 Okay, hopefully the microphone is on now. Like, do you thank the organizers for inviting me to give these talks? So I'll be talking about hodological rings of, well, mainly of MGN bar, modular space at stable curves, but also sometimes MG may be other things. A few words on that in a minute. So the basic objective study here is the MGN bar, lean Mumford-Modsch life space of stable, end-pointed curves of genus G, arithmetic genus G. And there are some words I'm leaving out here that technically one should have like connected curves and I'm always going to be working over the complex numbers to make our lives easier. And the goal of these talks is more or less goal to understand intersection theory of this very fundamental modular space. And that's a pretty large goal and unfortunately we still don't have that great understanding. So I'm just going to say goal, understand intersection theory of MGN bar better. All right, so today I'm mainly going to be defying tautological ring and basically talking about what a tautological class looks like on MGN bar. But before I get started on that, just a couple of notes on the history of the tautological ring. The very abbreviated history begins in the 1980s, Mumford defined tautological ring R star of MG, tautological ring will always be R star. So this is MG, modular space of smooth curves of genus G. It's a context in which Mumford first defined and context in which it was studied for quite a while. And then around 2000, maybe the early 2000 study of this tautological systematic study of the spring for MGN bars sort of began then. The definition was sort of floating around at the time is my understanding, but Faber and Panterapanda wrote down definition of R star of MGN bar. This is the sort of historical progression started with MG then at some point decided to study MGN bar. And more of the recent developments have been about MGN bar. So that's a history and the reason I'm bringing this up is that in this series of lectures I'll be going about things in a slightly a historical point of view. I'm going to start with R star of MGN bar. So I'm going to define tautological ring. Tautological ring for R star of MG will then just be the restriction of R star of MG bar. Then start with basic properties here, then go back and discuss what's known about this classical setting R star of MG and then return to MGN bar. And for a lot of applications, MGN bar is a lot more relevant for a lot of other moduli problems. Girl, wouldn't theory we want to think about moduli space of stable curves. But MG is a simpler space in a lot of ways. And so there's somewhat more known about it and we'll discuss theory of this probably tomorrow. That's going to be the progression here, just a note about that. Okay, so start with a few words about what MGN bar looks like for those who might not be as familiar. So MGN bar, moduli space points, what do points in it parameterize? Points in it look like you have a curve and then you have endpoints x1 through xn in the curve. And here, c should be, again, there are words like connected, that I'm going to omit, also all my curves will be complete. But the important things in terms of singularities, the only singularities, simple nodes, marked points and marked points labeled one through n on the curve should be smooth points of c, so away from the nodes. And what's more, they should be parallelized distinct and then there's a stability condition. The conceptual way of thinking about the stability condition is that you want your curve to have only finitely many automorphisms. The way the marked points come into that is that you need to have your marked points and your nodes sort of pin down the curve and off so only as finitely many automorphisms. It turns out that there is a more explicit combinatorial condition that this turns into. So just write it here, stability condition is that any irreducible component of your curve of c, of venous must have at least three distinguished points. Distinguished points meaning either nodes or marked points here, it's been done a lot of morphisms. There's something you need to worry about a bit here about how do I count self nodes? I count those as two distinguished points to each side of the node. These are the conditions on the stable curve. Maybe I should also write here somewhere that again n here is the number of marked points, g, is the arithmetic genus of c. What did you say? I would need that if I worried about the case of m1, zero bar, then I would have to worry about that. But the point is that maybe I should write the condition here, I'm always going to take two g minus two plus n greater than zero. My curves are also always connected and this means you don't have a problem with elliptic automorphisms like that. Sometimes it's said like that from this sort of combinatorial point of view that I'm going to use in these lectures. I really just have this one condition and I want to avoid the case of genus one with no points. So, condition I have at the top here aside from this one case of m1 bar, this is just rolling out the case of like m02 bar, m01 bar, those cases of these stability conditions can't be met. You don't have enough marked points but you shouldn't have to worry about that always. G and n will be larger, there won't be any issues with that. We'll just have stability condition that rational components must have at least three points pinning down the automorphisms. Okay, so I don't want to say too much about this modular space, I'm going to assume that we've constructed it or someone's constructed it. Basic properties of it get into that in a second but what