 for introducing me. Yeah, so I know my name is quite hard to say, so my name is Paweł Borówka, which is Paul Blueberry, if you prefer. So you can call me Blueberry. Yeah, because in Poland we have this kind of funny names and surname somehow. So yes, okay. So the idea is to talk about the print varieties and the lecture by Angel Ortega. And I prepared some notes. I'm happy to answer any questions that you found during the talk and the lecture. And today and then tomorrow, I think, and on Friday I will have like one more hour. So please feel free. I think answering the questions is the most important part of the exercises somehow. And at least for me, it's usually that only like after like half an hour after the lecture, I start wondering about the details and I find some interesting questions. So that's why, you know, you have time to do it. So for sure, if you have questions, this is like a priority for me to try to answer the question, what Angela had in mind. Yeah, okay. So starting from the beginning, I just prepared the references because, you know, the first lecture was like very, very technical. So the references is the like the biggest or the best book in the theory of complex abelian varieties, namely complex abelian varieties by Bigeha Kalanger, BL. There is a lot of things inside. So yeah, so most of the first talk was can be find here and those are chapters like, I think chapters one to four, one, three and plus then the chapter, I think 11 about the Jacobians. Yeah, okay, so yeah. So if you find yourself, yeah, I want to say one thing that when I was a student, I was told that the theory of abelian varieties is this kind of a strange theory that you want to do some geometry and it's really useful to start reading this book, Bigeha Kalanger, starting from the chapter four and forget about, you know, don't read those technical details before and only if you really need them, go back and read them. Obviously it's not easy when you want to have a lecture. So that's why those first lectures on abelian varieties and primes are usually so technical, yes? And there are lots of questions and especially, you know, you shouldn't be, the people who don't know much about abelian varieties shouldn't be ashamed, it's usually like that, that you start and either you know everything and it's like boring technical stuff or you don't know anything and then it's like crazy magic in some sense. So that's the thing. We've prepared some exercises so you can find them on the Trieste webpage. I think I should, maybe I should be able to actually put them also in the chat or somebody else maybe can find them and put them. Download, there if I can, how does it work? File, so I will try to maybe send it. Yeah, so there are exercises. The bad thing about exercises is that they usually, we assume that you know what the prime variety is in most of them. So you need to wait for the second lecture of Angela to see the, to understand the question sometimes but I'm pretty much sure that there are many of you who already know what the prime variety is. So I really would like you to go with the exercises. In a second I will just try to say a few words about the exercises and what's going on there. Obviously, because this is not a lecture, I hope this is not a lecture but the exercises or the tutors or however you call it, the seminar, I would be really grateful if you start talking because communication is I think the crucial one in the studies. So if you have any questions please do not hesitate to stop me and to start talking. Yes, okay, so maybe I will say just for those who do not know much about a billion varieties like once more what's going on there. So, okay, so in terms of what's interesting about or what's interesting about a billion varieties and the line bundles on them. Yes, the ample line bundles. That it's really, that there's really the, you know, the CRM that says that if L is ample and L prime is a line bundle, bundle sorry for my writing actually. I usually try to write too fast. So C1, this C1 of L is C1 of L C1 of L prime, one and the second if and only if L prime is the translation by some point of L, okay. So it's really, it's really, you know, when Angela was talking about polarization, yes. In principle, she defined it as a line bundle. I prefer to call it polarizing line bundle because polarization is the chain class of a line bundle. But, you know, but the information is more or less the same, yes, you have a line bundle and then you can just translate it and this is the only thing you can do. So in the course, there was this question about this short exact sequence, yes. Big zero of X, X and then there's a very class of X, yes. And there was a question about if it splits somehow, yes, because there was, we wanted to talk about big zero to see that it is also an obedient variety, the dual one. So actually, yeah, it somehow splits. So it's not only that we have this, this statement that the line bundle is defined up to translation, the class. So if you make a little bit more technical fixing statements, then you can find the so-called characteristics and then you can somehow distinguish the line bundle with characteristic zero. So this X, this X, somehow if you have L and X, you can call this X, this X is a characteristic of a line bundle. A-ca-ra-ter-ish-t-x. And