 Okay, okay, let's continue. So, yeah, let's, let's, I'm going to keep this part of the proof. You can prove it, use, then use, use, you can do as an exercise use as a property, property of frames. And then you have to compute to compute prove and then you prove that the prove that it has twice the the class of this, what I'm calling this map up C. Well, I think that the class of the push forward or the class of the image of C. Sorry, but it's still this here. It's twice the, the, the minimal class on the plane of the other one, the class G, G minus two over G minus two factorial. Well, it's, this is a general construction. The genus is correct. P, not G minus one G minus the dimension of P. I think it's G minus two. Yeah, it's correct. By the generation, you can use the same, the same, the same method by the generation. Yeah, but I'm working in G will cease. Okay. Let's, let's, let's do this globally. So let's define M six, the diagonal. Coming back to G no six is, is the model space parameters parameterizing the pairs of a curve and one G four. We'll see is in M six. Okay, that's a model space. Now, you forget the G one four. This is in general, is what I made mistake of yesterday. In general, a curbs of general states will have only five or two one fours. And this is so do have degree five here. This is a finite, a finite map. Now, now by base change, you can extend to our six. So you have this R six goes to be six from two eighth. No, it's not big six. Sorry. This is also the forgetful map over M six. You forget the covering. You have M six, the diagonal. So you have an option to find it here. I call it R six. Okay. So that the diagonal construction keeps a correspondence. Okay, I have to tell you what the correspondence is. Of degree 10 10 to two over here. So what is this correspondence is a subspace in the in the in the Cartesian product of our six such that I don't know. I'm excited to give a name is a correspondence. They are pairs. See till the CF and their pairs. Ct prime F prime C prime F prime. In our six. This is elements in our in our six the diagonal, the diagonal, such that they are the diagonal relate. So I can go from one to the other by this procedure. And this does define a correspondence is the degree to two, because when I project project to one factor, the fiber has has degree two. Yes, for one, I have two others that they can construct in this way. And these two to because it's symmetric, you have two projections right that projection one projection two. And this projection is to do what. Okay. And at the level of m six. You have a have also a 1010 correspondence on a call it when I call it the diagonal in our six. And actually you for you. You forget that you want for just just to skip the coverings that I want to relate. And but this is 1010. And this is 1010 because because of the fact I told you about the lines in the cubic surface. Every line in a cubic surface smooth cubic surface intersect at $10. So this is the geometry. So I want to write the theorem to to the nagi. So the correspondence, the diagonal correspondence, the diagonal on the in the inverse image of a of a generic a billion a billion right of dimension five. This is is isomorphic to the incident correspondence of the 27 correspondence on the 27 lines on a non singular or smooth cubic surface. Mm hmm. That means that means the monodromy the monodromy of the map is given by monodromy group of our say our six in a five. So that means the galore group of the galore closure of this map. So essentially where do they leave the, the monodromy of this day, the on the fibers is is the vial group. The six, which is the seed, which corresponds to all the symmetries of the symmetry group of the 27 lines that preserves the incidence yeah symmetry group of the 27 lines. Mm hmm. This is very, very, very concrete and actually the different of the different law size. So we saw the monodromy group can be a little bit smaller. So for instance, that we saw it very clearly over the cubic three falls over the cubic three falls we saw this group. The monodromy group is w a six, but then the Jacobians, there's some the generation of that on the Jacobians. If I. Corvians. The group in the fibers is w d five. So this is the, the symmetry group of the. I should say symmetry group of the incident corresponds. Symmetry. Symmetry group of the 27 lines, but when one line is is is a mark line one is fixed. Mm hmm. Symmetry group of of lines in the cubic with one mark one. With, with, with other words is the stability, the stability of a line in the w six. So it's a subgroup of Ws. It's a little bit smaller. Okay. Okay. And there is another one that I didn't say at all but it's very, very relevant for all their purpose is the intermediate Jacobians of quartic double solids. Corvians. Of quartic double solids. And this is the monodromy group is covered by w a five, the viral group of a five, which is isomorphic to the symmetric group six. So I must say, this is also the symmetric group of 27 elements is inside. Yeah, because you are moving the lines on 27 so is a subgroup of the