 To pa počuje, da se napravila, da se najboj nekaj inčen očel tega modulača, in to je početje, ko je nekaj inčen, če je izgleda, če je to nešto, če je je početje, če je? So, kaj smo zelo izgleda? Tukaj sem tudi bila na mgn. Zato ne bi se vzelo, da je to modulaj spas, očeč genus G kerz, z njim njim vzelo, Markzpojnje. Zato, da je bilo, da je zelo, da je to modulaj spas, zelo, ki je za njim, na vse relacij, kaj je zelo Markzpojnje in genus, ko je zelo. tako, če je to? To je modulaj spas, n plus 1 tapas, c p1 up to pn, where c is an unsingular projective curve of gsg, and p1 up to pn at distinct marked points. So, we will consider it not so much as a modular space, like a collectionisomorphism process, but rather as a stack. So, we will consider the fact that it parameterizes families over a given base, in particular it has a universal family. So, for the kind of invariance I am interested in the fact that this is a stack rather than the scheme is not particularly important. There is something to do with the fact that if you fix a curve and the number of marked points on it, there may be automorphism. And this is the kind of additional structures, which is encoded if you look at the stack rather than the associated course modulaj space, which is just parameterizing the isomorphism classes. The only nice thing of working in this set up is that it ensures that there is an existence of a universal curve. And this gives some kind of canonical way to reconstruct to the point of the modular space, the curve with its marked points, and even to fit it into a family. So, if we fix a base scheme s, can you read here, then the close points of mgn over s are flat families of gsg curves and to be able to give n marked points over each element of the family, we need to have n distributed sections. And I have, of course, to be at least still. So, if there are no marked points, there is no need to find the sections, but this is the way to look at the points of our space. And as I said, if one prefers to look at the scheme, one can look at the associated course modulaj space and modulaj, I mean, it does not fit well in this more formal definition of what the stack was, but modulaj space of isomorphism classes and pointed genus g curves. I guess I forgot to write smooth here. And, of course, then one should do it over s and then it could be curves over s. Well, this is just a scheme of the same dimension. And as I said, if you don't worry about how many marked points you have, you don't have to worry too much about this scheme versus stack business, because if you have sufficiently many marked points, then there are no automovils on your curve, so these two objects are going to become eventually the same. Moreover, this is not a particularly scary space to look with, because it can be actually realized quite nicely as a quotient. Exactly in the way it was originally constructing, using geometric invariant theory, and then realizing that there is a global space, paramethizing pluricanonically embedded curves and then dividing by the appropriated automovils group in the objective space in which the curves are embedded. So, what I was trying to convince you, yes. So, something I forgot to write is, when this guy, M.G.N., one of the advantages of looking at it as a stack, it is a smooth stack, because of the fact that all information of curves with automovils is coming together here, this scheme may well have singularities. But anyway, because of the fact that it comes from this nice smooth stack, this guy just has locally quotient singularities. And then the course module is based. So, this is all a very good theory, but we have already discussed today that, of course, smooth curves can degenerate to singular curves. There is just no way to prevent it, because if you have a curve, there are families that can't be filled by adding a smooth curve in the missing points. So, if one wants to be able to complete families, one needs to allow singular curves. The other problem is, of course, that we have distinct marked points. So, even if the curve does not degenerate, we have to find some kind of replacement for the phenomenon in which two points are coming together. So, if two points are coming together, let's say that p1, we have p1 and p2 in the family approaches p1, so we have a family parameterized by c itself, we need to be able to decide what happens in the case in which p1 and p2 coincide. And we don't want exactly to say, well, but we could as a first approximation say, well, we don't care that the points are distinct, because if we want to have a good modular space, we always want the automorphism group of the objects we are considering to be finite. And this was exactly the condition that was ensuring finateness, but then if the points are coming together, then the fact that we have n points when we actually have less, so this part of the definition is not going to be satisfying anymore. So, we are lucky. For curves, there is a nature that way to find a nice class of singular curves that can be added to compactify our modular space, our modular part. This leads to the linear manifold compactification. I guess it has not been in existence for 50 years yet, but the closure. So perhaps we should track down the first paper and organize a birthday party at some point. And this is the modular space and pointed of genus G. So, what are now the elements of m bar g n s, a family, a flat family of curves with n sections. Well, we have to look at flat families, because of course we want things to be invariant under the formation, so this is our guarantee that there are no strange jumps in the kind of objects we are taking, but this