for principle polarization, this characteristic is uniquely defined, so everything is nice. If L is not principally polarized, that you have to be careful because this is the kernel of the polarization. So this kernel of the polarizing isogenic, if I want to say so, because you have these five things and you have to live with a thing that, when I say polarization, I may think of any of those five. So there is this kernel of polarization. So there are points X as that those guys are actually isomorphic to each other. So the characteristic is then defined up to this kernel. And obviously again, hopefully we'll not go into details, but if you have a line bundle, you can, ample line bundle, you are interested in the space of sections and they are defined by so-called theta functions. And if you have seen theta functions, it's good. If you don't see, it's also good because you don't want to see them. It's like a series of exponents. So there's lots of things which is in some sense hard because you are really in the analytic world. And for example, finding the zeros of a series of exponents is far from being easy. So that's why we really want to go into the algebraic point of view. And that's why the Jacobians are really important because for the Jacobians, we have this apple Jacobi map and we can think about this tori as an algebraic objects and as a divisors of degree zero. So, okay, so here, yeah, so here you can think about characteristic zero line bundle. Up. Yes, may I ask a stupid question? Obviously. Sorry, but I'm not sure I understand really what do you mean by the C1 of L is equal to C1 of L time. So that they are linearly equivalent. No, no, no, no, no, those are all concentration classes. So this is in the Neurons' Severi class, yeah? So you can think about those guys as a this Hermitian form defined by the line bundle, okay? So this is in the Neurons' Severi class. Yes, so you're really mean equality. Yes, I really mean equality. Yes. Yes, so in some sense, okay, if you are here X, yes, if you are here and you go with the term classes and then you have equality here. And obviously, again, if you think about term classes, you can think of different things, but on a billion varieties, term classes can be seen as just the Hermitian form, positive definite integral on the lattice, yes. So you have equality here. And then you are wondering what's going on in the line bundles here, yes. And the thing is that it's really the same line model up to translation and it works for ample line bundles. So this is CRM, actually, I can check the statement. This is obviously in the Birkenhake-Lange book. Yes. Chapter for, I don't know, not from my head, but I know I can find it within like 15 seconds. Okay, so... Sorry, another question, sorry. Yes. And if you look at the first term class as the divisor, the divisor to which L is associated with, so what you get, you get exactly the same linear system on the a billion variety. So, okay, wait a second. So if you have a line bundle and you want to think about line bundle as the divisor, yes, then you can move divisor by a translation on a curve, on an abelian surface. And in general, those two things, so the divisor and it's okay. So wait a second. In general, they are not linear. They are not linear equivalents, yes. So in general, they are not linear equivalents. They are linearly equivalent if and only if the translation is in the kernel of this polarizing isogenic, yes. So they are only algebraically equivalent, yes. So exactly, this is, so in general, they are in this, it gives you the same class in the neuron savoury, yes. And sometimes, depending on the abelian variety, you have a linear equivalence. Okay, okay. So in terms of divisors, the second condition is the same for the associated divisors, right? Okay. Yes, thank you. Okay, so, okay, so maybe, okay, so this is what I want to say somehow about the line bandos on abelian varieties. And obviously, now we have the Jacobians, yes, the JC. So JC is, you can think about the... Wait, could you repeat please the last argument because I didn't get this part was X. So does this, the second part of the theorem pull for any X or just this chosen X? No, for every X in X. Okay, thanks. Because if you move by a point on an abelian variety, you get something which is algebraically equivalent. So this, yeah, for all Xs, if... No, okay, so there exists X, yes. In this statement, L prime is isomorphic to TX for some X, yes. Then they are in the sum X in X. Yes, so if you move by some X, then you get the same turn plus, the same hermitian form, yes. Okay, maybe now I will go into the exercises and I will try to persuade you that the exercises are interesting, fun, and however you want the exercises to be. And there are, I already, I want to say stress it. They are on different levels of... You know, there are some easy exercises and some harder ones. And actually there is a question which I don't know how to answer. So maybe you can actually help me to solve some problems. Okay, so exercises. So the first exercise is for people who don't know much about Apiglian varieties is just to show you that, if you have a complex torus with your lattice, yes, so C divided by lambda, then obviously you want this torus to be algebraic. So what are the condition on lambda