 So components, one locus, or maybe just corresponding for partition lambda, and this component geometrises the following. We have problem E, that is split as a direct sum of unknowns, by restricted to one of the EI goes to the biggest one. So E, by restricted. So this was the description. And so, because of the thing that is injected, the ranks are, this rank must be integrated with this rank and so this is a partition bar. Okay, so this was a situation. And it also seems so one. So my scores and contribution here, good one. Yeah. So. Over the first fundamental class corresponding to lambda, one over the order, or number, which I mean now just like this. The zero, unless the partition has a very special form, mainly, we have it. The same number. I didn't say anything about the proof. I did not. They also showed, they also showed that, and that I sketched, like the initial number of modelized space of sheets. Notice that by definition, I didn't talk about stability, but stability is it the component corresponding to the partition plus R. It's just a modelized space for sheets. This is equal to minus one. The virtual dimension was going to be one, this is the virtual dimension of the modelized space. So this was minus one. I'm just saying the corresponding invariant for this whole essentially from the difference. Right on the sketch. So. So in particular. So this part, one would call it horizontal. The one corresponding to partition one. This would be called the virtual. So that, according to what you wrote that you can write this as the sum of all lambda tissues are corresponding to the partition. Just a little bit over that component. And so in particular we see that are the prime number. So we have two conclusions. So now, maybe I want to talk about structure here. Then I have to explain some of it. And then I have to talk about it. And this has this then involves so-called nested table schemes. So first time we can write down practice. In a simplified way in which I was not. So that I don't have so much baggage. So. So we can. Well, I mean, so as you can invent this name. I don't know if I've done it before once and somebody called lucky cases with my board. But anyway, so that's, but the thing is, you know, you are an S you are, I think, just for response to the fact that we look at. And so somehow the structure groups is you are S you are, I think that's all that is met. And in principle, in physics, otherwise, one can also consider, you know, it's for bundles with other kind of fun with other kind of structure. So this I would know how to do. So looking at bundles with some extra structure with it. So as you are, you can have a mission metric or something. And one could have some tactic or whatever. And in each case, what would try to write down such a thing. And then if one knows how to do it, one look at the other version. Yes. For me, as you are just looking at fun. One could look if I also do the generalizations with other structure. So we take, you know, so then the statement is there exists some power series in explicit formulas, but I will not. I will not prove the explicit formulas. I will give an explicit normalization but I don't do this. I gave an explicit formula for a, and like I said, but I will not, not show that more technical. But otherwise we have just been C I J. One smaller. So these are all power series. So the surface surface. Like on it is about so each one pushes you to something. So this is over all amines one of the classes in the second one. This delta. The first chance set here. And the sum is one over I. Power series C I J. Our intersection number of these days. So this is the general structure. I had the last time set what this means but just for a and P. In. In that you mean that a D. A minus. C one is equal to the sum of these classes. So this is the model of something. Because that's what I think that is this formula. So, because I want to put something here, but here is product. So this is one of those I was written. I will review what it's like to have this. You might fly by the power of this. You see by itself. This is an infinite sum. But it is the fact that on the assumption. There are only finite many classes in the chronology of the surface such that there's other written event is non zero. So the sum is actually fine. As we are using this and they play a role. I want to briefly review what this is. This case. No, no, I don't think anybody or everybody will be aware of it. So by itself. Are some C infinity invariance of four minutes. But we need to be able to do that in algebraic geometry. How many foods means for that? This is an algebraic surface with the, say, EG of S, bigger than 0, H1, one of S, S to 0. Then they are come true using algebraic geometry. There are also different conventions in algebraic geometry and differential topology. So what I would call cyber-critical better, somehow differential topology, cyber-critical of, now I don't know, besides which one it was, something that is better, minus, it has in different quality when it was this day, cyber-critical is associated with a characteristic homology class, but here we associate any