 So, thank you very much so, for those who have survived the school pictures, so I, yeah, so, so. So, So, Remember, these things have an obstruction theory of expected dimension zero. And addition over there, not a contact. They have a C star action with five points and not every C star action such that it is compact. So, this formally applies localization. And as I said, one justice. So we have two special components. So, And we have seen that this is equal to minus one to the virtual dimension of the modern life space. Which is the same as the answer. I mentioned this model, so M was a monolid space, which. It's a single, but it also carries an obstruction theory expected dimension, which was just this number. And so, in particular, that the horizontal, but with an event just the virtual one that we see. So now we have last time. So, And we use for this localization. So, The abstraction theory to rather the dual of the abstract. So, what we have seen was just so we have. I assume that it does not exist, one can still work, but it's easier to take this case, then the dual of the obstruction theory is. And trace the X. So, that's, we want to compute this virtual intersection number on this modern life space on this virtual. And for this, we want to use this. And that's precisely that. The general. Now, which allows you to compute basically any intersection number, any version of the section on this model is based on the section that was able to section on this space. This is the right within range of them. Maybe I should. Maybe I can first start. So what kind of classes, what can we compute classes on what life space. So, for simplicity, we stick to the case of the records to is a formula for any range, which is similar but it gets more and more complicated rain goes up. But it's always. It's always more complicated way it's complicated up as is. So on end. So you want to see what classes do you want to stay here. If I write it just see what's going in the ranks to. And we have here, our universe sheet. So that's it for classes so far. For more. See how I push forward. The one life space. The ice journey class. The universe sheet. And the multiplying make complete sense of it anyway. This will be then class. So here maybe I always take, but don't write positions and say always keep. So we have these classes. And then the claim is we can take any polynomial in these classes. But. It's very well possible that this actually is the whole format of one life space anyway. And so, and so we want to come to this. We have to similar picture. We basically want to write out any polynomial. Write down the polynomial in some classes. It's such that you get the same. So we have. Similar picture. We have things that universal chiefs which we had seen before. And we have the universal keys. We were considering. With the best. This is just in points. So that this is the idea. It's an idea. And then for. This picture but not these things. So I think in this same day. I first take the new sheet. So there's one such thing here. So. We push it. Yeah. Which would like to see it. And basically put it back for three of them. And this was. So this is the sheet. On. Yes. Wow. The other things that we have, which we may not have. We have again. It's asking. Which has its projection. This is a second month. So in the same way. Oh, yeah. This is the full back corresponding bundle from the factors. This is actually this one. rotation. We have a. That we took. In such classes. We have to replace it by something. So we're not also. Yes. One. She says. All of them. Minus. You know. This is. Minus. One more. So we already had this. So. It's a very good. It's very good. You're going to ask. Equal to. So. So what it really means is that S is equal to the first term class. First term class. S. Which is a model. So it's a trigger line model. With a. With a non-trivial C star action. And S is the first term class. And then. And then. This is your best. So. CK. Your best. It's just. Corresponding. And so. Calling. For. A. One. S. Is. The line. Of the one. Is. This one on the. First time. This. Is. This one on the T. First idea. She. A1. Minus S. This means. First. By a. That. My. Plus. These. So here we have these. I do. And. We have. We have. We have. We have. We have. We have. We have. We have. P. And then this divided. The. S. S. And. In the end. We have. This is second term. Is. And then. It's a. Actually the world. So what does this pee mean. So. Just. So, he just needs this we do the same thing here. So, now, instead of. So this was. You know, which was you take the, on S times M, take the universal sheet. And you might like to take the ice turn class of the universe chief, and you might play with the pullback of S, and then you push forward. And here. Now, he was the same. So you have one. Yes. You might play with the pullback of alpha and you push forward to the part of it. So you do the same construction. And you place every instance of time I saw alpha in this polynomial by the corresponding expression that you get here. So alpha was only. Like this. The same. The same thing. The integral of this thing or there was points. This is some over all. One such and then such and one person. And take the integral for Chizuki's. I don't say that this. The whole more than zero. It's up to you. And then. He is again to see. Use this phenomenon. Is equal to one is a sum of classes. See one. A one times each smaller class. This is the form. Maybe I can't say. This is a, as you see it's rather complicated, but completely explicit. So you have this simple expression. So that's maybe some simple thing you want to compute here. So it becomes something very complicated in terms of the universe she's on the liver's teams, but it's completely explicit, and one can compute. So, but this condition is not very serious. Because they know when you if you twist. So if you have the same line, then the invariant will not change, but this condition will change. So you can. So, normally, this will only fix. See one. Fix see one. This will only hold up to a certain seat too. But if you twist by bigger and bigger powers of an empty line