 Mešljagovo malo bilo dokonflatoruousje z pleženju, ko je je vse strata vseh, da je vseh nek데čnej vizivnih. Mislim, da je vse 3 pravje dnečnje vizivnih, vem za tatozeneh klasek. In we actually were discussing how to define the mbar gn, so we had the side classes that were coming from the marked points, so if there are no marked points, they don't exist. But then by taking the push forward by a side class on the universal curve one gets the kappa classes. So these are in codimension one, these can have arbitrary codimension. And the lambda classes, that were the chain classes of the hodge boundary. And then we said, well, but these, there are things coming from the interior of mbar gn, so it's something which makes sense in mgn itself. So if we want to have something which is coming from the boundary, well, at least for sure we want to have the boundary starter classes. The fundamental classes of the law side of curve, of stable curves with a fixed topological type of view. Prefer the components of the space, of the closed subspace of mbar gn, in which we prescribe to have at least a certain number of nodes. And the last thing we were discussing was that if we want to identify a boundary starter class, well, we can do it in a unique way by giving the associated dual graph. It's a graph with a lot of extra structure, because if you remember, we had to put labeling on the vertices to keep track of the general of the components. And then, of course, we also had additional structure coming from the mild points, if there are any. Whatever, there are some restrictions on the types of the graph. Of course, it has to be connected, because of the fact that the curve is connected, but there are more restrictions coming from the stability condition. So if we gave an arbitrary graph, it may correspond to a class of curves which are unstable, because they have too many automobiles. So what we get is what we call a stable graph. Formally, we need a set V of vertices, a set H of half edges, because for each edge we need to remember that this starts where it ends. And then, we need a set of leaves or all the legs of the graph. They will give the marked points. And then, of course, as I said, we need to label each graph by its genus, which anyway will be a natural number, and you should give the genus of the normalization of the corresponding component, so geometric genus. There is some kind of attribution map that says, well, for each half edge and for each leaf to which component it is attached. So if we have a node, it's connecting two components, so it will be a self-intersection of one component. So it will give an edge, joining the two corresponding components and an edge is just to identify the half edges. So for each half edge we need to know where it starts and then we will need the information of which half edges we have to join. So each half edge should start somewhere and, of course, also each marked point should lie somewhere. We need attribution map from the set of half edges and leaves to the set of vertices. And then we need to know which half edges we want to join. So this is equivalent to 0.3 involučion on the set of half edges so we know which half edges we have to identify. And I guess, well, we also need so the set L should be canonically identified with the set from 1 to L. So the set of edges is just the set of orbits of this involučion. Once we do this, the graph should be connected and we need to state what the stability condition should be because once we are in this situation what do we need to do? Well, we need to count for each component the number of special points which are on it. So we need for each vertex to count the number of half edges attributed to it together with the number of leaves attributed to that. So for each vertex in V, the number of special points on the corresponding component is something we can denote by in V, so this is the number of things which may be either half edges of leaves, the start of the given point. And this is exactly the thing that played a role in the stability condition. So we just rephrase the condition on the genus and the number of marked points for each component. And then, of course, so it's clear how many marked points there are because that's just the number of legs. Yes. What happens if the graph is just one point so the curve is irreducible? How do you have a fixed point free involučion? I should be the empty set, H. If we have no, no singular points, then H should be empty. This is allowed. And then, of course. So yes, absolutely. So this corresponds to the marked points, so the number of elements of the number of legs should be N, so this may be this is empty with the other marked points. In this guy, of course, to have fixed point free involučion we always have an even number of them but then it may well be empty because it's related to the number of singular points Yes. And then I was saying it's very easy to tell from the graph what the number what the number of marked points is but you may wonder how we can reconstruct the arithmetic genus of the stable curve corresponding from that to that. So the idea is that the genus of gamma is just defined as taking the sum of the genera assigned to the vertices. So if this but then of course also the existence of singular points make the genus higher but actually it's enough to add to this the genus of the graph corresponding to our just of the topological realization of the graph. So in this way we can identify all graphs that are the correct genus. In such a graph we have an automorphism group which is just the elements of automorphism that the bijections of the sets V, H and L defining the graph that respect of the rest of the structure to take automorphism of sets that respect the additional structure. So once we have such a stable graph of genus G with n leveled leaves associated to it the locus of all curves in NGN bar with this dual graph So here the dual graph