and those conditions are called Riemann relations and you can look at it. Obviously for many people then you want to go into a construction similar to what you know from the elliptical, so-called Ziegels space. So you define the Ziegels space, so the matrices that we, which are special. So the symmetric matrices with imaginary part positive definite. So like a straight generalization of a Ziegels map a half plane and then you will know by construction that the resulting torus will be algebraic. So this is the first exercise. And then the second exercise is for you to see because okay, we haven't seen the definition of the prim map on the lecture by Angela but we already seen the definition of a prim map for double coverings on the Iran's lecture. So the idea is that, okay, so the idea is that you have the Jacobian here. We have a covering from C to C prime or from C tilde to C. So you have a Jacobian in the middle, then you have the smaller Jacobian here and then you have the kernel of this norm map, well known. And there are two, so the definition, the one of the definitions of a prim map is just the connected component of the kernel which will be one definition. But the other definition which you can think of if is by so called complimentary Abelian varieties. So if you have an Abelian variety, you can think, it's just a torus, so you have CG divided by some lambda. So if you find a sub variety, Abelian sub variety, so some smaller vector subspace, then there is a construction which shows you that there exists a complementary subspace. Yes, so this is quite a nice thing. Actually, this is technical to show, but there exists. And those two things will be called complimentary Abelian sub varieties. And actually those two notions of a prim, so one would be that it is just the zero of the connected component of the kernel. And the second one will be complimentary Abelian sub variety to the image of the smaller Jacobian in the bigger Jacobian. Those two things coincide. And this is exercise number two. The exercise number three is also very important. The universal property of a Jacobian, I will write it down in a second. And then exercise number four, this is again the classical statement of a prims. So if you have an etal-dabre covering, you don't, you know, when I said the connected component, this is really crucial because the kernel of the of a norm map has two irreducible components. And actually we can say what they are. And you have this description P zero and P one. So you have to show it. And actually you can generalize this statement if you have a finite covering between smooth curves, then the pullback is not injective. And especially this means also that the kernel is not connected if and only if F factorizes via a cyclic etal covering F tilde, I guess, of degree at least two. So this is also important because, you know, okay, so in general for the branched, for example, coverings, you have no problem. The Jacobian, the smaller Jacobian is embedded in the bigger Jacobian. And then you have the kernel, which is connected. And this is the prime variety as Iran said. But if you have, but if you have a covering which is not necessary branched or it is like a composition of two other coverings, then you have to be careful. If you have a cyclic etal coverings downstairs, then you don't have injectivity. Okay, six is one, the six exam, the six CRM is also very, very classical. So using Riemann singularity CRM, you need to show that the singular locus of the CETA divisors. So the CETA divisors on the principle polarized Avian varieties is defined, is well defined up to this translation. So the singular locus is also defined up to translation. And you can compute the dimension is G minus three for non-hyperliptic curve. And seven, I say seven is important because this is also what Iran was talking about also. So in general, the prime map is generically injective. But if we are in the locus of hyperliptic curves, we don't have injectivity at all. And this is somehow the exercise seven says this. So if you have a covering of a hyperliptic curves, that the prime map is a Jacobian of another hyperliptic curve. So you really, you know, you can think of it that the Picard group of the curve upstairs is built from the two Picard groups of smaller curves. And this is quite nice statement because it gives you the ability to work with those smaller pieces somehow. And yeah, and the question is how to find this curve and then if you find the curve, then you will see that the Jacobian of a hyperliptic curve is a prime variety for some covering and there will be lots of coverings that will give you the same hyperliptic curve. So you will have not injectivity. Exercise number eight, this is what I've said that I don't know the answer to. So the first part is in some sense easy. It's already, maybe you can find it also in the Iran stock. So you have, so why the prime varieties are also interesting because by the universal property of a Jacobian, the curve is usually embedded also in the prime. So you have this Abel prime map. So really thinking about the prime maps is also similar question. So the prime theory is also similar question to, especially in the very low dimension, namely for the surfaces is the same about asking what are the embeddings of curves into