do these look like? I mean, this is the technical conditions that we have from the topological point of view. Okay, we have some components. They are like Riemann surfaces of some genus. So this is a genus two component. Maybe we glue onto that a sphere. Let's say that the sphere touches itself in one point. So it would be a topological picture of a curve. Let's maybe say we have some marked points. We've point two there and point one here. So this would be a picture of a element of m32 bar. Thing that leaves is simple nodes. It's arithmetic genus three because this is genus two, that's genus one but then the self node basically adds a genus. So that's the topological picture but we'll mainly be interested in thinking about this from more algebraic point of view of two irreducible components. What does a curve look like? It looks like a curve. Say this is genus two. Then this meets the other irreducible component at one point and this component has a self node. Then I have the point two there and the point one somewhere on this curve. Two different ways of talking about the same type of curve. I mean, obviously when I write down this picture I'm not telling you what the complex structure is of this genus two Riemann surface but these two pictures are two ways of talking about the same stable curve. What points in the moduli space look like? Of course, most of the points in the moduli space don't have any nodes at all. MgN is open dense subset of MgN bar and a point in MgN just looks like you have a curve of genus G and you have n points on it. Okay, that's what stable curves look like. We're going to be the main topic of these lectures and just some basic facts which I'm not going to worry about too much but will be relevant. What is MgN bar? Well, technically it's a Deline-Mumford stack. If you want, you can think about this instead as being like a complex orbifold. You really want to lie a little bit. Just think about this as a manifold but it's going to be complete and smooth, right? Smooth, compact and mention 3G minus 3 plus 1. Complex dimension. I'm always going to use complex dimensions because thinking about this algebraically. All right, so MgN bar is this nice space. It's this high dimensional orbifold and the goal is to understand intersection theory of it. Okay, so what do we mean by intersection theory? We mean, well, usually we mean the child ring, A star MgN bar. But in intersection theory it would mean knowing the structure of this child ring. So this is algebraic cycles, modular rational equivalents. If you aren't that familiar with child rings, for the purposes of what I'm going to be talking about this week, it's actually pretty safe to work with comology instead. I take comology, whatever type of comology you want, so a singular comology of that with rational coefficients. Generally, child ring comology are quite far apart. They're quite far apart in this situation too. Generally, you just have a map from one to the other, cycle map, neither injective nor surjective in general. Why am I telling you that it's sort of equivalent to study these things for the purposes of these talks? I still haven't told you what a total logical ring is and in a little bit I'll actually define it, but what it will end up being is a subring. A total logical R star MgN bar is a subring of the child ring and the philosophy here dating back to Mumford is that you should think about this as the subring consisting of the classes that actually show up naturally in what we're doing. I mean, the full child ring or full comology even are gigantic objects with lots of classes in there that we don't know much about, but in practice, what we think of this as R star MgN bar equals classes rising naturally in geometry. And this is not completely true. I mean, it's a subring. There are constructions of non-total logical classes, but some of the constructions are even things that you could see as being fairly natural, but this is mostly true. If you describe some sub-variety of modulae space of curves, it's almost certainly the class of that sub-variety is almost certainly in this sub-ring. That's the idea here. And also define total logical comology. Give the cycle map a name, I don't know, Faye. This is just the map to an algebraic cycle. You associate to it the corresponding comology class. And R H star MgN bar, this is total logical comology is the image of R star, the usual total logical ring under the cycle map. And it's an open question, probably a hard question, whether or not the cycle map is actually induces an isomorphism between total logical comology and total logical ring. This is why I say that for purposes of this talk, think about comology or think about child ring, because as far as we know, the corresponding sub-rings are the same. I mean, they might not be the same. So there are some things that we prove, which technically there are some things that we can prove are true in just in comology, some things that we can prove are true in both child and comology. So there's some difference in what we can prove. But in practice, it seems to be the case that even in situations where we start out with some comological construction, we've pretty much always been able to eventually find some algebraic way of lifting that. Write that down as just open question whether it be from R star MGM bar or H star MGM bar is an isomorphism. And when actually proving things, sometimes you have to think about where am I actually proving things, but from the purpose of intuition about these spaces, you can think about comology if you want, you can think about the child ring if you want. Open