. So we have an isogenic. You have always an isogenic. We define as isogenic yesterday. From the product, so you take the invariant part times the anti invariant part. And you saw, it's just to the song. So both, both components are inside of the Jacobian. So you can, you can solve the elements. And this is an isogenic, because both sides have the same dimension. And it's subjective. Okay. Good. Okay, let's stop a little bit here. You have any questions? Okay, Angela, this last, so it can be covering, right? Which one? The last one. Yes, the last map here is subjective. So when I say isogenic, you know, maybe I should recall here as so many means is subjective. Plus, both are the same dimension. So in particular, the kernel is fine. It's a fine. Okay. Now, from now, let's assume that we are in the tall case. In the tall case. We have a kernel. When, when it's not the title, the kernel, the pullback is injective is an embedding. Yes, but let's assume. This is the title. So when I said the title, I asked, is it on ramified covering. And in this case, one can show that the kernel of the normal map. It has two components. Our prem, maybe I should be zero, but this my prem and another component. So this one. This one's contains the zero and not the other. So this, yeah, this one is in an actual a million two variety. So to irreducible to connect with complex. Okay. And one way of distinguishing there's a way of distinguishing two components, but one way is, you can see P. One prem. So let me call P of P, P of pi as the image. As I said, of this endomorphism, but of the pixel. And the other. You can see as the image of this endomorphism, but not of the pixel, but the big one. Because I am using. Okay, so this one, again, this was natural. The Jacobian. And this, this big one is isomorphic. You see, all this big G, et cetera. They are isomorphisms to the Jacobians are so morphic. The difference is that the Jacobian has zero and not the others. But when you, when you, when you take a line bundle of degree one, and so let's just to make it clear what to see. I have one Yota and take a line bundle of degree one, you know, C tilde. This is just to take L that means so in notation of line bundles is L tensor pull back of the Yota inverse. So the new line bundle will have now degree zero. So it is, it is, it lies in, in, in, in the Jacobian. Okay, but it's the other component. Okay, this is one way of these things. And as I said, as I said, when the C tilde, you have a kernel. It comes always with this two torsion point in the top. And in the other sense around is also true. So, okay, let me, let me, let me now define a one for all what is the two torsion point. So this is a notation for the elements eta in the Jacobian such that twice this is eta. So, you see, I see it as a line bundle is trivial. Or if you prefer, let me see, let's put the peak zero. Yeah, make it go. So this is a small Jacobian. And when you put in the Jacobian, actually stood this data is zero. This is the same under this identification. So this is the two torsion points points. Similar, you can define n torsion points. This is similar to the n torsion points. But I wouldn't need to know mostly since you just put here is that you need to take some there and roots of the trivial condom. There are roots. And this is a finite subgroup. And we know the cardinality. So this subgroup will have two to the g elements. Okay, and you can count that because you can look at the lattice the lattice, those points will be half of the lattice in any direction. So you can count how many points you have. Okay. Okay, so every time. So every time you have a curve together with the two torsion point, no trivial one. So I don't want to see one. You can associate to that. And it will cover the covering the following way. So system. So you have an isomorphism. And this is almost in those. The sum of these two line bundles with the with an OC algebra structure or a ring structure we need with ring structure. Okay, so this is so I can I can multiply elements on OC with here with elements of C of clear is clear or we see me data, and I can multiply multiply elements at the square data via this isomorphism. So I have a I have a ring structure in this in this shift. So, you actually the city that is defined as the spectrum of this ring. And so, and it also has an injection of OC here of the driveline bundle. So that induces a projection of the spectrum. Yeah. See, and this is our C. So this is by. This is a way of seeing a. Okay. So let me define. We think with it with this this equivalence in mind. Let's define. Another modular space, RG, the isomorphism classes of their C data such that C is this is a smooth curve of GNG and data is to torsion point minus zero. So I don't want the trivial, the zero or the trivial bundle because it leave a bundle will define something that is disconnected. Yeah, so, yeah, when I say no what you get is to twice the copy of C and this is not. Yeah, I want us a connected curve right they don't want this one this case. So this this is the space. This is moduli space. Which parameter I see. Tracing. It doesn't work. Coverings. And the. Okay, so you have naturally. You have naturally a map from RG to FG is the forgetful map. And so you take this pair you forget the torsion point. And as I mentioned, this is of the grid to do G minus one. It's fine. It's fine. Okay. Yes. You have to take a five minutes break. Oh, no. Thank you. Thank you for telling me. Okay. Then let's stop here. I think it's a moment to stop. Yes. Yes. Forget it. In your writing. Yes. Thank you. Yes. That is a sign that I needed the break. Thank you. Yes. Okay. We forget it. So okay. I leave you the thing here for five minutes.