Abelian surfaces? So what kind of curves can you find in the Abelian surfaces? Maybe not necessary embeddings but by rational maps, yes? So if you have a genius free curve, you can embed it in Abelian surfaces, in Abelian surface, if and only if it is a covering of an elliptic curve. So this is a statement by, if the genius is two, it's already by barf. No, if the degree is two, it's already by barf. And okay, so there are some curves, genius free curves that can be embedded in the Abelian surfaces and others are not, they cannot be embedded. So we know that the genius free curve, a general genius free curve is a quartic plane, okay? So you have, you know, you start with your favorite quartic plane and the question is, can you embed it into an Abelian surface? So obviously because we are in the 2021, the question is, if you start with your favorite quartic plane curve, can you tell me if it is a covering of an elliptic curve 2021 to one? Because there is like three dimensional family of such curves in your, in the, in the modulite of quartic curves. And the question is, can you actually, you know, find the explicit examples, somehow explicit example of those. So, you know, this is something like finding not rational numbers, yes? Because obviously the covering for, you know, if this number was not 2021, but two, then it will be like easy because then you will have automorphism and so on. But for 2021, actually, if you don't like 2021, you can have anything which is bigger than what the, let's say 10, it will be already similar problem from my point of view. But it's quite an interesting question because on the level of Abelian surfaces, the question is easy, yes? Can you find a curve that can be embedded in the one 2021 polarized surface? So this is like an obvious question, but on the other hand, on the level of MG, I don't know the answer, how to say it. Yes, and then the ninth question is also like playing with those, with those prims. So in general, the classical prim is the two to one unbranched, so et al. And then two to one branched, but then you can work with any degree in some sense. And obviously, as Irene said, then you will almost immediately go out from the principle polarized world to the whole world of Abelian varieties. So the question is when the prim variety is still principally polarized. And there is a theorem in the Birkenhake Lange, which has a gap. So that's why it's interesting to do it because you will go to Birkenhake Lange, you will read the proof. And if you believe it, you are wrong. There is a gap in the proof. And there is one more case, which is not covered by the Birkenhake Lange. But actually it is filled by the paper of Lange or Tega, which you probably know. Yes, so that's the exercises. So I would like you, even like next to exercise sessions, I would like to find people who will be able to say a few words more or less, hopefully the precise words, how to solve these exercises. So that you see the constructions and you see the methods, how to play with a prim. Because Angela wants to go with a theory and play with prim maps and the geometry of the prim maps. But then those exercises are like down to earth, precise, explicit examples. So that hopefully some of you who don't know much about prim varieties will be able to do something or at least find some references for it. Okay, so that's more or less about the exercises. Can I ask a question? Obviously, yes. So in the exercise eight, you said that basically it's hard to find explicit examples, but we know that there is a three-dimensional family. Could you comment on why? So I imagine this does not depend on the 2021, this one for anything. No, no, no. Okay, so give me a second. Let's go back to my tablet. So the answer is very easy, yes? So that's maybe this. So the answer is very easy because you can see it from the other perspective. You start with your elliptic curve, okay? You take four points, you choose your degree, and then you know that there exists a covering C E with a branched D branched at four points. Obviously, this covering will be non-Galewa, yes? But it exists. So this curve by Riemann-Kurwitz will be a genius-free curve, which has such an embedding, yes? And now if you think about the norm map, you will see JC, this will go to E, okay? And then you will have the prim of C divided by E, yes, so yes. And then because this is branched at four points and 2021 is, no, it's not prime. I don't know. Who knows, 2021 prime? It's not prime, it's the first thing I checked. It's a product of two prime factors. Anyway, E is isomorphic to E, so... It's non-Galewa, you can assume it's irresistible, so it doesn't go via, because, okay, yeah, no, it doesn't go via any other, so you have like that. And then you have here P of C E dual. And now you can compute, so this is, the restricted polarization is 2021, because it's 2020, it's 2021, so you have here 1-1, so here you have 1-2021, yes, polarized surface, so because, yeah, because you have here a covering, so here C, which can be embedded in the Jacobian, will... Wait a second, am I sure that it will be an embedding or no, obviously it will be an embedding, sorry, sorry, sorry, this will be just a birational map, yes, because this will be my rational map to, so let's call it whatever, alpha, alpha of C, because it