also is giving us an indication of what kind of genus we need to consider, because if you remember, one can define the genus in several ways. There is a geometric genus, which is the genus of the normalization, and this is what you find by looking at global differential forms, generalizing the definition for smooth curves in that way. But anyway, what we need to look at if we want to have a flat family, would be a flat family of arithmetic genus G. And because of the fact that we want the genetic objects to be as easy as possible, we are lacking in this case, we can just, it's enough to assume that we are working with nodal curves, so we simply need to allow the easiest type of singularities. Now the genus we have to consider is the arithmetic genus, which in itself already ensures that the family is flat, and the curves we consider may be reducible, but they have to be so possible. What I want to say, they may possible be reduced, but they anyway have to be connected. Then there is a condition on the section, they still have to be disjoint sections, but what we want them is that they can't hit the singular loci of the curves. So they have to stay inside the singular locus, and then we want to have a stability condition that ensures that the automorphism group of the curve is finite. So when is the automorphism of the group finite? Here is that this kind of condition, which is actually the Euler characteristic of a complex smooth, what I want to say, this is actually the Euler characteristic of a smooth curve of genus G with n points removed and then multiplied by minus 1, but the point is we need to have, this is exactly the condition that ensures the finiteness of the automorphism group, so for each reducible component we have the genus for the component 2 minus 2 plus the number of special points, so we take the preimages of the marked points and the singular points in the normalization of the component, this should be positive. Sorry, this was stupid. Or the equivalent one could take the Euler characteristic of the component, remove from it the set of marked points, not all of them may lie on it, and also all singular, and this should be negative, because if you try to understand why I'm writing this, oh, this is not minus. The reason why I'm writing this, so this is just the same formula we have here multiplied by minus 1, the reason I'm writing this is because it has some more intuitive interpretation if we think over the field of complex numbers, then our curves, then all components, or at least their normalizations, give Riemann surfaces, and the idea is that the automorphism group of the Riemann surface with some marked points is finite if and only if we remove from the surface all marked points, we get a hyperbolic curve, so a curve of a negative genus. The idea is that the compact Riemann surface has an untrivial automorphism group when there is some group acting on it, and this sort of flattens the metric on it, and this cannot happen if the Euler characteristic of the curve is negative. If we look at this formula, just 2g minus 2 plus n is bigger than 0, we see that actually this can become negative, well, the only thing that can be negative is minus 2, so this is only significant, if we have g equal to 1, in which case we need n to be at least 1, or if we have g equal to 0, in which case we need n at least 3, so actually the only additional condition if we assume this to start with is that every time we have a rational component, there are at least 3 special points on it. So each rational component, so each component of a genus zero, and the points should be counted on the normalization, so if we have something like a self-intersection, then it counts as 2 special points. And again, this compactifying space has all good properties that the previous one had, so it should again be reducible, the linear manifold style, but now it's complete. And if we look back at the history of the subject, this is the space that may realize from the construction, from the geometry invariant theory, and this ensures that we can actually define this directly over z, so we can make sense of this over any possible field of ring we may wish to consider. And if we just wish if you don't, for some reason, don't wish to maintain all information about automorphism groups of the elements, so if we are just happy to work with the isomorphism classes of stable clouds parameterizing this way, then we can work directly with the coarse-modulated space, which is of course a scheme. So what are we going to do? Our lectures with these places. Well, the idea is that most of the time we work over the field C of complex numbers, so we are actually working with modular spaces of reman surfaces, or this kind of the generations that are created by patching together pieces of reman surfaces by identifying points and creating the nodal singularities. And we will look at the co-mology of the space. So this is the most flexible but significant invariant that is attached to this. So why is this interesting? Well, for some cases the constructions will be very close for what one can do with our groups. In other cases we will try to discuss classes that have nothing to do with cykers on the modular space. What is the motivation mainly? The first motivation is I guess how the one of the main motivations to study the modular space of curves to understand it's a numerative geometry. So to be able to count how many the generations and on which time they are in each family. And this of course is some kind of a numerative information which can be found by taking the intersection product in the charting but in many cases the relevant information