 Oh, and that's the given scheme. Maybe I can still use where we are. We have this component of the big one. So, of course, it's about the mechanics. And so this will indicate and I minus one then introduce so we have and we also have in your system then we have the next some in so so you it's okay so we have an instance obviously that's something but now the ones can show that we see that this is isomorphic for the union of some of these I mean in theory all the locations so that I use it afterwards but then as I will wipe it out you know I just so we have this one I I some some section number to the union but such that you get correct no it's of course some like that and the formula first congruent and the second some and I that's the condition first of this of the like this what are conditions conditions and then we have a a mask and in fact the one actually does is push forward push forward this is this is some so and this is in terms of universal ideal so we have the universal we have the ideal we have the ideal and we have the ideal and we have the ideal and We take all the parts of the r star vector i with the vector part 10 to the s, 10 to the o, whatever you say. So this is a trivial vector bundle, so this is f thing minus, so I should maybe say here I have this, and then we have this right here. Here's a pullback in sm vector and factors. And so we take this difference, which is an indicator of the whole. And this is not quite it. We might try this. So we are here on as an. This is not in sm. And so we see here. Projective space. So in some sense, this thing just lives by itself on this product. So we have this one. So this is one. We know what the fundamental part of this thing is. Then we have to see. So we have to compute. Instead of working on this to pretend we can be pushed forward to product. So to push forward to this, we just. We still we have to just take the part which is the key zero in these coordinates. It doesn't get killed by my point. But what we need to compute is not just the integral of one over this virtual fundamental class, but we have to use this localization. We have to compute all the number of virtual normal. Normal where this virtual. And moving part means it's the part where the action of C star. Where the weight of the action. So, therefore, have to determine what this funding is, or what the product is, or currently, as this is anyway on the moving part, we have to determine what this restriction of the tension. And so, in some sense, it's quite, it's a. And you get something. Right. So basically, what should the tension boundary be. I mean, basically something like here. Maybe not quite because it should be minus. Or something like that, maybe trace free. And so we have to write down what is the universal sheet and we know what the universal sheet is because it's just written. I mean, we have basically written it up when you have to make it to globalize. Okay. Okay, so. And I see. I mean, it doesn't matter. So this is the line on. So this means that pullback, the li. And a pullback is one from here so I get the line on the program of these two and then on the triple product. It's a nice. Right. Pullback. And then. So, where the li. All right. And the sun from time. Is. This basically just, you know, white out this just described what these bundles are that are there in such as globalize them. I do achieve. And so, and so the team. Is a. Just means that we have a trigger line on this, which is not going to trigger. So this is the universe she and then you can just know, but. So now we have this equity, which is no. Globalizing this with chaos on this day. I don't think I see, but the claim is there is a total logical such map comes from the fact that you know, this has a logical section. So minus one has a logical section if I did it right, then there's a top logical. Yeah, so which does come from the universe. So we have this, and obviously he has me for that. And then can. There's some formula. This. Virtual. This is equal. So what is. In terms of the thing we had before. And with the abstraction theory. Precisely that they have. This is just. And this. And so, I can just put it together. We have all this data so we can write down the formula, what we need to integrate all the more fun. That's to make it. Yeah, we have here. This is given mostly coming there's this stupid. This L is that's the line model of the surface of this easy to see the same current way. But they're not doing the thing is this universal idea sheet that we have on the table. And you always have to look at. So what we're going to say is a direct sound of such ideas. So what you would be interested in is the X sheets of one universal idea sheet with another or with itself. Before you compute this, we can compute everything. So, I write this down as an existence. So we have to just put it together. If I want to compute this generated. We can generate the function to respond to this potential. And then this is something. So, something. Things. And. There was a shift with the second chance corresponding to the queue. So we have a. And. And it is some of the nice. And then the product. I want. And then. And. It's just. Nobody knew. I have to see what this is. And there's a, I can write down. For that. Which is just what we had before. So. And. The denominator to have this. The moving part of the central bundle. So this is. Of. And. So this is just a bit in there. And we are supposed to take however, the moving part to where the weight of the, of the action is not. And in the numerator, we have what I have written down before what gives us the fundamental class, which was the subject because one. Then. This is what basically what we have for the tension. And all these things that you can see, I just get tension. So. We can now start. I mean, I want to. I want to. I stated the structure is that. So this is basically. I think