will live in the 1-2021 polarization, so the genus, the arithmetic genius will be 2022, but the geometric genius will be free, so this will have a lot to do, but the normalization will be C, so you have three dimensional family of those, yes, that's... Can you say why three dimensional, because you have four points of an elliptic curve, which has three moduli, and then you have one moduli for the elliptic curve? Wait a second, no, so it will be four dimensional, wait a second. No, no, it might be that I didn't do the deformation, theory calculation, it's just that the way you write it, it looks like four moduli, and then maybe there is a reason why one of them doesn't happen. No, no, no, no, no, it will be, yeah, it will be four, it will be four, I don't... Yeah, obviously it will be four, obviously it will be four, because yeah, we are in the genus pre-case, so any, okay, so there is one more, one more construction, we can go with like that, okay? So we start with any Arbillian variety in A3, yes, we can assume, so construct A that will be something like E cross X, so this is just one, this is like a surface, yes, so this is like E 2000, not cross, sorry, not cross, I want this to be this plus, so E that will be isogenous to the product, X 1221, so you have a moduli which is one plus three, I thought about the surfaces moduli, but obviously you have any E and any X, so this is four, and then by construction, you know that A is in general the Jacobian of something, yes, and because it contains E, with restricted polarization, you have this map, so again, yeah, obviously four is the correct number, I forgot about this one, so even better, you have six-dimensional family of quartic curves, and it's just to find something of co-dimension too, yes, and actually there are infinitely many of those, so it is dense somehow in the Zarisky topology, whatever, so you should be able very easily to find something, and it's really, unless this number 2021 is small, I don't know how to do, I don't know how to do, yeah, so that's the thing, okay, but we really went for the boundary of the questions of what we know, so let's go back maybe so that, yeah, are there any questions for the first lecture of Angela? I know, yeah, there was one question in the lecture, so I tried to answer with the splitting somehow, so yeah, and there was another question which I haven't heard because it was in the break, I was away, so maybe there are some other questions. Sorry, I have a question. Okay. How can we see that covering you wrote before it exists, this covering, it's easy, it's easy, but for... This is the Riemann's existence theorem in some sense, this is like... Oh, okay. This is the topological thing, yes, you take four points, you take the, you take the, you find the group in S 2021, which is transitive and you know, anthropologically you can construct it, so it's, if you want the reference, I think the Miranda maybe a good, no, sorry, Miranda goes with the P1s, but I think it's Yeah, Miranda, you use the P1, yeah, yeah. But still maybe, yeah, but I think somehow... Yeah, maybe I have to remove four points. Okay. Fundamental proof. Massimilial, okay, so the thing is that, okay, maybe, you know, if it does not exist, you can go to the second construction, which I prefer, yes? You start with any, with your jackal, you know, with any ABN freefolds, wait a second, what I wrote here, IA free, yeah, any ABN freefolds that will be a Jacobian and then, you know, for freefolds, there exists those guys that are isogenous to such things. And this is for dimensions. So those, because this is for, and this is for, it has to exist somehow, at least, you know, finitely many of those should exist. But I think it's just the Riemann's existence theorem with some, you know, possible changes, but I don't, yeah. Thank you. But it was a good question. So I think, yeah, I assume that Riemann's existence would be like enough. Okay, thank you. Okay, any more questions? Yes, I have a question. Okay, okay. Because Angela said that when we defined Jacobian, the map from H1 XZ to H1 XOX, which is by certain reality, same as H1 X Omega X duo, right? She said it's injective. But, well, this map isn't used by long exact sequence, right? From exponential sequence. And it's, I believe it's not injective in general. So it's true that you may embed, well, I mean, this homology is a lattice, but it's embedded as a full lattice and H1 X Omega X, which is C to CG, right? But this map, I believe it's not injective. For instance, I mean, you may have two classes which are homotopic, I mean, homotopic and... Wait a second. You are in the, you are in the homology and... Okay, you want... Wait a second, wait a second. I will just look at the, what do we have? Because what you're asking in some sense, if I understand correctly, is that the construction of a Jacobian, yes? That the Jacobian is the H1 of XZ. I think that's what Angela said is, is that the statement was correct and it is the Abel Jacobi CRM, but I'm not, wait a second. I just want to look at it. Where is the, okay. I will just go with them. Okay, so let me, again, share the, now this one. Okay, so unless I'm wrong, the Angela wanted to tell you this lemma 1.11, 11.1.1, okay? So that H1 is injective, or I'm wrong. And if you want, there is... Yeah, well, okay. Now