survives to co-mology. So this is sufficiently interesting. And then there are two kind of motivations that are sort of autonomous. So because of the fact of the way in which the geometry of curves work co-mology groups in themselves may have interesting interpretations coming from number theory in spaces of modular forms. So in some cases studying co-mology group of these modular spaces is going to give us some kind of geometric interpretation of spaces of modular forms that are coming from number theory. And you know it doesn't matter whether the input is coming from number theory and we can give some kind of geometric interpretation for it or if the thing is going the other direction we can use geometry to prove something that people in number theory can see for themselves but the point is there is some kind of interaction which can be very useful for both sides. And then I must say I would say most of the application of results about modular spaces of curves so far in the last decades has been to mathematical physics because they are related to the construction of co-mological field so this is like when looking at Gromov written in writing. So as I hinted at before both the modular space of smooth curve and the modular space of stable curves are actually quotients. So the idea is that we don't want to make this so very explicitly but the idea is that anyway globally it has been constructed by saying well there is projective variety that parameterizes all possible embedding bendings of the stable curve in a sufficiently large projective space now I am not trying to add the marked points in this in this story but then of course one can add them as well by taking some kind of incidence correspondence and then reshenting it by the action of the automorphism group of the space in which it thinks we are embedding so the idea is that this is globally a quotient and this means that when we are looking at its homology we can reinterpret it as a equilariant homology of this space and this is the way to define it actually it would even work if we wanted to work with homology with integral coefficients and this kind of construction would work even with integral coefficients the problem is that everything becomes much more delicate and in that case the results just in very small genomes I think nothing further than genomes since we are working on the sea there is even a way to realize n, g and bar globally as a quotient of a smooth projective variety by a finite group so this group business can be done very one can even reduce to the case in which one is working with a finite group so the idea is that to do this one needs to put some extra structure which is called in this case a level structure we perhaps touch on the basic case of this which only works for smooth curves anyway and in previous I think it is too low and on the compactification for n equal to zero and then to the students in the beginning of this century so where is this nice well if we have that our mgn but it actually works with any stack the idea is that I didn't want to write it in too much detail but the idea is that if you want to parameterize curves you want to find some kind of intrinsic way to embed them into a projective space so natural choice is to look at the canonical embedding and then you look again and you say but this is not going to work for hyper elliptic curves because those are the exception so one needs to look at least at the canonical embeddings but actually if one wants to be able to embed all smooth curves without any trouble one needs three canonical embeddings ok, that's good but then one wants to apply geometric invariant theory and make sure that the kind of the degenerations that one takes is independent of how high the embedding is because one says well this is actually a procedure that stabilizes so we need to take a pluric canonical embedding but we want to be able to be free to take a higher power without changing the kind of stable curves because of course there are clear theoretical advantages in doing this also the other point is that if you take canonical embedding which is to otherwise we will not get our stable curves we will get something slightly slightly different than with curves which are not stable but just pre-stable so I think that one needs to take a five canonical embedding at least to make it work so it will not be nice to write the equations for all possible five canonical embeddings of stable curves but it's a problem that one can approach theoretically and look at which kind of stability conditions that are there for the action of the of PGL and this is the kind of embeddings first if one wants to take marked points into the story one needs to add them into the projective varieties one needs to take again an n plus one tamper in which the first part is the embedding of the curve and then one needs to take all the curves and I must say if you are interested in the GIT construction with marked points actually this is very not explicitly done in any of the historical references so one can find much more recent work perhaps 10 to 15 years old can give you reference about this so what I was trying to say is if we are lucky and what we get is simply the quotient of a variety by a group as I said already a story of a sea so if we want to look at the chronology of the quotient with rational coefficients this is actually just the same thing as looking at the chronology of the variety and then take the G invariant part so if one is working with a finite group one does not even need to look for the for the equivalent chronology it's just the invariant part classical chronology this is the reason why the