 So yesterday we start to talk about the boundary of these double coverings and I talk a little bit about what is supposed to be, we will talk about Cartier devices and start to work about coverings, coverings between between stable curves. Yes, so yesterday I had a stable curve and so this is the the compactification by stable curves of actually much more general as 2 g minus 1 I think and c was in g compactifications. Now we have this on ramified so it's going to be on ramified double coverings outside of the nodes it would be ramified in the nodes yeah so double coverings. In such a way in it I need this little computation that you have an involution here that if the involution doesn't exchange the branches the node descends to a node that was the just the c other okay under the and that was the mobile condition so and so we started from that so and we so we we we also define a norm map now or between the the blind bundles in any degree from c tilde to c that leads us also to to consider the norm map of pi at the level of Jacobians when you identify the Jacobian to pick zero this is also well defined. So here we are already in the world of generalize Jacobian and this is this is a lot of work have been done in this sense in generalize Jacobian you can look at the papers of Uchiyaka Coraso and so I it's too much to say now but we talk about how what does it mean to specify a line bundle or any on on a stable curve yes so we have a compact a normalization so you have a normalization of of a stable curve normalization map uh and so you have to to to specify a line bundle is a line bundle on c give a line bundle c the data is equivalent to say what is the pullback of this line bundle on n this is n now it's a smooth curve might be disconnected but we know what is a pullback a line bundle there plus descent data descent how do you write that yeah descent the data sorry the same data so essentially you have to specify how do you how do you glue glue the line bundles further than north or more precisely when a line when a section of the line bundle in on on c this sends to a line bundle this is the pullback of a line bundle from from from the normal from the normalization okay let me do an example and the example I'm going to give you is the example we're going to treat later is let's consider that that has a name so famous covering the Virtinger cover okay the Virtinger cover was discovered by Virtinger and essentially let's take x so I'm doing a little bit general x a curve in mg-1 any curve in mg-1 genus minus one and then you can you you take two copies of this curve x1 x2 copies and then and you specify two points over the curve so first of all you glue you you identify these two points p cool so this is c this is going to be x p and over this you take one copy just color so one this is x1 and you glue it but in an opposite directions here so x1 and x2 they come with also with copies of the points p1 p and q so let's we call it this is p2 p1 but you glue it in the opposite direction so yeah c tilde c tilde is gonna be let me make a space well where I put it yeah c tilde is x1 x2 the union modulo you identify p1 with q2 and p2 with q1 in such a way that you have an involution the involution on c tilde involution on c tilde is is just to exchange the cover so you take is identity is identity but is with it exchange the covers say in sense x1 into x2 yeah so this goes to this but the notes the notes are not fixed the notes are exchanged also because of the way I glue them okay so that is yeah they still satisfies what we're conditions but it has it has more components that we expect let me so what do we assume for instance to give a line model on c we have to give an identification on the note so let's call it s here so you have to identify yeah essentially you have to you have to give a amorphism in the note identifies as you have a line model a peak c so you have to take the pullback in the normalization so this is the peak c tilde the normalization the normalization on dc is the x in cells so then is the curve x in cells it's just yes so you have to you have to give a let's call it let's call it l tilde so you have to give a way of gluing the fiber over the point p and it's almost you know with the fiber over the point yeah that's and that will specify the line model okay so but essentially this is this is just a multiplication by a scar okay but after choosing some trivializations after choosing a trivialization trivialization of this line model l tilde around the notes okay so more generally more generally you have an exact sequence that tells you how to how to how to describe all this line model so you have this is the gluing data there's a multiplication by scalar b times times the Jacobian this is the generalized Jacobian and you can go to the the Jacobian of the normalization of the curve by taking the pullback map okay so what it was saying every element in the generalized Jacobian is is is equivalent to this to data the the the descend data proves what you get after pullback to the normalization but this b is also very very simple calculated this b is the first betting number first betting number oh so you can I'm just making a shortcut but this is a betting number of the graph the dual graph of c so what is the dual graph this is a very very important notion in in can I yes can I yes I'm a bit confused so uh yeah n is the normalization yeah