it looks believable. Abel, can you just make it a little bigger so we can see? I would like to, oh, okay. Now I will be able. So yeah, there is... Thank you. Yeah, because I have the zoom, your pictures and it was, okay, yeah. So it seems that, yeah, it's, you know, this part is injective and it's, you know, it follows from the instances construction of the Jacobian. Yeah, so okay, so, you know, from my point of view, again, somehow the motivation of thinking about Jacobians was to compute those integrals, yes, so the analytic theory, but the Abel Jacobi theorem says that we are, we are in the algebraic setting, namely, that the Jacobian is already, it's also, you can think about it as a divisors, model of equivalence, yes? So this is very nice. So, but this actually, again, gives you the question, if you start with an Abelian variety, can you tell me if it is Jacobian or not? And this is the Schottky problem and it is still unsolved for the general dimension. So it's really hard question, again, as I told you, because this is like, you know, analytic versus algebraic world. Okay, so maybe I can say now a few, oh, if actually, if we are here, I can find the universal property of a Jacobian because we are already in the correct, in the correct chapter of Birka-Hakalanga, for example. Obviously, Birka-Hakalanga is not the, oh, no, this was a year, no, it's not. Yeah, so you can, you know, you can find a lot of things and obviously there are other books that are on Jacobians and curves. So, the universal property of a Jacobian, now you see the statement. So what's the statement? Is that if you have that the Jacobian is, it has universal property, namely if you have any other by rational, rational map from the curve to any other a billion variety, then there is an induced morphisms of a billion varieties up to translation because everything you have to in for, for a billion varieties, you either you think everything up to translations or you fix a point and you take, you put zero to zero. And again, by some lemma, this is more or less the same information. Either you have a homomorphism of a billion groups or you have any morphism and then by translations you get the homomorphism. Yeah, so the universal property of a Jacobian, why it's important? Because again, this is, you can think about the Jacobian or actually about the dual to the Jacobian as an Albanese. Yeah, so the solution to the Albanese problem as a habilionization of a curve somehow. So yeah, so that's the universal property of a Jacobian. Is there anyone who wants to say a few words how to see this, how to see the proof or maybe I will go. Okay, because today it's the first time and probably you haven't seen exercises. So okay, so today I will just talk but starting from tomorrow, I hope that I will find some people who wants to share some solutions to the exercises. Okay, so please, I really encourage you to say a few words about your favorite exercise from this nine and that would be nice. Okay, so in terms of a universal property, what's the proof? The proof is more or less, we take the Abel map from C to J. Okay, so the J is the peak zero and so we have to fix a point C and then we have a map from C to J which is just taking X to X minus C, yes? And this is the peak zero. So what's the, how to, so if you have a map phi, then the map phi twiddle is like, it can be defined in the only, like there is an only way to do it. Namely, you just take the whole points here, P one up to any point in the Jacobian can be written as the sum of P G points minus G, remember G is the genus and the dimension minus the G times C but C goes to zero. So you just, you don't see it. So this map is just phi of P one plus plus plus phi of P G minus G phi of C. Again, if you assume that C goes to zero here, you don't have any, you don't have this part. So you have something like that already. Yeah, so this is like the unique way to write it down. And the only question is if this is, if this is, if this works, if this is correct. And the idea is that firstly by rigidity theorem, if you have a map from C to X, which is rational, then it is everywhere defined. This is a special property of an Abelian varieties. So you have actually not by a rational map but by a morph, you have a morphism. And then again, phi twiddle in general would be a rational map and actually bi-rational because those guys are of the same dimension. But this is also a morphism, yeah? So C to the G and X are by, this is a morphism and here we have a bi-rational maps. And then, so therefore you can find this morphism T and the diagram commutes by the definition of this map. Now, phi will be a homomorphism because it takes zero to zero. And you only need to check that if you change C to some C prime, the map will still be the same, yes? And this is like a computation. So it's really the universal property is really not very hard to show. You need some properties of Abelian varieties and you need to know that C generates Jacobian and this is exactly because you have this Abel map. And I think you will see even in the Angela's lecture that the theta divisor can be seen as the image of WG minus one. This is also the Riemann singularity theorem. And so the universal property of Jacobian is says that what you think it's correct is actually correct. So