question, I don't know if we have great evidence for it, it's making a very strong claim to say this is true for all GNN, but it's certainly true a lot of the time. So I should start moving towards actually defining the sub-ring and not just saying that the class is arising naturally in geometry. So do that, I have to go back and talk about the structure of MGM bar more. So you have these curves, which the sort of generic curve in MGM bar is just some smooth curve genus G with n points marked on it, one through n, distinct points. The general thing, general element has some complicated graph theoretic structure where you're gluing together these curves in various ways, you might have self nodes and so on. One way of formalizing sort of a formal structure for thinking about the serve thing can be useful to talk about some things which are sometimes called tautological maps, I'm just going to call them talk about maps between the MGM bar. First you have the forgetful maps for MGM n plus one bar wrap that to MGM bar by forgetting about one of the marked points. So I'm going to sound simple and know if you just erase a point from here. You have to worry a little bit about what if the curve is no longer stable since you have the stability condition. So one point, then restabilize the curve if necessary. You can think about why the restabilization is well defined. Basically the idea is that remember stability condition is that you can't have a genus zero component with less than three marked points or nodes, distinguished points. So if you forget about one point you might drop from three to two. What you do in that case let's say you had two nodes and then you had a marked point. If you forget about that marked point like point I forget about point I then you need to contract the unstable component. So in this case think about this as being tangent here that that's not really treated to the geometry though it's more true to the picture we shouldn't think of this as just the simple node between them now component here and other component here. Let's say g1, g2, g1, g2 contracted the unstable component. That defines forgetful maps which are nice morphisms between these. In fact you want to really think about these as modular spaces of curves over each point of mgn bar you have a curve the curve c. Pieces together to universal curve of our mgn bar it's going to be isomorphic to mgn plus one bar. This is the map of the universal curve. That's another way of thinking about the forgetful maps if you want. Then the other type of maps I want are gluing maps. So if you have a genus zero component you're talking about the situation where you have a genus zero component in two points here. Then you contract and you move the marked point onto the these are the only two configurations that can happen because you have connected curves. Okay gluing maps which we also have are a little bit easier to define because you don't have to worry about this restabilization. There are two types of gluing maps though. Two and three though they're in some sense the same map. mg1 plus one bar cross m sorry I want some marked points n1 plus one cross mg2 plus one n2 plus one bar this product of modulated spaces so we have two stable curves one of gsg1 plus one then one plus one markings other one gsg2 sorry why am I doing plus one here that's not necessary. mg1 n1 plus one bar cross mg2 n2 plus one bar will map to mg1 plus g2 and one plus n2 what's happening here is that you're choosing one marked point here and you're gluing the curves together. There you have two different curves i and j and if you're gluing together i and j then you got no longer have any marked points i and j they've turned into a node. That's why the total number of marked points is decreased by two. In the final case this was gluing together two points and two different curves second one is mgn plus two bar maps to mg plus one comma n bar so what's happening here is that we have two points on the same curve and glue them together so generically this is a self node of course points i and j might have been on different irreducible components to start with and then you wouldn't have a self node. There are no subtleties of three stabilization here so you can easily see because remember distinguish points you can have at least three distinguish points, distinguish points count nodes and here the marked points are just turning into nodes. These three sort of families of maps were two and three are really the same if we were doing modular spaces of disconnected curves and two and three would actually be the same. Two is just three where the curve happened to be disconnected but I'm always going to have modular space of connected curves. These are sort of the fundamental maps between mgn bar, of course there are other maps that you can define more complicated ways like choosing a way of taking a double cover or something like that but these are the most fundamental maps to define and it turns out that they're all that we need now to define the tautological ranks. So by sections of forgetful map yeah and so those sections will correspond to you replace a marked point with a component like this a rational bubble with two points on it. So that's the same thing as taking a gluing map where you glue on M03 bar. M03 bar I didn't mention this before but by dimension formula it's dimension 0 it's in fact a point so the map does talk about it is precisely a case where you take mgn plus one bar across M03 bar take this top map here. So let's define a tautological ring I don't need this anymore. The idea is just going to be a tautological ring and the