permutation group that preserves the incidents. And this is six. That corresponds to the group that the symmetric group of lines, lines on a nodal cubic surface. Okay. Okay, now from the oldest law size the only one which is actually a divisor is this one. This is dimension 14. That was 12, and this is the smallest one 10. Notice that by our, our discussion, the, the, the pre map fails to be injected a finite face to be finite over over the Jacobians. Yeah, we have a finite fiber so you have to blow up positive dimensional fibers and over over the other one. Mm hmm. So So it depends on the definition of branch local but where it fails to be fine that is the branch locals if you call it be be branch locals. Locals off of p six one can show that this is irreducible. It contains these two guys. Five, five, sorry, the locals of Jacobians, the cubic three folds and intermediate of those of total solids, they are all there, but this is irreducible and this component you have a component of dimension four that is also reducible. This, these are the same. And these two are the generation of this one. These two. They are the generations, the generation of intermediate. So I just want to finish that. That the branch locals of p six is also known, and it contains all these guys. Okay. So let me see. That finishes my discussion of p six. The original planning was to talk about the other other covering the father's case. And but instead of that. Okay, I can tell you two words about favors case. Fathers case so is triple secret coverings. So there is no reason that why one should restrict ourselves to to double coverings, you can consider triple secret one secret ones and let me let me define some some notation. Let's put our G D is going to be the mobile space of a double coverings of the green T that where C is in MG and is sickly. I also restrict myself to sickly for simplicity. In the non sickly case I have no consider just over two minutes to put this is it has completely different flavor for many reasons. Okay, so there is an interesting case, our case. Our study before three the premap that associated a triple secret covering over a genius for curve. And this goes to be six. I want to tell you know what it is. So you have some coverage. You take the kernel of the normal map, the connected component contain the zero did that defines a billion so bright of the Jacobian city. I call it also clean variety although it's not principally polarized as a polarization, but this polarization is of type. So let me see this is this is of dimension. This is genius for T tilde is 10. And this is dimension six this is why a dimension six. And the polarization is of type 111333. So it's not like no it's not principle. Dimension six. This is the type of polarization. Okay. Now, what, but you don't get all the all the polarize that type know the image of this premap has an extra an extra ingredient, namely the automorphism group. But since it's this guy a cyclic, they come with an automorphism of degree three. And then sigma is also an automorphism of the Jacobian, but, but also as an automorphism of the prem. This is an automorphism. So I did not be six by the set of all a billion varieties of dimension six with this polarization 111333 such that it exists. An automorphism. Precepts preserve the polarization. So when I take the pullback of the polarization, I get algebraic equivalent the same thing. Okay, is what they call the, or fabulously calls the basics. Okay, so just, let me tell you his result. His result is. Well, in this part, why this case, because in this case, both this space and the image have the same dimension as it's natural if you have the same dimension is natural to think of if it is generically finite or not. And if it's generally kind of what is this degree. And if, if you know the degree, it has some nice structures like the diagonal one. Well, it turns out that it. Yes, the answer is just to that. So, I see if you're in my father is that, okay, before three from our three to these basics is generically finite of degree 16. And, yeah, so both, both are of dimensions. Okay, let me see. It has the same dimension as the model space of course of genus four, because it's a finite coverage of them for. So this is nine. Yeah. Yes, dimension nine. Okay, both have dimension nine. And just to tell you briefly. So essentially father follows the steps of the nagging Smith. Sometimes is. And, and so he chooses is a very tricky choice. He chooses a special fiber where to compute the degree. He computes the degree. So the computation goes on the fiber over the following object. So take a generic curve genius tree. And then he defines P as the kernel. So you take the Jacobian. Three times. This is very tricky. Three times the same Jacobian, and you take the sum of the sum map, you can take one of that to some you get here here. This is a nomomorphism of groups. You can consider the kernel, the kernel is naturally a super idea of the triple product of the street, the Jacobian with themselves actually does is isomorphic as variety is isomorphic to the Jacobian