homology. This is our convention. Now I haven't said what it is, but once I told you what this is, you can see what the differential topology is. So how does one, so what, how are these defined? First and foremost, they associate those class, say, A in the second homology, the number in algebraic geometry. So to S, we have assumed this H1 of S to S to 0. So in some sense, so in this particular case, so we also, so in order for this invariance to be known as 0, we need that this class is of type 1, 1. So it would be a subgroup. And so, assuming it's of type 1, 1, so thus we want to identify corresponding line, 1. Type 1 would be 0. What we find about that is defined in line, 1. So, so then they have a quick modulate space. So the optimization space of section, the space, we somehow deserve a quick invariance. It's a number of computations in this modulate space, which is just to protect the space. And now this modulate space, which I call M, first term class of the homomorphic line bundle, then type 1, 1. From this analysis sequence here, you just get an injection of the invariance to write. Yeah, yeah, yeah, yeah, yeah, yeah. But somehow it turns out that the modulate space, so the modulate space will always be something like H0 of SR, that is assumption. And so if L is not a line bundle, then this would be easy. If it's not a homomorphic line bundle, then our homomorphic section, this would be 0. So invariance will be 0 if it's not a type 1, 1. That's what I said. I kind of... Anyway, so this is our modulate space. And now this ML is an obstruction theory, which of course, first we can remember, we have here S, the projection, S times M. For instance, you can first, you know, one knows what the tension space, so projective space at a certain point in projective spaces, and we can globalize this. So where did it write? So to let 4 minus 1 be the top logical line bundle, a bundle put for this Q up a star, which is just up a star of the space of sections. So we have obviously, it's a projectivization. We have, it's just a projective space. We can look at 4 minus 1. And then everybody knows that the tension space, projective space at a point corresponding to a certain line is the homomorphisms from this line to the quotient by the line. And so one can globalize this and say the mentioned bundle is the one by this one, which just means that hyperwise we have homomorphisms from both the bundle to the quotient. And then, and so I can write this simpler as the constant of this 2 star minus 1. We have now, here we have centralized the fourth one, which is so, so this is the tension bundle. And the abstraction bundle is just, as it's often the case, the tension is something like H0, the abstraction is H1. So this would be the abstraction bundle. Okay, so this is something very simple, right? Is it? And so one can obviously put also compute this trigger bundle and then we do this for everything. And so if we know, now we have that our modular space is a projective space with smooth and the abstraction bundle is locally free. So then we know that the virtual fundamental class is the Euler class. Of course, this Euler class, because it's physically, we can easily compute this. I mean, it's an exercise to compute what you see. And I will just give a few consequences of this. This is just in, so I call it a position. It's basically an exercise, but I have not done the exercise. So assume the virtual fundamental class of this thing is not so easy. So this corresponds to the fact that this abstraction bundle does not contain a trigger sub-bundle. Because the Euler class, if something has one, if there's a factor O, you take the Euler class, it means you multiply everything with the top-turn class of the trigger bundle O, so it becomes zero. So that basically what this corresponds to. And this is, so this is the case that first, and the virtual dimension of this modelized space is equal to zero. This is actually in the differential geometric context, such a statement is an open projector. Or it's not here, but it's two, you would say. So this is the translation to say all algebraic surfaces with Pg bigger than zero, it's one R of simple type. The second statement is that it will actually well affect the state of this line bundle. This is equal to the final multi-point effect. S for equivalency, S squared, no for a line bundle on the surface. You have one half S squared minus LxS plus LxS. And the third one is that H0 of S L must be bigger than zero. That's here because otherwise the modelized space is empty. And you also need that H2 of S L must be bigger than zero. So which is the same as saying that H0 of S, the S means F must be bigger than zero. In this case, if you work out how did this define, define an acoustic compute, this is a written invariant corresponding to F, which is sine, the dimension minus one, take the geometric genus of S. So all this is in some sense an exercise that people do. And so in particular, and besides state it sometimes before, if S is connected to the moment of device, there's a written invariant of L to zero, and there's two S for L. For this argument to be zero, the curve itself must be