situation is much nicer there but this only works when taking rational coefficients because the map going from here to there has something to do with the fact that you have to take an average I'm afraid it's only over the sea I should have to look it up but I mean it's made using level structures so I do think that they are truly taking structures on the chronology for classical level structures if you want to frame chronology so this is one of the reasons why it's nice to work over Q the other reason is that from the point of view of geometry it's very nice to be able to work with the stack because we can make many more constructions here because of the fact that as I said we have a universal curve over this so it's not just that we can we have some kind of isometric classes below it's just over each point we have some kind of natural way to recover the curve but for many constructions it's also very difficult to keep track exactly that the kind of construction we are using is respecting the stack structure so it's very practical that if we work with rational coefficients we don't need to make a distinction between the chronology of the stack and one of the cross-modulate space why is that? the point is that the information which we are forgetting has something to do with the automotives groups which are finite, they are torsion so they can't produce anything interesting at the level of chronology as soon as we take q coefficients this is the definition of the rational coefficient so I know I am being vague here so actually what I wanted to say here I am just taking the scheme related quotient of a scheme by a group and I am not worrying about the stack structure but the point is that in any case it's not the definition if you want to take a definition you need to work with equivalent chronology or if you want to have charrings with equivalent charrings that's the good way to define the chronology otherwise you could go back there are some alternatives but mainly the best choice is to work directly with equivalent chronology with these kinds of spaces what I wrote here was already thinking I was already thinking with a variety in mind because this would not be the standard notation for the quotient stack quotient stacks are written with square brackets but I don't want to make a point of this because as I said if we are not behaving so nicely with respect to the stack structure actually for what we are doing it will not matter so I try to think about the stack as often as possible because the geometry here is much better but if we replace by mistake our stack with a different one with the same coarse modulate space we are not in time we can do it all the time and if one is topologically oriented our way of constructing mgn is not particularly natural actually there is some kind of topological or analytic construction of mgn so how does one construct mgn when one is not in algebraic geometry depending exactly in the way in fact your modulate space is going to give you some kind of topological stack or just manifold or before perhaps so if you are topologically oriented and you want to parametize all possible compact Riemann surfaces of genus G when you look at them and you say well but actually topologically they are all the same because there is just one there is a morphin's glass of orientable surfaces of genus G compact orientable surfaces so the idea is that we can fix a surface of genus G and then mg parametizes all possible choices of a complex structure on it so for instance we can denote it by sg everyone obtains in this case it's a very large space the so called Taich-Müller space the space of complex structures and it can be realized itself as a topological as a smooth manifold and the important thing is that this is contractible so to take a choice of a complex structure on a smooth surface maybe a complicated business but anyway we can there is a contraction of this guy to a point so from a topological point of view it's not a complicated thing the problem is that of course some complex structures can give isomorphic reman surfaces so we have to divide it for some group this is what is called the mapping class group so the idea is that we have to divide by diffeomorphisms which are orientation preserving so this is just part of the definition the point is of course this would out on tg and the quotient will give the isomorphic classes we want to have but two diffeomorphisms may define the same structure they do this exactly when they lie in the same connected component of the group I've just written so one does not need to take this one needs to take the group of connected component of the group of diffeomorphisms on a fixed reman on a fixed orientation so this means that one needs to divide by the component of this group that contains the identity the idea is that you can deform a diffeomorphism to the identity then it's actually also trivial acting trivial on the trivial on the complex structure so if one takes the stack quotient here one gets again our mg in a different incarnation we want to stress that this is now a discrete group because it's a group of connected it's a group of connected components of a larger group and this and this gamma g is called the mapping class group this is true but let's say as topological stacks or if you want to take the quotient and then you take the course modeling space so if you just look at the isomorphism classes so at the quotient then you get the course modeling space and gamma g is called the mapping class group so the reason why this is done is that if you have fixed your orientable surface then you are free to make many to put a lot of structure on