 Okay, so yesterday I stated Fauber's conjectures, sort of the first conjectural description of the full structure of tall logical ring Femjee. I said the status is that the first two parts about the behavior and high comological degree are proven, the vanishing and the one dimensionality with the explicit proportionality is in degree G minus two. And then this last part is, well, it's true for G less than 24, but for G greater than 24, the greater than or equal to 24, it's still open. So today I'll be sort of discussing some of the reasons why this last conjecture is, I mean, it's still open, but it's getting less believed in. So I want to start by first mentioning that there are versions of Fauber's conjectures for other modulite spaces of curves, where these versions were stated by Fauber and Pantarapanda several years after this. Well, which other modulite spaces do we want? So I'll have to define a couple of those. So MGN we know is modulite of smooth curves. So far we've mainly been talking about MGN smooth curves when we have no marked points, but we could have marked points also. The other end of the spectrum we have MGN bar said a lot about modulite stable curves. In between there are two important intermediate modulite spaces. There's MGN rational tails and MGN compact type. So there are inclusions in all of those. So what are these intermediate spaces? So again, this is RT is rational tails, BT is compact type. And it's going to be easiest for us to define these by posing constraints on the dual graph of the curve. So remember with smooth curves the dual graph is very simple. You have a single vertex, you have no edges because I just correspond to nodes. Whereas with MGN bar, the dual graph can be arbitrary stable graphs. For these intermediate ones, the rational tails start with that. That's saying that it's stable curves whose dual graph contains a vertex of genus G. So this implies that all the other vertices have genus zero. They're rational curves. So the way to think about this is that you've one main component which has the full genus and then when marked points come together, remember marked points are not allowed to stack on top of each other normally, but if they come together they bubble off into these rational tails. And you can get these sort of trees of rational tails, trees of rational curves sprouting from the main component. And this is often more convenient space to think about than MGN itself because MGN rational tails is proper over MG whereas MGN is not. Of course, if N is zero then MG0 rational tails same thing as MG. So this really generalizes the case MG that we talked about before. Compact type, stable curves, dual graph is a trait. So this contains the rational tail space because the rational tail space always has to be a tree kind of loops because you've one vertex which is all the genus on it. So these chain of four moduli spaces is sort of the main things people talk about. You can define other things of course, but. And the versions that Fopper and Ponteroponda stated were taking Fopper's conjectures and replacing MG with MGN rational tails, MGN compact type, or MGN bar. I should maybe mention that the reason why this is called compact type is because this condition, so it's saying the dual graph is a tree. That's equivalent to saying that every node in your stable curve is separating. If you remove it, the curve is disconnected. And that you can see is the same thing as saying that the Jacobian of the curve is compact. That's the compactness, compactness of the Jacobian. Okay, so how do these versions work? Dang, by any condition on the graph, these are just the most natural geometrically in some sense. I mean, people sometimes talk about, for instance, MGN irreducible where your graph has a single vertex but can have lots of loops. That's something to talk about. But there isn't a version of Fopper's conjectures there. There's also a reason why this chain of, so I mentioned briefly that the explicit formula, which I haven't bothered to write out again today for this isomorphism to Q, that explicit formula really comes from the Groveland theory of P2. And in the same way, in compact type, corresponding theory is related to Groveland theory of P1 and MGN bar, it's related to Groveland theory of a point. So there's some sense in which this chain, rational tails, compact type, MGN bars is a, it's really a natural progression that shows up in the applications to Groveland theory. Okay, so what are these versions of Fopper's conjectures? So MGN rational tails want to replace G minus two by G minus two plus N. So this G minus two here, the degree of the top dimensional non-vanishing piece of the top logical ring, we call that the so-called degree because as a Guernsey ring, this sub-dimensional piece would be the so-called of the ring. From G and rational tails to modify Fopper's conjectures, you just want to shift that up by the number of marked points. And again, MG zero rational tails is just MG. So this is sort of the simplest generalization. You have to modify the explicit formula, of course, also. You have more classes, you don't just have polynomials in the Kappa classes, but the conjecture is still, and I should say for rational tails, it's the status is very similar to MG, it's still open. Is that you have this Poincare duality type ring with top degree piece and degree G minus two plus N. The explicit formula is a different one. It's quite