twice. Yeah, it's the can. But not as not as. As polarize a million varieties. Yeah, so one has to be careful, because this has a principal polarization, but it's not a polarization that coming from from from the restrictions so that this has a principal polarization. And then you restrict the principal polarization, such that the good part is that the restricted polarization is of the of the required type polarization of the principal one on the product three is is of type 111333. Wonderful. And now, moreover, since you're taking the product of the three objects, you have the symmetric group acting on it. This is cool. You have S3 acting of this permutated and and this, this is restricted to the kernel so something that is in the kernel is permutated is still in the kernel so you have any particular you have an automorphism. An automorphism of order three to take any more than one actually take the cycles of a length three. So permutation, you have an automorphism on P, preserving this polarization of order three. Okay. So he goes to to the same steps that we did it. So, first, he extended to allowable coverings but now I tripled once. So the conditions are a little bit different. But you know the structure of the proof is the same. And what are the local covers in this case that extended to that is this appropriate map. And then you go to the study all the possible, the possible coverings over this, this particular Jacobian. And he found some sort of version of vertigo covers but with three. So elliptic tails that doesn't count. So it's very similar. And in the end the structure is this 16 has an structure as well. Okay. And also, that the structure of the fiber of the fiber is the 16, call it 1666, Kumar, Kumar structure. So as a structure of the 16 lines. So you have a Kumar, so what's the Kumar, so you have taken an abelian surface, basically polarized, you can make the quotient by, by minus so this has singularities on the two torsion points, but then you just the singularized that. Yes, this just to take the the singularization. So over, over these two double points, you get, you get the line, you get so you, you have 16 nodes, 16 nodes, and there were 16 nodes you have 16 lines here. 16 lines living in this desingularized Kumar surface, and this is 16 six, because the incidence of these lines in us, it can be described by Faber puts it very nicely to consider four lines that it is a grill of four horizontal and four vertical lines. So each, each node in the grill is a line. Yeah. That represents a line. Okay. And, and the incidence is given by this, for instance, this line intersects all the lines that the lips in the same column and the same role. So this one intersects all the six lines. These ones is this is this is this. And this is this little description gives you all the incidents of the 60 lines here. So it's also, it's also the petso, the petso of degree four, that's a surface. Okay. That's right. And the automorphism group. Of these lines. This the incidence or the symmetry group of the lines is symmetry group. In this case, of the 1660 structure is as four times as four semi direct product with said to I think this for one is reflected by this for four intersections so you can, you can permutate that one column and one role and the dosen't change the incidence. Okay. Yes. So I am now I want to finish with some open question is, is, is there any questions about this father case. Any, any, anything you want to know. Particular. Just one thing. Yes. upstairs you haven't written that the sigma to the third is identity so maybe. Yes, yes, yes. It has to be of degree three. Yes, this has to be here. Correct. Mm hmm. Thank you. Yes. Okay, now let me show you. Okay. So, when I look at this paper of father, actually, at the beginning of power, try to be a little bit more general. And say, okay, let's consider all the possible. Degrees and and. Yeah, genius and you can have more of these cases where this. You have a you can expect to have something generically fine. So the idea here is that you have to compute this be. Right, because when you consider maps of higher degree acyclic maps of high degree, and you have an automorphism. So the images is always something is a particular image of the billion, the model of a billion varieties because we have an automorphism in this automorphism will preserve the polarization. So you have to compute the dimension of the image to know when the image, the dimension of the image coincides with the dimension of the of the of the coverings. Yeah, so that's just one thing. Let's let's try to be a little bit more general. So, so you consider in general. And as I said, this is the general case that we consider have acyclic one. See, and, and then let's allow ramifications. So the degree of P is T. And let's call two are a ramified or not this is the degree of the of the ramification. So it's a pair. It's a pair number. And because it's acyclic, it will be always totally ramified all the ramifications are totally ramified. Yeah, if he is priming can be separate. Okay, but let's just give it like