effective, the line bundle must be effective, but it must not be too effective, it must still be smaller than economic contrast. So this is a very strong restriction. So this is for this case R. So we have R, I'm very slow. Okay, now we want to at least, I want to talk about this next difference. Continue the version of the written invariant, I'm trying to prove this. So let's recall this invariant. So it goes to the end, C1, 1, prophesies, direct sum rank one, she's I is to zero, R minus one. And T goes from EI, EI minus one, and so S forward, so check this and then relates it. So we will see more points, because if you have a torsion free sheet of rank one, then it's standard that this is the ideal sheet of the zero-dimensional sheet times the line one. EI is of the form, it's a derivative of points, variation. And then obviously here, we have always a map from EI to EI minus one, 10, so S. So that means that E, that LI minus one, minus LI plus DX plus 10. So now you can therefore imagine, you can describe this kind of thing in terms of Hilbert scheme. We know somehow Hilbert schemes, Hilbert schemes of points, normalizing these and Hilbert schemes of curves, hybridizing the subsets, the line one of us, the corresponding map, the image of the two. So for the tuple to the end, which is zero, so R minus one, and that's EI, points is integers, plus beta, which is an element minus one, which however corresponds to the effect of line one. That's the product of these zero-dimensional points. The zero-dimensional points I have to do with like this, with linear system corresponding to beta, is the product R minus one of the linear system, that's one that always the linear system means the perfectivization of, and then we have to correspond these. Necessity of a scheme, which is a certain incidence correspondence in the product of this, and this. This is going to take some zero-dimensional substance, such that this whole thing is in the product S, and what we want, the ideal C, C-I, just that the minus C-I is contained at a certain zero-dimensional scheme, but we want that if we twist in the addition of curves, it's contained in the next report. This is a, and as you can imagine, this is a rather tender scheme, I don't think it must be quite singular, whatever. Anyway, we can first say that we can identify this monolid space for the virtual property invariance, which is doing such things, and then we have to see what to do with it. For all C2 of the end S, besides moving, that's the deal with it. So, because you can see ideal genes as they are, we always have a map from one ideal sheet, tensor with something to an ideal sheet, tender with something else. There should be a morphism from then, which means that besides C, if there would be a section of i minus 1 minus l i, such that the ideal sheet of the zero set of that section, tensor with 1 is contained to the next, and so on, precisely this situation here, we just put it together. So, now one has to please remove this, but I can at least write down what it is, and it's a bit ugly, but anyway. So, if I put it all together, so we have to see how does the second-gen class of the sheet come to pass, how does the first-gen class come to pass, and what are these conditions, we put it together, we have the ns. So, if you put the right description, number q of a1 to the minus 1 is this crazy sum, j. So, this ei and second-gen class of minus 1 is the union of these things. So, some numerical foundation, first c1 must be equal to the sum i equal to 0 over 1 minus 1, i, and the s minus eta i, 2 must be equal to n. So, this is just the sum of the n i minus 1 divided by 2. This is just, so if you just look at this, so if you write such a thing, and you have these corresponding maps, then you know that the corresponding c i is always, so it's this one, i minus 1 minus li plus s, this is the class of the c i lies in that corresponding beta, and then, so the first-gen class of our e is the sum of all these li's, this way we take this as that, and the second-gen class is, you know, we put it by the union of the second-gen product formula, and this will give us this one. And so, we have just a different description of the product space in terms of these necessary results. Now, maybe I don't think, so maybe I should say, maybe I should stop here. I mean, it doesn't make much sense to, I mean, it's a little bit, so the next thing is that, so as I said, this space is very terrible, and so it doesn't make much sense to continue with it, but now one can define the virtual fundamental class space. These spaces are, you know, certain components of this monolith piece of vertical property. So therefore, if one takes the virtual fundamental class model obstruction theory, one has, on this thing, coming from the localization, from the virtual localization, this gives us an obstruction theory on that. One can actually compute what this is in terms of the data, and then one can say, okay, this thing lies inside this much nicer space. This is a product of the virtual point times the product of protected spaces. So one can, instead of saying what this virtual fundamental class here is, one can