similar in this case for rational tails, but I mean, there are also more classes than just the Kappa classes. So the top logical ring and all of these, I said I'm always defining it by restriction from MG and BAR. And MG and BAR, we have these basic classes corresponding to every graph that appears and you push forward some cap and psi classes. So in the MG and rational tails, you actually have non-trivial graphs. You also have some psi classes. So you have more topological classes than just the Kappa classes anyway. So the formula has to be different because you have different inputs. I'm not going to write out the explicit formulas in all of these cases because of that. They get more complicated, especially in the last case because you just have more topological classes to pair. But the basic idea that this should be a Gorenstein ring is going to be the same in all of these. MG and compact type. Replace G minus two by two G minus three plus N. And then we have our MG and BAR replaced by three G minus three plus N. And yes, we have to plus change explicit proportionalities part two. The new explicit formulas, I mean they're actually in rational tails and compact type if you just pair together Kappa classes, for instance, they're very similar to the MG formula. For MG and BAR, it gets more complicated. I mean the question of what does for MG and BAR, this is actually the top degree pairing you're integrating classes on the full MG and BAR for the six proportionalities for the third version of the conjecture. And that's given by Winn's conjecture, and proved by Konsevich and others. So that's sort of a much more complicated combinatorial formula of recursion. You actually need a recursion for, I guess, for MG and BAR. But the basic structure of these conjectures that you have a top degree piece is one dimensional, top degree happens at a specific location and that you should have the sponcary duality. Those are the features which are the same between the versions of these conjectures. I think that they originally stated as speculations rather than conjectures. Yeah, so the last one is interesting because if you took the full co-homology, it would be trivially true. Yeah, so I'll get to that in a second. But yeah, the last one is in some ways more plausible because I mean it's not an obvious fact that the top degree piece of the topological ring should be one dimensional and top degree piece of the child ring is certainly not one dimensional in all cases for MG and BAR. Top degree piece of zero dimensional cycles when our points equivalent in the, as algebraic cycles much lower rational equivalents on MG and BAR, you expect the child ring is going to be enormous. So it's a non-trivial theorem that I should say, parts one and two, the fact that you have this one dimensional circle at the top of each of these rings and those degrees have been proven, but they're non-trivial even in this case of MG and BAR because we're dealing with child ring rather than co-homology. So if you want, you can view the one dimensionality in this top degree equal to the dimension of the modular space as evidence for what I was saying earlier that topological ring really looks like it's the same in child and co-homology. There's no obvious reason why they should be in this top degree piece because again, top degree child of MG and BAR is going to be gigantic. Top degree co-homology is one dimensional. The topological ring really picks out one dimensional piece of the child ring. That was, I think first proven by Grayburn Fakiel. Okay, so this looks great. There are these other versed in the conjecture just changing some numbers around for these other naturally defined modular spaces. The explicit formulas for top degree proportionalities come from Groven Dere in a very natural way. There's a problem with these versions which was discovered in the last couple years which is Peterson and Tomasi. The R star of M220 BAR and R star of M280 compact type are not Gorenstein. Gorenstein conjectures are false in all cases other than possibly MGN rational tales. Which again, MGN rational tales is the natural generalization of MG where you add in marked points. So these results are pretty recent. For really, I'd say half of the reason to doubt Vauper's conjectures for MG since, I mean the Gorenstein property, it's true in small cases just as it was for MG. Like for MG, Vauper is able to verify it for G up through 23. And then after that, we don't know. In the case of like M2N BAR, I believe they also proved that it is Gorenstein for M2N BAR for N less than 20. So this is really the first time. So it takes quite a while. You might also notice 20 is remarkably close to 24. There could be some relationship there and this isn't known. I should say these Gorenstein conjectures, they're analogous statements but there's no logical implication between any of them. Between like MGN BAR, MG. There's no easy way to get from one to the other. So these counter examples found by Peterson and Tomasi, they don't imply anything about MG and rational tales but they are suggestive that the Gorenstein property was some accident that happened in small cases. I speculate that like there are some actual geometric reasons for the one dimensionality at the top and for this intersection formula there. And the fact that you have that pairing is sort of making it somehow plausible for the ring to be Gorenstein. But once the ring gets large enough to be