it and to find very explicit parametrizations so actually depending on the problem which one is interested this may be a very a very explicit description it can be useful but it has nothing to do of course with algebraic geometry and of course one can produce a variation of this idea that works with marked points so it simply one needs to add more data here and to be preserved there and even with with boundaries which is something that does not make sense either in algebraic geometry but makes sense in a topological setting so I will just show a picture so we all know what the marked point is it's just a fixed point so that's not the problem but something which one can do if one is working just on surfaces is to make a hole in them so instead of marking a point to cut a smaller circle into it one can use this kind of circle to attach together other reman surfaces or other compact surfaces with the boundary removed so this means we can take another one it works more nicely if we parameterize the boundary so the orientation is giving us in which direction we can move around the removed boundary and then we give an explicit parametrization of it so we can take another one and then sort of attach them together so in this kind of world there are natural ways to transform a surface with boundary of certain genus to a surface with boundary of a boundary with a higher genus if you start with a say a rational curve and we mark a point we can't simply say well now we have a standard way genus one with a mark point but in this kind of world one can say well if we fix the thing we attach to it perhaps we need to and we have boundary here we can always attach it so it's a different way to look at it but it's also show us where this kind of approach is good for studying the dependency on the homology on the genus homologists have been more successful recently in comparison with the algebraic geometry yeah you can I can look up a reference about this for you if you wish but to me it looks very unnatural so to me it looks like they want to have mg and bar so they are sort of artificially gluing things on the boundary to produce exactly mg and bar but still the procedure of the generations is working well because somehow degenerating a curve from a topological point of view how do you create a node on a linear surface the idea is that you fix some kind of loop if you are truly into the business of course you will fix a geodesics not just anything because of course they are hyperbolic surfaces with a flat metric making the kind of things and the idea is that you can fix a parameterization and make this in another way so in this way you get the stable ribbon surface you expect as a limit so somehow if you want to produce mg and bar out of this construction for mg and what you need to do is to parameterize to add to the data the length of the inappropriate geodesics and make it go to zero and this explains to you how to put together the elements in the boundary but anyway when I submitted my proposal for these lectures I thought that I decided on the aim and my aim was to try to illustrate to you as many ways as possible to construct homology classes on these modular spaces to illustrate how we reach the homology theory but also which kind of meaning these classes may have for the geometry of the modular space so of course if you want to answer this question and you have a fixed candidate you may be in good shape as long as you have some nice explicit description of all curves that occur then you can try to parameterize and so on like in the case of m2 where there was just this polynomial of degree 6 and somehow all information on the isomorphism class of the curve was like that but if one wants to be able to construct homology classes in general one needs to have something that works independently of the genus anyway the first one to work on this problem in the 80s in 1982 where it actually published 1983 paper about the homology m2 bar the idea is that if one wants to have a universal construction one looks at the prototypical example of a modular space which is the grasmanian of vector subspaces in a fixed vector space so in considering it as a grasmanian matter of what then let's see k-dimensional subspaces in cn so it can be considered as the modular space of such subspaces and it has again a universal family and since it's parameterizing vector subbanders the universal family is just that universal subbander as for subbanders the idea is that s lies in the trivial vector bundle of rank n over g and it has of course a quotient so s has rank k and the quotient has rank n minus k so then from this construction where first of course we are not worrying too much about what the homology of the cell groups are because g has a nice cellular decomposition as you know of affine spaces you can stratify it by affine spaces so in this case the charting is the same thing as the homology ring it's just generated linearly by the classes of the cells and this leads to the theory of Schubert cycles and so on but in this case what I wanted to say well the generators are just 10 classes of the subbander so the thing to look at is the 10 classes of the universal bundle in this case of course we have no universal bundle in our case and then if one wonders about how to get relations when we can use the sequence to say because we know that the rank of q is n minus k then it's 10 classes up to in degree larger than the dimension have to vanish and actually this is enough also to give all relations so if we look here then we have to take what we have for s and invert because of the sequence so take the sun then we invert and then we will say