bundle. This tool you can that will continue. So this is somehow something one can see up. So this will therefore, always. And maybe one should see. So maybe one word. I mean, the proof itself is, you know, is one. pages, but the basic idea. It's really not so terrible. So one considers. Another life space. I think it's a she together with this section. This she, this one life space now has a kind of star action. This point locus of this C star actually consists of certain parts on and stands. One is something related to sort of written in the answer to this point. So something's one locus. I look like this. Other things. The other things. This is my life space. And now you integrate something over the one life space. We integrate something over this master space. Which of which you know it is zero. I'm just going to be zero. But when you compute it, you get that the fixed points, the fixed points corresponding to this will give you this. The fixed points will give you this contribution. And so therefore zero is equal to the difference of these. But it's normal to set up so that. In the trip is somehow not new, but. And then how I want to buy this. So in this case, we take the obviously to be the term class like this. Now, you know, the virtual tangible as a, as a complex. We can, and one can compute this by so we're again. Yeah. And so, and so that's what it is. The thing is, in some sense. Now, one is done because now everything works the same way is in the vertical case. So I can just catch. Because we are basically a thing we're indicating it looks very similar to what we were integrating in for the work with my friends. And so the methods will be precisely. The other round is the horizontal one. So how to modify. So that you just maybe for the C prime. This is. This is. Generating. We just like a, but the key is the one which corresponds with a one scheme. So, particularly can consider the part where n is equal to zero. So when we look at the zero points here. So this will be equal to the coefficient of q zero of the same. And something which I call our series start with one. And we have, so now we are looking at integrals with the systems of points. And this, we have to use some types of this is what is there. It says that if you have anything reasonable, then express the expression in terms of intersection numbers on the virtual surface instead of. You have a nice generating function. For such numbers will give you a map ticket in form where just is a product. So, so power series. Take it to the powers of the intersection numbers on the surface. And so you can do this program yet. And so, and if I have policies. So, we have this is a one. That's that I've sketched it. So, given this out. If we know these policies, we can compute the result of a bit later. For this power series, we only have to be able to compute this generated function here for certain examples of it. Some examples of that because only backup numbers. So in the same way, so this allows it to be. So, for example, we construct from these policies are some secret. The meaning is these are sort of services with an action of C. So, we can start with six points, you can take nine models on these, which are a brain for this action. And so then one can continue by the basic. You know, the fixed point on the action of civil schemes by the politicians, but to bits of petitions. And we have this fact. The other time, what the action is from the next group of these ideas she's, and it's even simpler to see what the action is on these top logical bundles. So you can just write down everything for the local contributions of the fixed points in terms of combinatorics partitions. And then, you know, this smart one could do something with it depends not so smart one in right program so you then so then. Yeah. So, so one has to be careful because one process to check off S. So the higher the key was the bigger this the ported S and one has to finally take the coefficient of s to zero so that makes this computation more difficult also for the computer. But it's not there. Anyway, we can compute this, and then one can look at it. And anyway, we find, let's consider this function. So we have the data function of the artist, the intersection form was given by this. And then we have the matrix, which looks like two on the diagrams like this. Now, we can also look at the data. They are like this. They are we just replace. And then we have the following lecture. So we even get a zero. Exist a lot of structure theorem. So, you know, the idea, once more, one, the version of the case and efficient. much of the following expression. So very similar to what we did for this event on the difference that they have of theorem. And then it's better function function for the minus one times these things and zero times the sum or better. So we have a sign. So this is the intersection number of that one. And then. Remember, this is precisely the same formula that we had. The structure. Instead of the data. Instead of the sort of unity, we had this data. It's better. I was very similar to this difference and they have this universal five just called C zero. C zero and C IG. And the relation to this is this is basically the formulation of this context. So, as we said, the ordinary function of all that shopper written invariance. It's nicely and the power minus one of a tower. So with the same thing, it says that C zero, C zero minus one of the top. This is because for the work for the vertical of the difference, and in the same way. And so it means the generating functions for horizontal and the vertical of the difference determine each other. And don't these DIGs and CIGs. We identified them here as modular. And so it makes sense to do this. And so we have shown this, that the range is two or three for four and five. This gives us this tells us this gives us predictions for these policies. It's a formula. There's actually very simple way how you transform the formula for the vertical of the difference into the horizontal by just like this. I mean, what this operation does. Okay, so maybe I will. I wanted to say, so that's Yeah, questions. I'm still