complicated then in these cases, MGN BAR, MGN compact type, it apparently stops. The first N for genus two, yeah. It's not known for, just for two, yeah. I mean there are some cases in higher genus where we've competed by constructing relations and checking that the Gorenstein that it's true. For instance, I think for M3 bar, maybe M4 bar, maybe even with a point or two in there, we can compute that those are Gorenstein. But they were using methods very specific to low genus. Basically they were using, they made a very detailed study of the spectral sequence for computing the co-emology of these modular spaces, viewing them as vibrations over M2 bar. M2 bar is a relatively low dimensional thing. They had some hope of understanding the spectral sequence. But it's much harder when you move even to genus three. We all just stayed out loud because it's certainly not, so but there's some speculation that some reason to believe that M3 eight rational tails might be non-Gorenstein also. But proving that seems out of reach at the moment makes genus three so much harder than genus two. I should mention that on the other side of things we do have topical proved that R star of M2N rational tails is Gorenstein for every ad. Yeah, so if R H is not Gorenstein, and that implies that R is not Gorenstein as well, it's a stronger statement slightly. Yeah, and they were working like I said, with these co-emological tools. So only able to prove things about co-emology, but it's actually stronger because things on Gorenstein is saying it's missing some relations. If it's missing those relations in co-emology, it's also missing them in child. Yeah, there's some experimental evidence and also some conceptual evidence that M3 eight rational tails might not be Gorenstein. But that's, I've talked about it with Peterson a bit about extending his methods there and I think he's working on it, but it seems harder. There's some conceptual reason to believe there should be a problem there. Okay, so this is the first main piece of evidence against Fauber's conjectures I want to state today, that there are these completely analogous conjectures for these other moduloid spaces. They're true to start with in small cases, just as they were for MG, and then they stopped being true at some point, at least in genus two. I should say that there are some means for moving these around. Like, I think if it's false for, if it's not Gorenstein for M220, but it's also not Gorenstein for M2N bar for any N greater than 20 or something like that. Or at least for all sufficiently large N. It's not going to be Gorenstein. That's just the first counter example, but they actually have infinitely many in genus two. I don't know if you can move the counter examples to higher genus, but okay. So the first seven things don't work as well for these other moduloid spaces. We know for a fact that they're not Gorenstein. The other evidence I won't say I hinted at before, which is that Fobber's original means of producing relations he used to prove for G less than 24, suddenly started failing for G equals 24. So maybe I'll do a little survey of just the different geometric methods have been used to prove relations between the Kappa classes just on MG. So a lot of these methods also produce relations in some of the larger moduloid spaces. Methods, constructing relations in R star MG. Lot of complete lists, these are some of the methods which have been studied a lot either by lots of computer search as in the case of Fobber's method. So Fobber's method used certain classical apps between vector bundles, CGD equals these fiber, the universal curve CG, it's MG1. So CGD means that you take a curve, you take D points on it, but as opposed to MGD, the points are allowed to stack on top of each other. You're just seeing the fiber product with lots of copies of the universal curve and define certain vector bundles over that by taking having the fibers be sections of certain sheaves on your curve and very classical constructions. This is Fobber's construction, I'm not going to say that much more about it right now. And that's actually the one method I'm listening here that we only have computer evidence here for what these relations span. First, the second, the geometric source, just listing geometric sources, virtual class of modulite space of stable quotients, some simple modulite space of sheaves. And if you know about modulite space table quotients, so to say this is specifically modulite space thing over P1, it's very close to modulite space of stable maps to P1. And by applying localization to the virtual class, you can get relations under upon them myself. Other geometric source, I'm just listing a lot of different geometric sources that have been used. There's Whitton's R-spin class, which very special class described by Whitton relating to taking R-thrutes of canonical bundle, a curve, it forms a comological field there. So when you use this along with the Given-tall-Telemann classification, comological field there is that relations. So I'll actually be saying more about comological field there is probably tomorrow because they're one of the main sources of interesting topological classes. A lot of things fit into this framework of comological field there is. But these relations under upon the myself and Demetrius von Cain. Two more sources of relations that I want to mention. So there is, maybe I'll just write geometry of the universal Jacobian over