and x okay x x is in my example okay in my particular example I construct c as as the nodal curve where you glued once and the normalization of that is the curve okay yes but it's not only that example yes i'm not okay okay here maybe sure right here more generally I said it no no no I was just confused okay thank you yeah yeah this uh yeah is is this diagram okay until here the example finish here okay so what is the dual graph of the curve so this is a very important notion if especially if you you like to do tropical geometry um yeah uh actually this this issues in the in the boundary has to do with the tropical tropical geometry as well um okay so so the betting number of of a graph in in general any graph is this Euler formula that they say the number of vertices edges minus the number of vertices plus the number of connected components components of the graph okay now let me tell you what is the graph you have a curve like this I just going to give examples I'm not going to give the full definition but you have a curve they have to the nodal curve yes the associated you associated the vertices for each component of c and for this intersection you you associate an edge an edge an edge is in between two two vertices means that these two components intersect in one point so in this case you have a loop here this is the graph okay that means that the curve intersects himself in one point so in this case the betting number is one with this formula so you have one edge minus one vertices plus one component okay and the other curve for instance the the covering okay the graph you have two components and each component intersects the other twice you have two loops like two two two edges and the betting number is you have two edges minus two vertices plus one connected component okay so this is disconnected because you can have disconnected the graphs but this is disconnected and then you have this for instance so it's essentially only this and the betting number here is to be one this is one edge minus two vertices plus one connected component is also zero okay so this is very useful so I can you can always try to do this exact sequence for every stable curve and you know how to describe all the line bundles in this way now why I'm saying that is any question I don't hear please move your microphone yes I'm here do you hear that no this is not me right is he's oh no one cannot hear him okay maybe you can write on the chat and I will look at the chat uh I had to remember I'm sorry okay okay we'll continue and I'll come back to your question okay uh yes uh okay so um so you know you still have work to do because I haven't spoken on the polarization but I'm gonna first of all there's a little demo one can um you have always in the digital coverage if you have a line bundle uh on pick zero pick up c tilde a line bundle such that the norm of the sale is trivial downstairs uh then you can always write it of this form m tensor this is the pullback of the evolution m vertices for some line bundles for some line bundles uh um m can always in peak c tilde but m can be chosen with multiple will of multiple multi-degree and as I didn't mention at all but zero zero zero zero zero zero or one zero zero zero zero zero zero okay so in other words the kernel of the norm map uh has two components also in the generalize Jacobi two components so when you can write it like this or like this so this multi-degree corresponds uh is the degree on each component so every line bundle on the unstable curve you can take the restriction to switch component and it has some degree so you have to work with the multi-degree and uh the dilemma is not so difficult one has to just to work out that is the definition of cartel divisors that you can always um normalize and multiply these bundles in such a way that you can you can concentrate all the degree on one component but I I I will leave it like this just to just the one to define the print variety in this case so p is the our print variety for the generalize Jacobian is well the component containing the zero as usual is such that the degree of m has multi-degree zero zero zero zero zero zero okay and this is a connected algebraic group okay I saw there I saw they connected algebraic group and this is the print variety so generalize print sometimes I don't yeah I don't I'm not sure if I want to use generalize print because I will see all their prints print and on the generalize jaco then we see the question uh oh okay it's good to do the answer uh there is a type of maybe q2 definition let me see yes there's a type of thank you p1 is q2 and p2 is q1 exactly okay so yeah I say that is a print but you still have to check that um why is that what is compact here this is not clear at all um so you need to to show let me at least explain why this will be an abelian variety because it is you have several components why why the why this p has to be compact so let's prove it well let's at least give an idea p is um it's an abelian variety of that ancient arithmetic genus of c-1 oh do you expect it so this this this proof at least works on the on the on the the assumption is asterisk assumption from yesterday so as I told you is that is the assumption that says um the only singularity start to fix the points by the evolution and the evolution doesn't doesn't change the the branches on the node okay so the idea of the proof is that you you write