full comology or full child rings are really large so we won't take a sub ring we want our sub rings to still have these maps between them rather still respect these maps I mean these maps induce via push forward or pull back maps between child rings or between comology. And we want the tautological rings to still have that structure given by these maps. The actual definition is that the tautological rings r star mgn bar simultaneously defined for all gn satisfying 2g minus 2 plus n greater than 0 which I wrote down before which is what's needed for these moduli spaces to be non-empty so how are they defined? Defined as the ballast sub rings here I mean sub rings with a unit element my rings all have a 0 and a 1 I want them to be sub rings with a 1 of the child rings closed under push forward forgetful and gluing maps okay so this definition the idea of it is okay you have these sub rings you start out just knowing that you have the element 1 everywhere element 1 the child ring just being the class of the entire moduli space as a cycle you start with just those but then they're sub rings so you're allowed to multiply things together if you just have one doesn't help you but you can take one you can push forward along some of these maps you can do this in a few different ways you can multiply things together you can push forward more and so on you take the ring consisting of all the things that you can obtain starting with fundamental classes in these moduli spaces and pushing forward using these maps okay so through some examples of what is in the tautological ring the first thing that we can do is we take so if we want to push forward a map under one of these things one of these morphisms push the push forward of 1 under forgetful map just gives you 0 by dimension reasons it can't give you anything but pushing forward under a gluing map maybe I should clarify that by being closed under push forward in the case of map of type 2 here I mean you take a tautological class here and you push forward their product over here and what you get here should be tautological by the definition start with well let's take the let's actually let iota be the map that we discussed recently mg1 bar cross m03 bar to mg2 bar so let that be the gluing map really one thing the gluing map I mean I need to choose a point there and one of these points here and glue them together and I need to choose labels for the remaining two points between one and two so there's some choices here but it doesn't really matter here it'd be a gluing map find delta 1 2 it'd be yeah there are two maps but for they're different but the push forward will be the same here of one yeah there are lots and lots of maps yes they do in general in this specific case it doesn't matter yes in general really here you have to pick g you have to pick n and then you have to pick ways of indexing the various points involved here pairing up the ones you're gluing together choosing which one you forget and so on you might also worry about how you order the remaining points you can take the induced order that's fine if you want you can use that point those maps instead if you want you can use those maps instead I mean all of these maps are basically the same thing if I wanted to write down technically what I mean here I should when I talk about forgetful map I should say there are forgetful maps Pi 1 through Pi n plus 1 each for getting a point and then I don't need to then using the induced order on the remaining points here then in the case of gluing maps here yes you need to use all of them I'm saying in the specific example I'm using one of these two maps here from MG1 bar cross M03 bar the same map because M03 bar is a point it's the same map so again what is this morphism this is I said this out loud maybe I should actually write this down this map from MG1 bar across M03 bar to MG2 bar we start with a stable curve has one marked point on it somewhere named 1 this is my element of MG1 bar I also need to take a point in M03 bar there's only one point there it looks like a projective line with 3 points on it and then glue together in this map and it ends this pair to something which should now have 2 points what is it doing I'm gluing that point to one of these and the reason why I'm saying it doesn't matter is that the curve that you end up with doesn't matter which one you glue it to doesn't matter how the remaining ones are ordered is that the stable curve you end up with is just the original stable curve but at one you've bubbled off a rational curve here and put the 2 points at the end so you have a pre-muted you have an SN action on the tautological ring given by pre-muting the points SN representation structure is complicated in general but this is a simple example and I want to start by taking the push forward of 1 under this gluing map so what is this this pushing forward 1 via this conclusion to be just the class of the image here so this is the class of the closure of the locus of curves with points 1 and 2 on their own component here closure just meaning you can degenerate this so this is a divisor this is just an example of one class that from the definition has to be in the tautological ring it's a class of this divisor could take other gluing maps get other divisors these are the boundary divisors of the modular space I'm sick with this delta 1,2 for a second so these things are sub-brings so I can certainly square this element and it will also be in tautological ring I want to so far have constructed one interesting element MG2 bar let's now take the let pi it's a pi 2 be the forgetful map from MG2 bar to MG1 bar