has to be gamma so for this as the number of singular points is not allowed to increase for this reason this is of course not closed it's a locally closed subset of NGN bar so it's open in it's a risky closure so this is the the closed starting associated to this and this is of course a closed subset of NGN So the question is how can we construct these spaces so this strata once we know what gamma is let's do it first in some examples Yes No it's not obvious I mean one needs to check that this is an open condition in the closure so what needs to do if one knows that these are the irreducible components of the locus of curves with a fixed number with a prescribed number of nodes then of course one knows that this is closed well this is also easy to check but then one needs to check explicitly that this is open but somehow here one can sort of use the fact that MG is open in MG bar and so on so somehow the only thing happening here is that for each component in the dual graph we can only attach a smooth curve so of course there is just one strata, but I mentioned zero this has just one vertex and leaves and this encodes this is the general strata if we take this to be our gamma then we have that M gamma is just MGM the open guide because these are the curves that are exactly this topological type and then of course the closure is going to be MGM inside but what does it mean that we are in code dimension one as I said is something about the deformation theory of curves that if we are in code dimension one then we have one edge so if we have one node and there are two types of nodes the separating nodes and the non-separating nodes so the idea is that a node can be a self intersection of a component and this means that when we resolve that node we get any reducible curves normalization that's the non-separated node and in the other case we get two connected components so that's a separating node so what does a separating and non-separating node mean this kind of characterization only works when we have just one point so it will not work if we have more nodes actually it can be more tricky to find out whether it's truly separating or not separating but we may discuss this later if you are curious but the idea is that a separating node if we blow it up we get something which is connected and of genus g-1 so we just take so this means that we have exactly one vertex because after the normalization we have just a component and the genus has to be g-1 because in fact we solve the node then on the resolution there are the two preimages of the branches of the node which will have to be identified and so this is the graph we consider in this case and then I didn't say anything about the math points but they are not allowed to do anything in this case because there is just one component in which they may lie otherwise if we have a separating node this means that the graph should have two vertices corresponding to the two components and the other priority is just any number between 0 and g and then it depends a little bit how many we need to consider because if we have no math points of course there is a symmetry between i and g-i so we can stop halfway but if we have a decoration we also need to keep track of which points were assigned to one side in which points to the other one so we need to take a partition in the two nodes and this may kill or symmetry in some cases so somehow it's clear that there is just one boundary component in this case but counting the number of components here is more delicate because it depends on the number of math points and the genus and then of course some possibilities are ruled out because of the stability conditions for instance if i is equal to 0 to g we need to check that there are at least two additional math points yes so the idea is that the half edges is telling us what we get in the normalization so in this case we have one edge so the one half edges are starting here the other half edge is starting so if we call this gamma 0 so the question is how do we construct the elements in gamma 0 is that geometrically the idea is that in the curve it has a node if we normalize it then we get a non-singular curve of genus g-1 at least if we start with the element in this open part and gamma 0 and we have two points the ones let's say one level by one half edge and the other level by the other half edge that get identified so this means the open starting so the non-compactified one what we need is to take all information that is coming from this graph is we simply delete the edge and we have a set of leaves 1 to n and the two half edges which is isomorphic to mg-1 n plus 2 here we want somehow to have as labels let's say 1 n and then two identifications for the half edges which are which we identify which in the general situation are not allowed to coincide with the nodes because of the where stable curve is defined is that everything will know because of course after we identify them h and h1 so we have to divide by an evolution this is my notation for the symmetric group this is the evolution interchanging and the idea is that of course geometrically we know that if we identify two points we don't know we can't we don't know anymore which one was the first one which one was the second one but more than that we can also say by in evolution the evolution interchanging h and h prime because this is the automorphism group of the graph so if you look at the graph an evolution has to respect everything but ok there is just one vertex so this is going to be fixed there are n leaves but there are labels so we can't move them because the label keeps them fixed but then we have two half edges and we can switch them part of the structure is the only part of the structure we have to consider is satisfied and they will the evolution interchanging them was part of the data but if you interchange them we get the same evolution so clearly this was the automorphism