say, what is this push for, for this to this thing, then one can compute here, essentially just the virtual point. And then this will be next time, and then we'll do this thing, and we'll also set up the stage for giving explicit computations for what is power series A, B, M, T, C, I, J, R, F. Okay, maybe I stop here. So I, yeah, I would, is... Yeah. Yeah, I guess, I have two questions completely unrelated. I mean, not questions. Well, I guess many questions are completely unrelated. So with this one you were just describing, this is kind of a local complete intersection here. Now, I mean, this nested Hilbert scheme, you can sort of explicitly say how it sits inside this much nicer smooth ambient space, no? Yeah, complete intersection. I don't know. I'm not sure. I didn't think about how it's actually, I mean, so... But when you push it forward to the, essentially you have to calculate its class right inside, well, I guess we'll see next time, I think the... Yeah, I mean, there's a relatively simple formula for the push forward of the class. That is, I mean, in some sense, anyway, so this thing is very singular, but it carries a virtual fundamental class and the nice thing is one can push it forward to a nice space where one can then explicitly say in terms of standard things one knows on Hilbert's schemes what the virtual fundamental class means. Yeah, right, right. But that seems, it seems like it's expressible in terms of, I mean, there's a, so this doesn't occur as a section of, it seems like it should occur as a section of a vector bound. I mean, that's how you probably calculated by push forward. So I must admit that I was now just wanting to take this result kind of at face value to say, you know, whatever, so now a miracle first and the push forward of this fundamental class is this. I have actually not looked into the proof, but you're right that most likely one should do, the way one will do it is that one finds some kind of section or something, but I didn't look at the proof, so I cannot tell you. Yeah, it seems like this is how it should go, but yeah, I'm looking forward also to seeing some details in that. But the second thing I wanted to ask, which is completely unrelated is what happens to this whole theory when the rank is zero, right? So when you look at dimension one chiefs on the surface, so obviously one still has the Geesecker-Mariama space, so that makes sense, but is there a way to, yeah, I don't know, what does this become? Does it, it's completely nonsensical when the rank is zero or does one get anything interesting and is there? So I think, I don't know what that has been studied, I mean, so I mean certainly whatever methods we have here won't work, I mean that's, but that doesn't mean that it's, I mean it could work, I mean one could hope that one could do something like this and one of your chiefs that somehow would be more related maybe to some kind of fixed things on curves, no, but it's some kind of, but I don't know, I haven't seen it done, so and I haven't tried to do it, so I don't know, but at least then, I mean, so in very, at least, you know, you would have some kind of, you know, you would have some kind of relative compactified Jacobian over some linear system. Yeah, that's right, that would be a relative, yeah. It's corresponding, it's a system or something like that, or some part of it, but I, but you know, how, together, how, with what happens at the, at the bed fiber, so I don't know, it has not, I don't know where it has been studied, but it certainly would be I think it has the potential to be very interesting somehow, because, but anyway, I don't know, I haven't, I haven't, yeah. Thank you, sorry, thanks. It's necessary, but I think so, but I didn't think about, I mean, you put the, obviously you could do something really trivial and say that these numbers are both zero. Oh, we are one there, we are one one. Yeah. Obviously, for the whole setup, it's also important to deal with the case when this is zero, it's somehow the kind of, if you wish the, for the separation part of the addition part or something like that, I mean, it says somehow the constant part, pull out, it's important to do that, and then obviously it's something to take about, something or some integral over some projected space. Now here, the case where you have one and one, I expect, yeah, I expect one to do it, but I haven't done it, I mean, it's, you know, it's a, you know, you're just saying that the situation that, I mean, I would think it would most likely be single and so on, I don't know, when the point flies on the curve and so on, it's not so much, but I'm not quite sure, right? And I certainly, I cannot, I haven't thought about it, I cannot solve problems like for, I'm not very good at this kind of thing, but yeah, I mean, it certainly is something, I think small cases one should be studied, I can try to think about it, but I haven't, and as I said, the case that it's n is equal to zero is kind of trivial, but we can easily do it with something about projected cases, and then, and that's important for, you know, actually want to know what the constant term is, is for next year.