well if we look at the part of this which has degree l which is at least n minus k plus 1 then this has to vanish and this can be an entity that's giving relations here so this is a completely explicit description of the cohomology of our Grassmanian so this is what we would like to copy the part about creating classes defining them can be copied very easily the business relations is actually much harder and of course there is no guarantee that what we get with this kind of constructions gives us the whole cohomology because m, g and bar is a space with an interesting geometry so we can't startify it by putting together affine spaces so for sure there is no reason why cohomology and Chow agree and no reason why it's generated linearly by classes that are easy to but anyway this was the idea so we look at the universal family n bar but actually the universal family m bar g n is the same thing as taking m bar n plus 1 because the idea is that well if we start with a curve with n mark points if we take a new point so the additional in the universal family we simply need to take an additional point and let it move on the curve so as long as it's distinct from the mark points but we have nothing to have if they hit so let's say that p hits the point n this is actually the image so we keep track of where the point n was and here we take the genus zero component and we put on it the p and n and this is the kind of thing that happens the way we have interpreted the case in which the point that we left moved on the curve hit one of the mark points if the point hits a singular point then we can sort of blow up the node to give us two distinct branches then there will be an exceptional device of the blow up which will be something of genus zero and then we put p inside and if you remember that we already had in this lecture in the previous one when we have three points on the rational curve then they have no moduli so as this is just encoding the same information the same p has moved to the singular point so now if you are looking at the universal family we can create some kind of universal some kind of vector bundles using the data so the idea is that I want to think about there is something which happens on the universal family but we think we can mg n bar g n plus 1 is just some moduli space in which we have at least one mark point so there is something in simply a construction that makes sense at any point as long as we have some mark points and we can say well some way of constructive vector bundle over mg n bar is to take to look at where the mark point is or the point on the universal curve if you prefer and look at this cotangent space to the to the curve we have a smooth curve and we have some mark point pj unit then we can consider the tangent line through this and also take the cotangent space which is be having more positively so that is what we want to take and then if we let the move in the moduli space then this is going to give us a line bundle pj one of the mark points if you think about the universal family we can take the last one for instance and since this is a line bundle on m bar g n we can take its first-gen class and for sure this will give an element of the homology in degree 2 and if you prefer of course this would also work in the first chart group then of course this does not make sense if n is equal to zero but then as I said if n is equal to zero we simply go and look at the universal family and this gives the so-called kappa classes the morita non-fog classes so what is the idea here where we don't know what n is it may be equal to zero but then we look at the universal family we always have a side class there the last one and we can take any power of it and pushes down to m g n bar using the universal family looking at higher powers so we will have algebraic classes of higher degree whereas we already discussed in the previous lectures there is a natural bundle over m g n which actually extends also over m g n bar because it actually depends on the definition of the universal family the Hodge bundle so let's see the idea where this again works the same whether we have marked points or not so point c of the modular space we can take the space of regular differential so this of course works well if we are working over a smooth curve but it actually can be extended in general the idea is that one takes the shift of relative differential of the universal family I am not writing n here because it does not really play a role and this of course lives on the universal family so one takes the one push it down again and now we have rank g vector bundle over m g bar if you wish also over m g bar and one we can take yes the Hodge bundle has this nice natural bundle definition we can find the line bundles but you just find the term in terms of their fibers that is also feeding a little bit yes with the universal families so the idea is that if one truly understands how the thing is defined then there is a natural way to figure out how to give the global definition and personally I am specifically worried about this because you see the dimension of the space of differential drops if there is no component of genus g so the term classes of the Hodge bundle are going to vanish in which there is no on which there is no component of genus g so this makes much less intuitive to think about this bundle over the locus of m g n of curves of m g n bar of curves with only components of smaller genus this in general but of course since this is a rank g bundle one has to stop but if one wants to have something don't give so so how what Maffel thought about these classes is something one should keep in mind when