group of the graph so what this suggests is well if we want to construct a staton then we need to take to isolate what happens at all vertices special points we have on them as a label and then take the product for all components and then divide by the automorphism group of the graph and if you want to take the closure well if something lies here this means that the only thing that can happen is that we have some further the generation of the curve in which we have additional nodes and creating additional components but still there will be all controlled if we take the blow up we still get a stable curve of genus g-1 with n plus 2 marked points over there so we don't know how to describe this but this is controlled by taking the closure by mg n plus 2 in the sense that we can use the identification of points to obtain what we call a gluing map which I will denote with m gamma zero which starts from m bar g n plus 2 and ends in the closure of the staton and these factors through the quotient by the automorphism group of the graph yes yeah thank you let me check if I have the same mistake on the other blackboard yes I see that also this one of course as I pictured it so if you forget the leaves this is the graph and we need to have g-1 to get a graph of genus g because the genus of the anthropological graph is 1 because we have this loop and then we need to add g-1 and we can do something similar for the case of the separating node and then we will need to glue together two curves and which one has some kind of additional mark point on it corresponding to two half edges that we have so we get if we denote the graph we have by m gamma then the two ingredients we need is to have a stable curve of genus i so let's say that we have let's take a simplified situation in which we have the leaves from 1 to m on one component and the leaves from m plus 1 to n on the other one so we have m i m plus 1 times m dot g minus i n minus m plus 1 and this is going to start on the closed one and if we restrict this to the open part here and there we get the open part and this second map is an isomorphism once we quotient by the automorphism group of the graph which I mean depends so to be the automorphism group in this case can be non-trivial only if we have no mark points because otherwise the mark points so I would say the automorphism group is non-trivial only when i and g minus 1 are equal and there are no mark points so this was the explicit construction we had in the case of codimention one but actually the kind of reciproviews can be applied more in general because as I was saying the data encode in the dual graph is exactly the behavior of the normalization of each component of the curve so we take a stable curve of the of the right topological type to be a degeneration of normalization of one of the components in the smooth case and then the identification of the half edges is giving us a recipe to glue together a stable curve of the topological type we need so these are the most important kinds of gluing maps but of course the stable graph will give us a gluing map what does it start as I said we need to pick for each irreducible component if we are taking just a normalization of a typical element we need to take something which is smooth of the genus given by the labeling of the graph with the number of special mark points that correspond to the number of special points but because of the fact that it won't allow degeneration actually we need to take the closure here if we want to dominate the whole structure and in this way we get a subjective map and I said this map factors to the action of the automorphism group which will interchange branches of nodes of the curves and possibly components and if we prefer to work with the starter so the locally closed ones the ones which are disjoint the typical type is unique if you restrict to the case in which we are just speaking non singular components then we get again a subjective map this type to the stratum but now it's exactly the same as quotienting by the automorphism group so I'm calling the M gamma starter and there is this is natural when we are dividing into disjoint locally closed subsets but the concept of stratum also sort of implies that when we take the closure of the stratum we should be able to describe this as the union of other so the boundary of the stratum the complement of M gamma inside M gamma should be a union of the stratum so we may wonder in which cases we get for which other graphs we get something which lies in the closure of M gamma or equivalently we may wonder in which cases M gamma prime bar is contained M gamma prime so let us assume that we have two graphs gamma and gamma prime and the closed stratum associated to gamma is contained in the closure of the stratum associated to gamma prime so when we wonder what this means unless they are equal this means that the number of singular points may only have increased because the way in which we characterize them as components of the locus with a certain number of singular points but exactly how can we recognize which stratum with a higher number of nodes may be contained in this one well the idea is that if one looks at the combinatorial data this means that the graph gamma is substituted at gamma prime we can produce it out of gamma prime by substituting to a vertex a stable graph of the same type and this is phrase by saying that gamma is a specialization obtained by substituting one of the some of the vertices of gamma prime with some dual with some stable graph with the same invariance so how does this work I think that we start with chosen graph gamma prime I've taken three vertices no math points three, two and two the general and here we have two loops this is a possibility so this means we need to pick some of the vertices and make them degenerate somehow so the idea is that here this vertex is giving us genus three with two special points so we may