this. Okay, so what I was telling you, you can consider always BT, the subset of the subspace in the modular space of, I don't know, the ability of varieties of the right dimension with the right polarization delta. So the polarization and the dimension depends on the parameter depends on the degree, the ramification, etc. Okay, we want to see six. This is this this has a cyclic, cyclic automorphism of order D. It will preserve such that it says that is the image. So the morphism of P such that. So, as Pavel says has to be the identity has to be of order T. This is the polarization algebraic equivalent. Okay, so the question is, when is when have the same dimension is the dimension of, ah, now I'm going to introduce the notation. Yes, R, G, R, D. I see plus the dimension of BT. So these are the coverings D to one cyclic ramified degree of the ramification is to R. Okay. So, so long, long and I were looking at the paper of father, father studied only to concentrate himself to the case or without ramifications. And he found some cases found, found, of course, the Nagi case he's on case and he mentioned another one very small and but we thought he has a mistake. And then we wrote another paper about giving this list, and I can give you the answer. Have five minutes. Okay. And I wrote this table because I knew that I want to have time to write it out. So this is what we're. Yeah. So I want to, I want to explain you so put what we did in this course in a more large picture and share with you my, my big question of the of of this year. So, so we did this case today. This is my list. Lang and I will work out that the complete list where these two dimensions coincide. And in those cases, the map is generically fine. We know generically fine. Okay. On those cases, and this is generically fine and always, and we want to compute all the degrees. So the first was don't I just meet the 27, then we'll come forward with the 16. Lang and I will work out the case of sickly covenants of degrees seven of vaginous to core without ramification. And then there is a very nice paper from there is two papers that show the same thing in a very different ways. And then you can imagine that they are very different methods. And they studied and the case of, and you see, double covenants over a genus to curve. So ramification is twice this ramified on four points. Yes, four. And then they computed the degrees three. This is very nice. Actually, I got a random claim that they have a 3123 that I want construction with five points, which is still done. I think it must be true, but I still don't. I wanted to write it down properly. And you have the polarization types. Then there isn't, we didn't talk about the cases where it's generically injective. And then there's this case of my coach in an angle over a genus to elliptic curves, double covenants ramified over six points as to our degree of ramification. And then this is this case is missing 233. So triple C click covenants over a genus to curve from five in three points for the ramification. So essentially you see three, three, three, three to one. So this is C tilde ramified over three points. This is genus to genus seven and the blim has dimension five with this polarization type. I mix up something. Yes, this is not here. Anything. Yes. This one, this is one. Okay. And I wanted to know the degree of here. And we work with. Yes. You have three ones and two trees because the dimension is five. Yeah, too many ones right. Yeah, that's right. Thank you. So the number of here has to coincide with the dimension of me. I hope there is no other mistake. Okay, now what is the right here. So when I, the first time I wrote down this table, I didn't write it in this order. I didn't know, I didn't want to have an order particular order, but then some, some, some person with a good observation, namely, Roy Smith, from the magazine. He told me I look at the table and you notice these numbers here are precisely the number of lines and other pencil surface of this degree. Yes. And if you believe in the conspiracy theories, this number that you're looking for should be six. Because I have an answer, but I don't have a group. So, this is a conjecture. So the conjecture is that this number is six. Based in this observation, but this observation means nothing, because so far, these the pets of surfaces. Well, yeah, they are present in the construction but they are present. In a very mitigating ways in a small fiber all over the place, you don't know exactly. So, my question to you guys, I mean, is, I want to find some common denominator to all these constructions that should be based on this observation of the number of lines to the best of surface. This is something bigger than missing. So so far, our biggest weapon was this tetragonal construction, trigonal constructions that you move around, but we don't have anything else. And, yeah. I think, I think I can prove it to the degree six, but I still don't have a clear idea how is that related to the lines on the pencil surface of, of degree six. So this N. I'm sorry, should be the degree is