down the the the diagram of the of the norm maps so you have you write the jacobian the cover goes to the normal jacobian of the normalization by the the pullback of the normalization map you have the norm map jacobian this is also a norm this is so well defined and then you have a kernel here so the kernel essentially looks like a c tilde no just yes so uh what are these cameras so the t and t tilde's are the groups of multi-degree 0 0 0 0 0 0 with singular support yeah as is this this data on the on the oops sorry this is a t tilde oops this also you have a restriction of the norm then you consider the kernel so the kernel uh the kernel of these guys looks like uh so so first of all the pullback you you look only only at the at the at the at the at the nodes the pullback and the nodes at the level of the nodes induces uh is an isomorphism between induces isomorphism yeah and also and also i something that the i didn't say but the should be more or less at least geometrically clear is when you take the pullback for instance you take a pullback of the point take the double covering and you make the norm map for this this of the of the of the fiber you get twice the points downstairs that's a very simple observation so at the level of the groups this is multiplication by two when you go up with the pullback and then you you apply the norm map okay so um and since you have these that means that the kernel so that means when do you the norm of restricted to the tilde as it's actually uh is subjective one is an isomorphism and others is subjective okay so do you have we have a zero here we have a zero here too here so um okay and i can tell you continue the kernel so the kernel the kernel of this diagram here is the tilde two i would say this is um b z two goes to r let's see okay this is the diagram we want so what is t two t t two tilde this is the kernel of the norm map restricted to t tilde is is simply the the points of order two of order two on in in this t tilde so you you you think of this t tilde as uh remember you have so you think of this as c tilde b yes and this and this normal map it will be in practice will be the take the power of yeah takes power two of of these elements here so the kernel is is is the points of order two then okay on this side this is a complete variety because this is this is the smooth side that we know this is complete this is a complete two variety what we prefer is a compact in the Jacobian of tilde so this is this is finite yeah and so that means that p is also is also complete this is ideal proof and p is complete compact let's put it here let's compute reduced and connect so actually this this two reflects that we have two components on the kernel of the norm but i want just p one okay so as remark as remark so here is the ideal proof and you have an isogenic there is always an isogenic between p and r so the Jacobian of this side so you have the same dimension it's an isogenic but in general are not an isomorphism with the kernel so the kernel the the cardinality of the kernel of this isogenic is two to t minus one where t is the dimension as the dimension of t so for t for t bigger than two for t bigger than two you have you have a kernel and then there's not this one is just a remark so let's let's come back to my example of the coverings since in this example uh c tilde have p equals two the node has equals p one and and the kernel you have a kernel but this is is finite it's finite kernel okay okay so let's let's talk about the theta divisor so you i will define a theta divisor theta n simply um you pick a line bundle i will say west land of us but um of all the line bundles in the Jacobian such that uh when you translate with this l m you have sections so you are in the in the canonical data device so l is a suitable t minus one so this is multi-degree well this is multi-degree on c and if you choose properly this l for instance you can choose l as the pullback of uh data characteristic l zero if you choose it okay um this theta is a theorem theta induces twice a principal polarization so yeah maybe it doesn't make sense to to speak more about that but uh so um yes so let me see so for a good choice of this l for instance what you can get is that uh as you have this this normalization map so how will translate the normalization map from n to c so here you have a you have a theta oh yeah let's see you have a theta divisor in the Jacobian of the normalization there's something that we we understand so you can choose you can choose uh this data this l uh in a good way such that the pullback this n the pullback the polarization inverse okay pullback pullback is from gc to gn so you have data here you take the universe of this one uh-huh is algebraic equivalent to this t l for a good choice for a good choice of l so for instance a data characteristic of l so for instance take take a such that twice the multi-degree of l degree of l this is multi-degree is the multi-degree of the canonical one so one has to be a little bit careful when one when we work with this generalized Jacobians but the the moral of that story is that everything that happens on the smooth side it can be extended to stable curves the normat the generalized data the generalized Jacobian generalized data generalized data divisors and it makes sense so uh that's that's now i want to just to tell you what an allowable allowable covering is so i'm