insert for this any other thing of genus three with two with two math points for instance we may substitute it let me think if I take two things points of genus one then one plus one is equal to two and then I join them by two edges so that the genus of them the live graph is one this is something of genus three then we need to put two math points somewhere for instance both of them on this component and I substitute this for that so for the rest I take exactly the same things and I join them using the same recipe this is some kind of the generation of course I did it for one vertex I could have done for more vertices for all vertices then of course gamma and gamma time are equal so this would be trivial so this is how it works but we can also so this is the way to obtain the graph so to obtain gamma from gamma time if we want to go in the other direction what we do is that we need to contact some edges so if we want to go from here to there what we have to do morally is that we take the two additional edges here if we contact them we need to substitute for this so if we choose which edges to contact then we need to identify which vertices were involved so those are going to collapse all together so one plus one makes two and then because of the fact that what we made collapse had was given extra plus one because of its fundamental of the existence of a loop we get something of a genus tree so the idea is that in the other direction we can interpret this as a contraction of two edges and of course if we contact something else we get some other kind of of specialization so as an application let's just classify all possible stable graphs of a given type and let's determine the specialization so we want to do this in case of genus tree and no math points so the idea is that there are things we always have something we always have is the general statum of course and then the dimensional m2 where this is 3 times 2 minus 3 so this is equal to 3 so this means that we have room for up to three nodes as I said it's easy to know what the co-dimension one strata are because there are just two possibilities a no separating node and then the genus of the component goes down by one and then we have a loop and then we may have a separating node but because of the fact that we have no no math points if we take a separating node there is something of genus 0 and genus 2 because we have no math points to stabilize this so this can't happen for stability reasons so the only possibility to have 1 plus 1 and now it's more complicated what happens if we have two nodes so this means that we need to create more gas by specializing some of the vertices so what can we do here for instance we can say well we can specialize this graph of genus 1 this component of genus 1 with two math points with something which is of genus 0 so rational but with a loop and then we kept this so this is a possible co-dimension one strata else we can do this for instance to pop up a new rational component because this is a possible up to here we get a possible curve stable curve of genus 1 with two math points a stable one so now we identify this so in here is a rather difficult to guess what is going on so morally if one wants to do it right technically one should think of all possible degenerations of curves of lower genus classify them and actually as far as I know this is the only these are the only two possibilities and now as I said so they correspond to curves with nodes with with two nodes and then it's clear that this is a specialization of this one because if we contact this loop we get this one if we contact this edge that one so this is a specialization of both here because of the symmetry if we want to know which graph this is a specialization well it doesn't matter if we contact one loop we get something of genus 1 with a single loop so this is only a degeneration of that so if we think about the geometry of strata this means that this stratom is some kind of self intersection of that one and then there are only only two possibilities so we have stable graphs with three edges so in the maximal strata of course all components will have genus 0 because otherwise you would be able to degenerate things further and if we look at all components there will be three valent graphs this has to do with the fact that in this case we are looking at strata of dimension 0 and the only m, g, n which has dimension 0 is m, 0, 3 so here if we contact something we always get that one this is just the degeneration of this one and this is the degeneration so it looks like in disguise we have eight possible stable graphs so let us look at what the strata are and you will say that again I think these are the only two possibilities we can get in disguise there are two ways there are two strata which are easier than the other ones the the co-dimension one strata is easy in some sense but in some sense also the strata corresponds to points is easy because one knows how many components one needs to have and one needs that everything will be three valent so the idea is that one can check that these are the only ones just by taking the two components with three mark points and thinking about all possible ways of joining them by three edges giving something connected in all these by symmetry they are just for two possibilities but of course it is a huge combinatorial problem if you want to do it in general combinatorics of graphs so what are the other strata some of them we know already so this has always an automorphism group so this has a user description and one-two in this case divided by the evolution interchanging these two points in this case is in which also the other device comes with an evolution because in this case of course we can interchange the two components so this is something like the symmetric product of two copies or M11 but then it becomes more complicated because so this means that we have taken a rational