forget point 2 what I can do is I can take the delta 1,2 class I defined before I can square it and multiply by minus 1 and now I can push forward by pi 2 then tautological rings are small strings closed and they're pushed forward by these maps anything I can obtain by multiplication and pushing forward repeatedly should be in tautological ring check dimensions this will be in R1 of MG1 bar so what is this we've taken some class of this divisor here we've intersected it with itself and then we've done this push forward and if we wanted we could compute what we got here we'd have to use a natural geometry compute the normal bundle to this divisor and compute what the first term class is of that normal bundle maybe I'll just say the answer is this class is more commonly referred to as pi 1 again a class of divisor in MG1 bar this can be computed again I think this is a self-intersection interpreting as a normal bundle then doing this push forward to the push forward basically gets rid of this tail here of points 1 and 2 can be computed to be the first term class of a line bundle L is the line bundle given by the cotangent space to the curve C at the point x1 the one marked point we have this is the first example of a class that we could have defined directly psi 1 is usually defined directly as the first term class of this natural line bundle you can define line bundle you can also think of it as the relative dualizing sheath of MG1 bar over MG think of that as the universal curve it's a very natural object here and this computation which I've just claimed here is why psi is in the tautological range as defined we can keep on going we now have psi that we can work with we're still allowed in constructing tautological classes we can so multiply things together push forward so we let pi MG1 bar to MG bar be the forgetful map only one in this case kappa i is defined as the push forward of psi 1 raised to the i plus first power so psi 1 was this class on MG1 bar could have mentioned 1 raised to the i plus first power then push forward for forgetting about the last point end up with a class in r i of MG bar so you might worry that you can sort of keep going here and get more and more complicated classes as you go using this definition start with just a class 1 we have all these maps you can work with so that we push forward by a gluing map to get this boundary divisor then we push forward multiply things together and push forward repeatedly to get this psi class and this kappa class so you can do this for any non-negative integer i and this will serve a limit to how complicated things can get to make that precise I should have enough time to do before I finish for today the main ingredients for an arbitrary tautological class will be sort of the idea of these three examples you have these gluing maps give you boundary strata have these psi classes you have these kappa classes so first I should say that I just defined psi 1 here in MG1 bar and I defined the kappas on MG bar but you can also can similarly define psi 1 through psi n in R1 of MGN bar in terms of the line bundle that will be the first trend class of the line bundle given by the cotangent space to the ith marked point in general take one of the marked points and you also have classes kappa 0 kappa 1 R star of MGN bar and really kappa i is always an R i of MGN bar these are again given by having one more marked point having the psi class on that additional marked point raising that psi class to a power and then forgetting about it if you wanted to construct those you just follow the same procedure that we did here except start with more points at the beginning have MGN plus 1 bar to start with and map to MGN plus 2 bar here and take the self-intersection push forward back to MGN plus 1 bar by forgetting about a one of the two marked points at the end do the same thing here you can basically carry the other points with you through this computation and you get end different psi classes because again of this issue about how you how you order your endpoints so to actually my goal is going to be to tell you a set of additive generators for the tautological rank so do this I need to use talk a bit more about gluing maps we're only using one very specific gluing map here basically I want to think about compositions of gluing maps and the easiest way to do this is to give a dual graph definition the x1 through xn is a stable curve dual graph is given by okay so you'll say give a name to its dual graph gamma is given by following correspondence vertices of gamma correspond to irreducible components of the curve C edges of gamma correspond to nodes of C each node has two irreducible components on each side so it corresponds to an edge if the self node then it will correspond to a self edge a loop in the graph and finally legs or if you prefer think about them as half edges of gamma correspond to the end marked points the half edge only is attached to one vertex it's attached to the vertex corresponding to irreducible component where the point xi lives so that's what the graph itself looks like it has a bit of extra structure that you want to remember which is that the vertices are labeled by the genus of the corresponding irreducible component geometric genus and the legs are corresponded by the label on the corresponding marked point so also vertices labeled with genus legs labeled by index of marking an example in this situation I had before where we had a genus two component and a genus zero component we had two marked points two one so this is C in this case gamma we'll have two vertices because we have two irreducible