going to extend a little bit this this definition of of mobile due to yeah to get more more more um coverings on the on the boundary so the partial compactification will be the following we said we says that a pair c tilde with an involution so c tilde in m to g minus one compactification and an involution is allowable if p is an ability well this is not yeah so p p is as as the please as i defined it here this p this p this complete that's the kernel map okay so the prem of this pair okay well that's i don't say much but what is good is that the that this equivalent this definition is equivalent to the following very numeric very precise data is equivalent to the following description of the covering so if so c tilde is allowable again we have the condition asterisk from before if the only fixed points um points of the involution are the nodes okay um where the two branches are not exchanging the two branches are not extended and and the number okay and the number of nodes exchange so you can have a change in note on the on the on the involution equals the number of irreducible components exchange it on the x exchange it on the sorry for my hand okay exchange so typically again the the birthing and cover you have this situation the birthing and cover is an allowable okay this allowable okay because you have two two nodes that they are exchanging and two components that they are exchanging okay so these two components are exchanging and these two nodes are exchanging just one and where this condition comes from well it's essentially uh just to tell you you have to write down a diagram like this yes and you search a way you want to control the b and b prime that appears here so the number of nodes here that appear in this t tilde such that the in such a way that the kernel is is is finite and if the kernel is finite the thing that you get in the middle gonna be complete because you have an isogenic with that this is this is the so there is an numerical way of of of of getting something that is so a description of something is compact okay no i just want to make a remark so boville they call it admissible myself i also like admissible but but but i realized that there is a compactification of double coverings by admissible coverings but is a different compactification so this is something bigger than what i am defining here and here here i'm restricting myself to those coverings that produce something an abelian variety a prime variety which is compact so uh i think this is why donagens meet prefers allowable i don't know now one can get confused but in the context is only one type of coverage that we are looking at and i will define like this rg i'm going to define this compactification are jiva are they allowable coverage i cannot even pronounce it properly allowable cover okay and is inside and another compactification yeah and then i will say compactification by admissible coverings which essentially you only you only declare you have notes goes to notes and well it's a little bit it works for other other other degrees not only for double coverings but and and you want to say how they um yeah what happened out of the notes so what they're what they're the the parameter that i had exchanged when when you you blew out the notes um okay so here's the theorem in which i'm not going to say more than just so stated that this is the good the good the compactifications for the pre-map so the pre-map extends pg extends to a proper map so pg bar from rg into a this is not embedding ag minus so we didn't need to you don't need to compactify on ag minus one it's just the compactification on the side of the coverings and then then you can start to compute the degree now you have a proper map so in the degree one has to take into account um allowable coverings it could be you could have we will see we'll have coverings that are in the boundary that gives you um principally polarized a billion varieties okay um so let's let's concentrate ourselves in the genome six case um so what do you do we we consider i don't know many things are are more general but for for the purpose of the for the course let's concentrate genome six and uh we are take um consider a Jacobian in the Jacobian locus of a six a five this is Jacobian locus so essentially um actually is i can take the closure of the image of the trailing map inside of a five so and let's say that this is a generic Jacobian generic Jacobian in the sense that the curve of genus five here is also generic okay so we will see what would you need about this generic yeah um so for for a smooth smooth smooth curve uh smooth Jacobian um Mountford gave a list of all the possible covers that gives you a Jacobian okay um and Mountford uh Mountford gave uh this is what Mountford then uh well this was for the smooth covers then come the negus meet and extended this list to also the allowable covers because there are also allowable covers that give you Jacobians and there are four um so for the computation of the degree are four for relevant cases so this is what the allowable covers uh in our six give that is the prem let's see p6 of this covering is the Jacobian okay okay so actually the list is longer but many of these low size uh get contracted in something uh um a smaller dimension than g5 so g5 it has so we want something that can can this has dimension the dimension of m5 that is 12 so if there if you have a low side that