curve with four points and we identified them pyro wise so in this case actually I think the automorphism group has order 8 and it's something like a station of s2 times s2 by s2 I'm trying to, I thought it was the same but I didn't so why are there so many automorphism the idea is that we can interchange these two half edges we can interchange any pair of half edges we have identified so this is going in an evolution but also but then of course if we wish we can interchange both half edges at the same time so we can interchange also the two loops so this is what I was saying clearly the automorphism group in which we have just fixing each one of the loops is giving this part of the automorphism is s2 times s2 and then we have an extension by another s2 I think the extension is actually trivial but I haven't checked anyway it's a group so this is actually so and then we can go on and do this kind of things so this is better because here we see there is just a single automorphism in the graph so this is actually m11 times m03 in which we are able to switch two points but m03 was the point so even if we cautioned it and then we have just two complicated guys but yeah we can describe them in terms of of m03 but anyway the strata in this case are closed and they are points and the idea is that one can do this in general but it's not the kind of combinatorics you want to have to do explicitly because it takes a lot of computation to get all possible stable graphs so in theory this works well but it's only useful if you are focusing on a specific problem in which the combinatorics is treatable if you want to get general formulas you can use this construction in a theoretical way but you don't want to use this directly in any explicit way but anyway the fact that we have a a stratification is something that will work well if we are working with some kind of additive invariant something like the older characteristic let's say with compact support of a stratum where this will then the one for the whole modular space will be the sum of the one for the open parts and the open parts have some kind of inductive structure in terms of something which is of smallest genus so what I was saying is the closed guy is more complicated so in some kind of additive invariant we can try to get it from the ones of the strata and the strata can be somehow controlled by taking the open modulate spaces which are, yeah are you forgetting the automorphisms there? this one? yeah well, yes, you're right I'm forgetting the automorphism so our points with automorphism so of course if we just want to know the homology the automorphism will not matter but for some kind of constructions I may be interested in prescribing something about the kind of representation we have if we interchange ideas for instance so it's not the automorphism we will play around and what I was saying, yeah but then we need the open parts so the genus will be at most G but how many mark points will we need at the end on a single component so it won't fit us a little bit and things realistically then one says okay the idea is that every time we lower the genus by one we may have to pay for it by adding two we lower the genus by one this means that we created an edge so we created two special points and we have bad luck and they all end on the same component we increase the total number of mark points on a component by two so anyway the number of mark points is increasing but not in a way we can control so these are the things we may potentially have to glue together of course this is an overkill because something like a single component the genus zero it may happen so I guess that's the most cases we will see so now the idea is that we have three families of natural classes so manfa's intuition was well these are the three kinds of classes we really need to do geometry most natural geometrical loci inside MGN bar will have a fundamental class which is just some kind of linear combination of monomias of products of these classes so where do they live all together so somehow if we are working on the open MGN we can say ok we may want to take the subding of the homology just generated by the side classes the kappa classes and the lambda classes so if we start to think about stable class we need a way to encode also the stable the strata classes into this so it's not so clear what the nature definition would be for the ring that contains everything we want to contain so logical classes we consider before can be viewed as elements of a nice subring either of the chowring or the homology ring and this is what is called the topological there is nothing topological about the ring itself is more something like the fact that the classes are coming from topological bundles over the moderate space when if I looked at the literature for some time people were not quite sure of what the best way was to define the topological rings but at some point Faber and Pandaripanda gave the definition which is now used and the idea is that because of the inductive way of constructing boundary strata by gluing maps the easiest way to define the topological ring this is actually an algebra is to do it for all values of g and n at the sign so if we are working homology we know it by m by g n and this is the smallest sub systems of collection of q sub algebras of the even homology and as I said we want to take the grading that comes from co-dimension so we will get twice the same degree we had before which is closed under push forward under the nature maps which are the gluing maps and the forgetful maps are just those that realize in all possible way n g n plus 1 bar as a universal curve over m g n bar please notice that there are n plus 1 possibilities so even if we fix g n n we are free to forget any point here otherwise we would get a theory which is not really symmetric and then the gluing maps because of the fact that we can always factorize gluing maps to