components they're labeled by genus two genus zero and there are two nodes here one of them between the two components and one of them a self node then finally the legs we have a leg here marked with a two and a leg here marked with a one for the two marked points so that's the dual graph of the curve that you are this is yet another way of drawing a stable curve the dual graph and this sort of picture convey exactly the same information so then the idea is that dual graphs gamma are in bijection with something a whole generalized gluing iotus of gamma which before the gluing maps which I've now erased they were between a product of spaces gluing together some of the marked points gluing together two of the marked points here going to allow gluing together more than one pair of points at a time going to be from some product mgi ni bar some mgn bar and the point is that the graph precisely tells you the information of which points to glue together and also what the sizes of these components are so in this case the corresponding map iota gamma will be from we have an m22 bar so we have two this vertex has valence two counting the legs cross m04 bar and we are gluing together two pairs of points here the graph tells you exactly which pairs of points and how to label the remaining points the result is an element of m32 bar so these pictures I've been drawing also give sort of the schematic for gluing together a bunch of curves generalized gluing maps these gluing maps you can also think these are just compositions of the basic gluing maps I drew before so in particular zoological rings should be closed and you're pushed forward by these maybe I should also note that if you push forward one under this gluing map then you get the a boundary stratum consisting of take all the curves of the given topological type take that locus, take its closure take the class of that up to a scalar which is the automorphisms of the graph the push forward of one will be equal to the class of that boundary stratum ok so now the theorem of Draper and Pontheraponda is that r star of mgn bar is additively by classes of the form so we pick some stable graph stable graphs for a given genus number of marked points I didn't emphasize this earlier but from the dual graph you can recover the starting g and n and it's just the number of legs g is given by adding together the genus vertices and then adding the cycle number of the graph so I want to pick a gamma for the specific g and n take this corresponding generalized gluing map and push forward a monomial in psi and kappa classes these psi and kappa classes I've explained why classes of this form are in topological ring it turns out that this is as complicated as it gets in some sense any topological class can be written as a sum of these I'm almost out of time for today I should just emphasize these are generators they're not a basis they're not even close to a basis I mean there shouldn't be a basis I haven't even imposed any degree restrictions on the monomial and the psi and kappa classes take psi under its power plus g is pretty large or n is pretty large that's going to be 0 so definitely not not a basis so topological classes can be written in this form there's still the question if you want one way of thinking about the structure of this ring is what are the relations between these additive generators okay tomorrow I'll say a little bit more about this and then back up as I said at the beginning today I wanted to case of our serve mg maybe I can just write down here at the moment that we've defined topological ring of mg and bar r star of mg is just the restriction of r star of mg bar I can define topological rings for any sub space and any sub variety of mg and bar they want just by restriction okay now it does need to satisfy the stability conditions because of what these components end up being I didn't go into this in detail but if you have a graph and you want to read out what the different components will be and what the different factors will be you have to look at each vertex and take its genus which is labeled on it and also the number of half edges incident to it and the stability conditions precisely that you can't have any m zero two bar m zero one bar two bar m zero that's the biggest thing you can do to be the best shake scaling factor at the end so you actually need the same stability condition generalist but did you say yeah so the definition just uses push forward something else I'll say about tomorrow is that grayburn ponder ponder established after Yeah so one question that you might ask is is the tautological ring generated as a ring by say the size, the kappas, and the boundary divisors and the answers no. You need at least a bit more and I think that you sort of need arbitrarily complicated gamma is my belief. Yeah I mean we I mean we know for sure the thing I said that you need more than that whether or not there's some better expression yet we don't know. Yes I will get to both of those things later in this week. Geometric description of which classes. RH-RMG is some subring but yeah I mean as I said psi is given by the Turing class of some line bundle. The kappas don't have a good description as Turing classes have a bundle themselves but they're pushed forward to powers of the psi classes which again are yeah in general if you have a class defined as Turing classes of some naturally defined vector bundle over MGN bar then it's it's probably tautological because these definitions are close to using vector bundles. I don't understand the question do these rings have extra structure or sorry I can't really hear you very well.