MG1. By geometry, I really mean some, there's an SL2 action on the chow ring here. And by using that, and at some point pushing down to MG1, you can get relations. This is work of Ginn. Finally, I mentioned that this is also related to Jacobians but taking powers of Theta divisor on the universal Abelian variety. This approach was really, first taken seriously I guess, although people were somewhat aware of it by Grzhevsky and Zakharov. There are five methods using different geometric sources. There are definitely relationships between and the result of all of these is that not only, so these are all five of these methods have been studied to see what do they give in M24 in this case where we have a missing relation or what do they give in higher cases. So what ideal of relations to these methods produce? All five methods produce the same ideal of relations. In order to be precise here, I should say in what genus they produce the same ideal of relations. So really this is known that the last four methods are meant to give the same relations for all G and the Fauper's original method does the same according to computer. Let's say, I forget exactly how high he checked this but certainly for G equals 24, G up through 30 and there's some question with his original methods and say priori he's producing infinitely many different relations in every single location so how do you check infinitely many relations by computer? But it seems by computer that it gives the same set. These methods are all relatively recent compared with Fauper's original method of constructing. It's basically Fauper's original method it seems less amenable to actual actually proving things about what the relations are than these more recent methods but they do seem to give the same results. Yeah, for every genus. So I should attribute that to some people but neither way is known. I don't think it's hopeless too but people haven't tried that much because there's not much gain for showing that it gives exactly the same as these other four methods. I mean it would be nice to know but there's some technical reasons why this is sort of harder to study the space. Having sort of the same general flavor like it's very plausible that it could be done but it hasn't been. I should say that the equivalence is I mean there are various people involved in it but the main name I should say is Felix Yanda under certain he proved the general result that in some sense any homological field theory gives the same relations and that with some work immediately shows that these two methods should give the same. They both have homological field theory and then with some more work you can connect that to here should also mention Clader and then the connection between these two they both serve involve Jacobian civilian varieties. There's another connection here due to yen so that's sort of the chain of equivalences possibly some other names I should mention here but sort of key result in combining all of these is the statement about homological field theories which again I'll be saying what a homological field theory is tomorrow. Okay so this is really the other half of the evidence which is causing people to doubt the Gorenstein conjecture for MG both it's false for MGN bar or MGN comeback type due to Peterson-Tamosi and the fact that we don't just have one method of producing relations which according to the computer is missing some we have all these different methods which I mean we know something about how they're connected and we can prove that they're the same but I mean it's a fair amount of work to prove that they're the same and they all give exactly the same relations not only to each other but according to this which we only have the computer to verify for us so these all give same relations all are missing relations for G greater than or equal to 24 it's a homological degree 12 R12 of M24 there's a missing relation so a kappa polynomial of degree 12 right so the missing relations we don't know what 36 or 30 yeah but we know from the pairing though since I mean 11 pairs with itself R11 of M24 pairs with R11 of M24 and we have enough relations there to get a perfect pairing which means there can't be anymore this is certainly one possibility so I will say that if Fauber's conjecture is true there's still something special about this set of relations I think it's fairly clear from all of this and there are sort of two main possibilities I see for that one is that this ideal of relations if it's smaller than the actual ideal of relations it's missing some relations then this is the sub-ideal of relations that are true in chal as well as co-amology and other relations are just true in co-amology so that would be the situation where Fauber's conjectures are true in co-amology but not in chal and these are the relations in chal that's one possible hypothesis certainly the other possible hypothesis is that these relations this ideal of relations these relations have the property that they extend to the boundary to MgN bar and now you might think you're guaranteed that relations should extend to the boundary in some way but they don't need to extend to the boundary in a tautological way so these relations in Mg all extend to tautological relations in MgN bar as well so you might think that maybe the missing relations are relations on Mg which somehow don't extend tautologically to MgN bar that's the other possibility certainly not obvious with