other gluing maps it's enough to do them for divisors so this is the one that is connected that comes from a single separating node we glue together to go across i and g minus i by putting so the sum of our large policies is equal to n plus 2 because we have two additional edges it is two additional half edges the same way you want to create separating nodes we start with curves of g minus 1 with n plus 2 points so the first case gluing something of g minus i with additional mark points separating nodes and the other one is creating something of g minus 2 with two additional points a curve in which the two points are identified that is a very compact definition rather intuitive that in this way we will get the starter classes because we put the gluing maps into the ingredients it is less clear that we are obtaining the psi and the kappa classes so for sure we get the kappa classes if we have the psi classes because they are created by taking a push forward under the forgetful map but do we really get the psi classes in this way so the idea is that we will see later the lambda classes can be obtained from the kappa classes so the lambda classes are actually polynomials in the kappa classes so if we can obtain the kappa classes we obtain that and of course we obtain the kappa classes if we have the psi classes in the push forward so the question is are the psi classes there so what is the take the idea is that the psi classes arise naturally when we have the self intersection when we are considering the self intersection of starter classes so let us start with the curve of g to g with n plus 1 because of the fact so we want n to be positive so this means that we can that if we join a g and we leave all other markpoints on g with the irrational components so this black dot is just the component of genus zero on which we have a special points the last markpoints in the point i and we take it square then the self intersection self intersection so if we intersect the starter with itself there are no other ways to produce so if we want to intersect starter we need to look at common generations but if we think about this this is the only the generation this graph is in common with itself so somehow this means that is one of the cases where we are making an intersection supported on the on the starter itself and actually this morally should say that we have to when we take the square of this morally should say that we have to take a psi class here minus a psi class here so we have to to start with the class on mg n minus 1 here which has a psi class here and we need to decorate here but this is so small so the punch line of this is using formulas for self intersections this is actually related with the psi class which has something to do with this fixed markpoint special point of the component of genus g so if we take push forward to mg n by the forgetful map of this so this is the same pi we have here this actually gives us minus psi i so there is a geometric way to obtain psi i as the push forward on the forgetful map of an intersection of starter classes now the starter classes were there so their intersection have to be that we have a system of sub-algebra oh they are not small loops near vertices let me try to make this picture a little bit bigger so the idea is that if we take the closest atom corresponding let me call gamma this graph just to be this is dominated by taking mg so n because we still have n because we moved out to one of the markpoints n plus 1 so perhaps we have n but we have a special point and then this times m0 3 bar which is not doing anything so if we take this gluing map and gamma then inside here we have the class which is taking the height side class so with the one which is these are the other points and then i is the one that is identified with the additional half edge here so I could call it h if we have n things we can call it psi i so if you want so let me call this h prime so this is the psi class h prime this is just the point so we can forget this and then what I was saying is the excess bundle formula for the self intersection is giving minus psi minus the image of this class as a contribution so what I was saying is if we take the stratum of so take sigma square by the self intersection formula this will be equal to the image I hope I did it right never mind but anyway so geometrically this means that when we push it down we have to get this one and yes, we should also get the contribution of this component because of the fact that this component is zero dimensional space so it's not giving anything sorry? I will explain it later I'm becoming skeptical of the fact that I will do it today but I mean it has something to do with the general topic of finding relations between total logical classes so what I wanted to say is by an excess one word so people didn't realize this directly if you look at all the papers they were asking explicitly if you have a system of survival that contains the psi classes but actually if you have the fundamental classes of stata this is enough so of course this is enough because we are working on the compact space and then I gave this definition for mg and bar but we are actually also very interested in open subset of mg and bar for instance mg and itself so for open subset we simply take the restriction of the total logic so the open subsets we are interested in and of course when we work to work on mg and but there are actually also two other of open subsets which are useful for the theory those are curves of compact types and curves with rational taste so what are curves of compact type? curves of compact type are curves whose only nodes are separating nodes so the graph has to be a tree so if all curves if all nodes are separating nodes the idea is that we have taken away the closed divisor of curves with separating nodes 0 closure of the star I mean I want this to be an open subset so what to do we exactly have