defining them so the geometric intuition is that the lambda classes are natural because they have something to do with the sheaf of differential and this sheaf is geometrically related to the fact that curves have a Jacobian and abelian variety and so how the lambda classes come from the fact that they are the natural kormodric classes on the modular space of abelian varieties so they are there also for the modular space of curves because there is a very precise relation but they are not not necessarily so natural for the geometry of m g n bar on the other hand these kappa classes are new and at least this was what Maffel didn't expect but it turned out to be they should have some kind of specific meaning for the geometry of the modular space of curves and indeed a kappa one is the ample one one is used to prove the projectivity of the modular space so far we have a collection of classes we will actually see that the lambda classes are not independent one could actually forget them from the theory so they are useful because they are geometrically natural and they are an example of classes that can be expressed in terms of the other ones but which other natural classes do we want to consider? well the point is we want to consider the boundary strata classes the classes the components of the locus of curves that have at least a fixed number of nodes can't remember which notation it was but the idea is that one moves at curves that have at least a fixed number of points of singular points the number of singular points on the curve is at least and actually each one so these are of course the reducible components of this guy are of course closed subvarieties of the modular space of curves so one wants to be able to take a natural thing to take the fundamental classes if you remember we stated as a fact that if we impose the condition that there are at least k singular points there is something about the formation theory then the dimension of these locus the co-dimension of these locus of all components of these locus is k so they will give a collection of fundamental classes co-dimension k means in degree 2 k this one can do the same and these are what we call the boundary static classes of course we can deal better with them if we understand better what the combinatorics of the boundary is and usually we do this by using stable graphs so the idea is that each such reducible component corresponds to the topological type of the degeneration so smooth curves are all the same if we are working with the nodal curve then we need then there is some kind of combinatorial information attached to it we want to know for instance what the general of all components are so how many components there are what the general is which points have been identified if one wants to take the map from the normalization to the stable curve every time we have a stable curve of course if we have a constant map points we can extract from it the combinatorial information about this topological type and this is encoded in its dual graph so we try to sketch an example so let's say we have a curve a component which is a curve of genus 3 and then there are two other components that have genus 2 and here we have a component genus 0 where we said that this needs to be stable so we need at least two map points here but perhaps we can have 3 oh yes, from this picture there are no so I will turn this into a component of geometric genus 3 with a node so the genus I write is just the geometric genus of the component it's a genus of the normalization genus labor in this picture so let's say the points 1, 2 and 3 same this genus 0 component and then for instance on this one here we have the label 4 this is stable because I put sufficiently many points here so which is the information we want to keep in mind so we want to attach to each component a vertex of the graph labeled with the genus this will be actually a labeled graph with also some open leaves so here we have 1, 2, 3, 4 components 1 of genus 3 2 of genus 2 and 1 of genus 0 and components of genus 0 are often just written as black dots because of the fact that genus 0 inside without having it looking very strange also good for print so then each node should correspond to an edge of the graph and since a node joins to irreducible component it should be an edge that joins exactly the vertices which are associated to the components for instance so this genus 0 thing just means one of the two genus 2 components let's say it's this one then the genus 2 component which is here means also the genus 3 and the genus 2 component so it goes like that and then let's check this is everything so the other genus 2 component also means the genus 3 1 and then we have one additional node this self intersection of the genus 3 component which is going as an edge starting and arriving at the same component and then we need to keep in mind where the marked points are and the marked points correspond to marked leaves of the graph which start exactly the corresponding vertices so here 1, 2 and 3 genus 0 vertex and then we have an additional one which is on the genus 2 vertex node and this is the dual graph of this curve but this is actually a kind of construction that works in general we start with the curve we can reconstruct the graph but if we have the graph just by peaking the components we can construct the normalization of the of the stable curve so how many edges of leaves are starting so in this way we get the normalization and then the way in which we constructed the edges tells us what we need to identify to construct the stable curve so I guess that this is everything for today I hope it was not too long