all of these methods that they extend but with some of these we know that they extend specifically the co-amlogical field theory methods naturally give relations in MgN bar not in Mg so there are definitely possibilities to reconcile all this evidence here with the with Fauber's Gorenstein conjecture I think that if this was the only evidence then we would be not as convinced but this combined with the other half of things that Gorenstein conjecture fails in MgN bar MgN compact type is it fails for tautological co-amology if it's something specific to co-amlogy then yeah that doesn't explain why it fails for MgN compact type or MgN bar aside from those being different conjectures okay so they all produce the same ideal of relations and time remaining I want to actually say what this ideal is give an actual explicit statement of what the ideal of relations is proved constructed by the last four of these methods and which seem to be given by the first method as well according to the computer what is this ideal of relations ideal of relations is all the relations we know to prove so this ideal of relations is generally known the literature is a Fauber-Zagier relations since the I mean the history here is complicated and I wasn't around back at the time rather wasn't involved in the field back at the time these relations were come up with but my understanding is that these relations date back to almost as old as Fauber's original Gorenstein conjecture that these were sort of this family of relations Fauber and Zagier came up with by looking for families of relations that are compatible the pairing into top degree in the Gorenstein conjecture so what are these relations so for any BD finite list relations in Rd of mg and they're parameterized by certain partitions parameterized by partitions sigma no part congruent to two mod three so only parts allowed are one, three, four, six, seven, nine, 10 so on and there's some bound on the size of the partition the size of sigma to be less than or equal to 3d minus g minus one and also size of sigma should be congruent to 3d minus g minus one, odd two parameterized by this finite list of partitions depending on g and d need a little bit of notation before I can give the formula for that formula for the relation given by gd and sigma so first some power series of the a series is some hypergeometric series summation n equals zero to infinity six n factorial divided by three n factorial two n factorial e to the n also the b series which is very similar in by playing some differential operator to that and summation n equals zero to infinity six n plus one divided by six n minus one times six n factorial divided by three n factorial two n factorial t to the n so these are just formal power series I also want notation for inserting kappa classes into formal power series and t so if I have a series summation cn t to the n curly braces outside with the sub kappa then this should be inserting kappa classes in by multiplying the coefficient t to the n by kappa sub n they should cn kappa sub n t to the n and final piece of notation that will make this convenient is something that I stated yesterday also the curly braces over a monomial in the kappa classes and this was this summation over l in sm product over c cycle of tau kappa a sub c a sub c is some of some of the a sub a's we extend this linearly as some as some key linear automorphism of the ring of polynomials in the kappa classes not a ring of automorphism so that's a notation I'll need to write down the fz relations as they're usually known I guess some other notation so sigma is some partition let's say sigma sub i is number of parts of size i that's not the ith largest part it's the number of parts of size i so given all that this is not how not how faber originally described the relations to me but it's equivalent the relation again parameterized by gd and sigma okay so I want to start by taking one minus the series a insert kappas exponentiate that some power series in t what's coefficients which are polynomials in the kappa classes then I want to multiply that by big b insert kappas raise that to the power sigma one number of parts of size one take t times a insert kappa classes reset to the power sigma three b sigma four sigma six and so on sigma only is part of your zero or one mod three parts of your zero mod three correspond to a parts of one mod three correspond to b some finite product because only finitely many of the sigmas are non-zero they take all of this apply this automorphism to all the kappa polynomials appearing now I take the coefficient of t to the d in this series this is actually some polynomial in the kappa classes so again there are a couple of equivalent ways to write this this is the most compact in some sense and then the so these are the relations originally discovered by fauber and zagheir and they I mean fauber was able to notice that these relations matched up with what his computer was producing using his method of constructing relations even genus greater than twenty four and so it's now theorem these relations are actually true so using the second of the methods I listed there the modulae of stable quotients punterabandan myself for able to prove that these are actually relations in the total logical ring of mg again for sigma satisfying those inequalities there this is I said this is actually the ideal produced by by all five methods in the first case experimentally