nine minus N, right. I knew this is nine minus N. And this is the number of points that you blow up in six MP2. For instance, these three is to blow up six points. Yeah, it's how do you construct the cubic surface by blowing up by blowing up a P two on six points. The 16 you blow up a P two in five points and so I think so. Yes, that's correct. This is number of points will blow up. This is N to blow up. Now, for once I finish on time. And do you have any questions. Is it monotony proof in this locus, like the wire group of the associated liar to them. I have that in those cases. Yes, but in this case I don't know I of course, once you have this observation as well I understand again, I obtained it by computation of the degree, but I didn't have this structure. Yes. And I should be there also this question but I don't see it. Yes, it's also a question. It's I don't know. Only these two cases are the well understood in terms of the structure. Fabulous cases also where. Yeah, I said this 166 corner. Yes. I'm also wondering what about the degree one and degree two. Oh, that's also a question why are not in the list. I don't know. There are too many lines I think too many. I think it starts to be very big and also hundreds of them, I don't know. None hundreds but too many. I don't know. Maybe appears in other forms. I don't know. I have no precise answer. Maybe we have to find something which is not cyclic or, you know, or that's right. That's right. That's a good guess. Okay, this list, that list. We restrict ourselves to the cyclic. I have a question when you talk about the cyclic. Do you mean simple cyclic or arbitrary cyclic. So simple cyclic means to say if you wanted to degree n, you take a divisor which is a multiple of n. But if, for instance, if you take P1 to P1, which maps a z to z cubed, this is not simple cyclic because it's ramified over two points, which is not divisible by three, but it's still cyclic, of course. So in this example, is it always simple cyclic? Yeah, simple cyclic. Yes, yes. Yeah, I had also this question. What happened when the points on the branch coincides? Yes. No, what I meant is really that you can have, you know, something which is cyclic, but it's constructed in a more complicated way. In degree two, there is only one possibility. Yes. But as soon as the degree is higher, then, as I said, you have more complicated pictures that can arise. And this is because there is this general classification in the PhD thesis of Rita Pardini of Abelian covers, and these in particular include cyclic covers which are not built in this straightforward way. Okay, no. I just wanted to say that before you go all the way out to non-cyclic, that is an intermediate step. Okay, that is a good point. Thank you. So you mean non-cyclic cyclic, okay. No, no, I'm considering all the simpler parts. Yes, yes. No, no. Yeah, because, yeah, I don't know how will be the computation of the image in that case. The dimension of this PD is based on the, well, has to look at the, how do they, the period matrices of the polar essentials looks like. So you have to, we have to study that. This is a Shimura paper of the types of the period matrix and study the automorphism there and compute the dimensions. And if the, everything is simple, simple cyclic, but the one is not, I have, I don't know, I have to look what, how, yeah, how to check what other cases. So just a method that doesn't apply in the more general case. No, no, no, no, to check. But, yeah. Okay, this is a good suggestion. Can I ask another question? Yes, please. So fully pre-map, we can consider compactifications of this A5. Oh, yeah. Is there any way to extend the pre-map to a larger total space? So consider more singular. I guess you can. I haven't done it. And by simplicity, but I guess you allow all type of coverings of admissible, so admissible is in the sense of, I mean, without this condition of being compact, the kernel of the normal map. Yeah, you will, you end up in the, in the compactification. Now we have to decide waste compactification, the one you want to consider. I don't know what would be the most natural in this way. But I don't know any work on that. So the pre-map extended also farther than this now. No, I don't know. Okay. Okay. Yeah, I think you can get all sorts of things in the image. So yeah, maybe also, maybe also, yeah, of course, dominant in the lower cases. Lower genomes. Okay. Because I'm thinking, for example, cubic three, four cases, we can be considered like a nodal cubic. Yeah. We can still project a lot from a line in the, maybe nodal, maybe a nodal, just the. Yeah. Yeah, and you get a lower vocabulary in that case. Yes. Yes. Let me see if it is that tone. Yeah, it was already considered in the work of Don Aguismith. They did a lot of things there, but that maybe not. No. It's because they started with something very generic. Yeah. So generic already in this loci, but you can go in smaller loci and look at the fibers there. Yes. Okay. Thank you. Yeah. Okay. Well, so I want to finish the course with this. Yes. We can stop the recording now. Yeah. And