that's the idea of a universal property of Jacobian. But it is very helpful because now if you have a covering of two maps, you have this norm map in some sense by the universal property of Jacobian. So that's the, this is how you can solve the exercise number what's the number, number three, yes? So the universal property of Jacobian. So obviously if you haven't seen the proof before there is some details to be checked from the Abelian varieties theory, but if you know the technical details then the idea of the statement is not very hard to see and to follow. Okay, so maybe I can just now stop sharing. Yeah, and okay, are there any more questions or maybe someone wants to say something about the exercises? Okay, so if not, maybe I will say like a few more generalities about Jacobians. And Abelian varieties in general for those who don't know much about the story so that you see a bit more. Okay, so as we said already, Abelian variety is C, G divided by some lambda, yes? So in general, in a very general, this torus is not projective. So we assume it is projective, yes, so projective. So in general, in general, the Neural Severi group of A is just Z. So you have up to a constant, you have just one polarization. So you are really rigid, yes? So there is this, so we have this line model L and what you can do, you can just take the cave tensor product of this line model L and obviously you can translate it by points in the Abelian variety. So yes, so this is one thing, which you should somehow know. So moreover, the endomorphism in general, the endomorphism of A is again, just only Z. So you can, what you can do, you can only if you have a point X, yes, you can multiply it's called, let's say N. You can take X in A and you can just multiply it to NX in A. And so in general, the endomorphism is also trivial somehow, so there is, and obviously there is this crucial automorphism namely minus one, yes? You take X to minus X, the involution, the minus one, okay? And in general, A is simple, i.e. does not contain Abelian sub varieties. Abelian sub varieties apart from the trivial one namely zero and the whole thing, yes? And this is obviously, this part obviously is over complex numbers, yes? Because if you think about it as a real torus, then you can always take it as S1 to the G and everything is not true. So we are really over complex numbers, yes, here, yes? So, and similar thing is for Jacobian. So yes, for general Jacobian, so, okay, so in general, so the dimension of the AG is G2 or G plus one choose two, so it grows quite a lot. For general Jacobian, the similar things happens. Yes, for general Jacobian, similar. So the general Jacobian, so the general Jacobian of a general curve is again simple, it doesn't have any endomorphisms and the neurons are very, it's also that. So if you have a covering, it's far from being general, yes? Because as we said, if you have a covering, then this curve is far from being general, especially your Jacobian contains a sub variety, the image of a smaller Jacobian, usually the Jacobian is also bigger. So we are really, you know, we're setting our story so that we can work, yeah? And that's the idea, yes? So then the Jacobian JC tilde, as we said, is the Jacobian of JC. Actually, okay, let's call this map F. Oh yeah, it's F already. As I said, some abelian sub variety and we will call it the prim, yes? So this is just the prim. And obviously we are interested in what can be the spring, yes? So as we said, for double coverings, the prim is of dimension G minus one, which we'll see the results of the Nagis for A5. For A5, yes. So then there is the prim map, which I guess will be defined again in Angela's paper. So if you have a covering, which will be in RG and then again, usually you define, you have some degrees and you have some branching because you can do it for any degree and any branching, and then you define this P of C tilde over C, which will live in some abelian varieties of some dimension and some, so here you have dimension and here you have polarization, yes? In general, for degree two, it will be like G minus one and the principal polarization, but in general, it can be anything more or less, yes? And when the prim map is injective is a good question or generically injective, it is very important to see that classically it's generically injective, but never injective. And as far as I know, actually, we don't know the locus of non-injectivity. So we know that the tetragonal construction gives you the locus where it is not injective, but we don't know if this is all or what are the other examples. So there are some still questions and some research going on when it's not injective, even for the classical case, but for non-classical cases, we know, for example, injectivity as already was mentioned in the Iran's talk. Okay, maybe that's all for today. I guess I should finish now. So again, I'm here for all workshop. If you have any questions, please do not hesitate to ask. And hopefully tomorrow we will go and try to work out some solutions to some of those exercises and maybe we will have some more exercises depending on what happens in the lecture. Okay, thank you very much. Last minute questions. If not, then maybe we have now our lunch break. Thank you very much. And remember that the gather is always open.