to yeah of course I need to move all further the generations but if there is at least one no separating node then the curve then the graph must be a specialization of this one so the curve must line there so if they have nodes so they may be smooth curves the curves are separating nodes so the underlying graph is a tree but it also means that the Jacobian of the curve is compact in the sense that if we define the Jacobian of the curve if we look at the line boundaries of degree 0 on the curve then what we find is the product of the Jacobians of the components there are no loops on the dual graph because the loops in dual graphs is what are contributing no separating nodes and of the curve is compact and it's in fact the product of the Jacobians of the component and curves with rational tastes are curves that are that contain a smooth component of genus G so this is the pre-image pi inverse of Mg under the forgetful mapping which we forget all markpoints so the idea is that we start with MgN by composing forgetful maps we can get to Mg bar and if the curve has a smooth component of genus G then the image that lies here lies in the open stratum Mg so we can take the pre-image of that so the idea of the curve with rational tastes is of course if we have a smooth component of genus G I mean this may will be empty if G is too small because this is only defined so this kind of definition is at most true is at least true so we may have n distinct points here but the curve is also allowed to sprout a rational trees of rational components on which the markpoints lie so somehow this is favorable because it has some kind of vibration structure of Mg because this is some kind of configurational points on a curve of genus G and of course if they have rational tastes then the graph is clearly a tree so it's an example of course of compact type and everything which is just given by a tree of curves will give a curve of compact type the restriction here is that all components but one have to be degree zero mainly the point is the tautological ring was introduced because working directly with the whole of the cohomology for some reasons but also working directly with the whole of the chow groups for some other reasons turned out to be incredibly complicated so the idea is that we want to have some nice subring where we expect to find all information we need and for applications that should grow more written theory certainly tautological classes and also there are several classes of geometrical subrises whose fundamental class lies there so this is nature but still if we want to find something for instance a relation between tautological classes how are we going to do this so the idea is that well this is very complicated combinatorially so first we check it on MgN then we try to say well how does it extend to the part of the boundary of M bar gn in which we have curves with rational taste so we check it there one may want to work here directly in many cases these two spaces are actually equal because of the fact that everything is controlled very well by Mg and then one tries to approximate the most powerful kind of result by allowing more and more types of the generations and I defined everything by taking homology classes because I like homology classes but actually everything we gave can be also defined the same way in the system of subalgebras in the charing so if one takes the same definition but as subalgebras charing of MgN bar what are the different definition which is how small powerful the definition of the tautological sublings which in this case denoted by R and not by IRH of MgN so one would expect that if one works in charing rather than working in homology there is more information there because there is a cycle map that forgets a lot of the structure but actually in all known cases these are isomorphic to the image in homology so it's an open problem but any additional information here which is not present in the image in homology cycle map in homology by definition in tautological class is always an isomorphism there is no reason why it should but there is also no counter example let me recall as theorem is more like proposition but has two parts some of the structure of tautological rings so as I said the definition with the with the system of subalgebra is very elegant, very compact but from that is perhaps not very clear how to generate the tautological ring as a cube vector space so what a system of linear generators would be and the idea is that the tautological ring is generated by what are called decorated strata classes so the images or products of tautological classes under arbitrary gluing maps I should give a simple example so the idea is that we take a stable graph and then we put monomials if this is the point mark point 1, this is going to be we take monomials in the we decorate it by putting kappa in psi classes if you want lambda classes so the kappa classes decorate vertices and the psi classes decorate either half edges or leaves so what is this kind of thing well this uses the fact that if we have the graph gamma given by 1 with the leave 1 and then 2 then there is m up m gamma from m bar 1 2 times m bar 2 1 this component and that component 2 m gamma bar which is contained gm and then here we can take arbitrary tautological classes obtained by product of psi, kappa and lambda classes so in this case we have psi 1 in this case and then here we decorated the only edge by psi so in some sense it is also psi 1 but we also took kappa 1 and we take this image gas let me call it gamma will be the push forward of the tensile product of these classes so the idea is that arbitrary gluing maps are going as ways to put arbitrary tautological classes on each product turn them into something and leaves on m bar g and these are the linear generators and are you suggesting that I leave the second part of this for tomorrow morning let's do it like that let me call it