they have dimension less than 12 is not considered because they are going to map up to something that is not generic on on j5 on the Jacobi locals so let me tell you what are the the relevant low side um so one um right this is not the corner I think um take a smooth six stick which is actually plain quintic plain quintic and and a double covering on this plain quintic but we are going to call both an even an even cover even cover even double cover so what is an even double cover so we have two types of coverings of the plain quintics depending on the parity of the following dimension so um so the plain quintic comes with the so you have a map the c is embedded in in in a plane and take l the push forward of a line oops one okay or yeah the push forward so this is a degree five line bundle on c yes uh so degree five means g in of degree g minus one so in general you tensor l tensor eta we can have okay and might not have sections so when this is um equivalent or congruent to zero model two we say that this is an even so uh so what is eta eta is the two torsion point that defines discovery so this is an so we say that eta is even it's an even cover and when this is covering to one we say that these are not covered that's the definition I'm gonna use it later it's very important distinction uh among the plain quintics so how many plain quintics do you have does no anybody know the answer I mean how many parameters you need to this to describe uh plain quintics I just I just ask in the audience have you seen that I can I would can actually the answer is uh is how it has the right dimension model count I mean how many parameters do I need to discover to describe uh curving p2 so you have to how many parameters you need for uh writing a polynomial degree five in three variables modulo and modulo the actions so the automorphism of p2 okay you that's right thank you someone computed exactly model account is 12 uh is 12 so is is essentially is as I mentioned is also the dimension of the our jacobi locus so this is that is a relevant loci and then we have another consider a trigonal curve c trigonal curve p1 three to one so again the model count is also so it depends it depends only on the branch points and you have a degree of the ramification of the branch locus of each is essentially 12 also and okay and over the trigonal curves that's all you can you consider every double cover is on a trigonal curve double covering rap on ramified so we're still in the smooth in the smooth locus double on ramified that was already known by by month four and actually it works for in any in any genes it's more general than the genus six and then on the boundary and we have the vertingen covers so those ones I'm not running right this is just I want to use it a lot vertingen covers and that's actually it's very simple to see that that the the when you look at the at the normalization of this you have two copies two copies of x and goes to x so the nor the norm map is the kernel of the norm map well well this is the normalization sorry but the kernel of the norm map is just to the it's isomorph to to the jackhole this is the print okay so it's kind of easy to see of course yeah yes remember this is betting number one this is also betting number one so the the the the the the kernel of the on the on the of the t of the t t tilde and the size is going to be finite so the print is isomorphic to the to the jackhole okay and the last one in the list in the boundary so called elliptic tails that also has the right moduli so actually here the moduli count moduli count is the dimension of m5 plus the choice of the two points where you glue so you have dimension two so is 12 plus two is 40 so it's even bigger than the than the just locals so you can actually we all gonna have here uh fibers of positive dimension the elliptic tails uh oh it should be about i just to give you elliptic tails and then we do the the break um they look like this so you have copies of of of a genius five curve so copies of the same x x same copies two to one now um and then on this side you glue with an elliptic curve at this point p you can always choose the zero to glue so two translations and then you consider double covering e tilde here any double covering of the which is also an elliptic curve so here it's also clear that the the the frame of this double covering is going to be isomorphic to the jackhole because uh on the side of the elliptic curves you you don't have contribution of for for the norm happens in zero okay in the moduli count is count is 14 as well so it's the dimension of m5 plus the dimension of the moduli space of elliptic curves that is also one as one and plus one of the choice of gluing choice of p so this is 14 so this is does tell you that these elliptic tails is a it's a divisor on divisor on r6 okay so before yeah i'm late for the break but just to give you the answers so the computation degree tells you that um the plane quintics uh contribute to the degree in one above uh the jacobian will have 10 trigonal curves double covers over trigonal curves above the the verting covers without 16 and the elliptic tails don't contribute to the degree so the sum is 27 so uh this is yeah i'll that's the next step trigonal curves dimension 13 uh i will i will do the count again and and we come back after the the break okay 13 yeah that's a good moment for a break computing come back you later