so and state that clearly all our methods of constructing all logical relations seem to give exactly the fc relations by exactly I mean have the same span probably not obvious from the formula but the these as I've defined things is the additive span of these is actually an ideal I'm saying that that it's fairly easy to prove that additively they they form an ideal I mean the statement here is that all the methods of constructing relations we've been able to to study in high enough genus to tell say genus twenty four greater seem to produce exactly the ideal for relations which is additively spanned by these relations and no more and that's a theorem that they all that of the five methods I listed that methods two through five there give exactly this set in the first case of father's original relations such as an experimental statement well based on this the natural conjecture is an alternative to the guaranteeing conjecture this is something that it certainly wasn't stated by f or z but I'm so in cult fc conjecture because that seems like a natural name is that every relation in our star of mg is a linear combination fc relations main motivation is conjecture is just we've never constructed a relation is not in this family including by father's original computer method back we know of several proofs that these relations are true several methods of of producing all of them the status of this conjecture is already like this for g less than twenty four both the gorenstein conjecture and the fc relation conjecture are true g equals i think it's twenty four and twenty five exactly one of gorenstein and fz conjectures is true for g greater than equal twenty six at most one of them is true somewhat silly way of saying things but the the content of this is that for g less than twenty four we know the full structure of the ring and both this description explicitly giving relations and the gorenstein description give the same result which is what we know so they're both true then for g equals twenty four and twenty five the discrepancy between these two conjectures is exactly one relation see they're true it isn't so exactly one of the two conjectures is true then for g greater than equal twenty six there's more than one relation discrepancy so it could be that neither is true one of them is true they certainly aren't both true these relations are not enough to make the ring gorenstein in general alright so there's some way there's some ways in which this conjecture is less appealing than the gorenstein conjecture the duality is very nice duality is very clean way of determining structure of the ring this conjecture does have some advantages that I mean it's an explicit list of relations and although this is somewhat complicated combinatorially you're actually sort of forced to have at least this level of complexity in the sense that these like the a and b series are really forced to show up if you look at the first couple relation so I should say some words about that so here remember size of sigma is less than or equal to 3d minus g minus one that means that there are no fc relations if 3d minus g minus one is negative these inequalities possibly I should copy over to the serums and I really do need them for these to vanish size of sigma less than or equal to 3d minus g minus one congruent on two first meaning in lowest comological degree fc relation for g equals feeling of g plus one over three and depending on the value of g mod three that relation might be unique you can two cases there will be two different relations in the third initial relations in the first case if you think about the case where g plus one is a multiple of three and d is g plus one divided by three then sigma has to be the empty partition you don't have any factors after here you have a much more simple you have a much simpler combinatorial expression in that case it just involves this a series that's a sense in which you're forced to use the a series and feel like look at some different parity in some sense forced to use the b series because the unique first relation involves the b series quite heavily I guess I guess what I wrote only happens when g is one partly because of this thing I'll get back to something which is the theorem originally I believe due to I know which is that r star of mg is generated not to take it's like kappa one kappa two up through kappa I guess I'll just write ceiling of g plus one over three minus one instead of figuring out what that's equal to that is you don't need any of the other kappa classes what does this mean it means that each kappa class after this point can be written as a polynomial in the lower kappa classes and if you stare carefully at this formula you can check that this formula actually gives a proof of that I know prove the theorem earlier using relations which are more or less the I believe that I know more or less use the first relation in this family by some other construction but you really do need all of these generators because I mean the first relation happens after this so up to this g over three approximately point the tological ring is free polynomial ring on the kappas then you start getting these relations and according to this fc conjecture those relations are precisely given by this formula we'll stop there for today tomorrow we'll be shifting back to mgn bar we want to talk about comological field areas and also how to extend these fz relations to mgn bar