that if you have an integral Hilbert's teams of points. Natural formula, the class, which are even given either given by. The universal bundles or push forwards from S times. The Hilbert scheme of universal ideal sheets, this is by some line bundles coming from the surface. This can always be expressed as a universal formula in terms of the obvious classes, which you can see. The line model. And so we want to use that. So. So what we really need to do in order to prove it, we need to know. So these are all just numbers. This is all very often. We need to know what you can say about these. And take a. Now I want to. So these are the things I want to know. So I put them into generating series and I want to see what I can say. One. Just to just do the constant part of this thing. And times. Over all, and. And it just. A. Maybe say here. So when we do. We have this. And this does what you get when you scan. So with this we have normalized the thing in such a way that the power seems to have to. And so as you can imagine, I mean I don't write it down. It's an elementary formula for this. So you should look at the formula here. You can whatever or whatever. In particular, if you push forward just to point you have some physical formula, which happens on the surface itself and give you some simple intersection. I can also write it down to that, which is no mystery. It's just a number. We have this. And now. So, we need to compute this GSA, or to say something about. As I said, by definition, this gamma a and he was expressed in terms of transfer of the universal edition. And then you put forward. And then he's not going to be I. So. So, so we have an expression for this. And so, if you just look it down, it's expressed in terms of this. So, so now they, what is in range that integral over in the numbers. They are all high and they are. So, is a universal point. That means there's a formula, which is independent ads, which depends only on obviously what this is. So, the original. was just for the case of the human points on the surface. You will have a product. And it already includes the universal edition. The statement is really adapted from the original. So, so this is the first part. So, this thing is just universally such. So what more can you say. And then the next thing is so this is so. So, we want to see how it depends on the state. And then. This is very nice. So. So now, what does one do. So, So, So, So, So, So, So, So, So, at the formulas, I mean, I write out what happens. How can I express the thing for S, and a, in terms of these, and the way it's done. You can see it somehow product of things over, over the past. So if you work it out to find out that it is. That is easy to check, but you have to look at the actual formulas that. So the third thing that we have to see is obviously, if I look at SN, what is this. I mean, here we have this tool for anyway, and it was just a human scheme of endpoints like this. There will be a sub scheme of playing and on this one union of two projects varieties. It's, it's, it's, there will be a sub scheme of some length and one on S one, something on something and two on S two. You have to go to all possibilities to do this, like this for the product of them and still to. So this will be over all and one and one and one. And then if you work it out, you find out that the interval. The only thing you can take the contribution over this product of the thing of a one and a two. It will just be the product of the one of this with a one and this with a two. It's not automatic and on the formula that you are given. But it is at least understandable. And so, from this it follows. And if I take G associated with each of you. And so, and this is just the product, the generating function is just the product. It means term by term you get product and then the generating. This G is one. So this was a one and this is. And now, now the point is that this tells you that energy function here must satisfy a product form must be product over contributions for the idea says each individual journal is a polynomial indeed but it's actually much safer. Because if all of formally this thing. So it only depends on the numbers but in fact it depends on these numbers. So basically, the point is the following. So let's do that. If you can associate if the numbers of your intersection numbers that we have here are those which correspond to the situation, then it's the product of this. It's a product of many such terms. The product of all the. Now, the combinations of intersection numbers that occur as products of the correct number factor, you know, as the second number of factors will actually be the risk dense in the space of all numbers. So the point that you have a product structure when this thing can be written as a union like this now a lot of tool but many tools that it must in general like that. This is a formal thing. Okay, we have one final thing to get the structure formula which was slightly better because you had a few more less generating questions we just have. Because remember that you had always to take this I will quit the event of AI in front of a written class means that the, that is our written off AI is non zero. And obviously we have only contributions in the self written on the ice on zero. So therefore, we can eliminate these terms absorbed in here. Yes, it's equal to I squared so I can take this thing with the I can guess and multiply to the term, the I squared here. There was one more thing that he found. So by specializing that these things are universal. So, if I can complete what happens in the case of the service. And in that case, one can so there is, you know, basic classes, but this term still there, he finds that a key minus from there. Okay, so here on print us explicitly do some geometry. Look what happens with the key service somehow. Look what happens most terms, many terms are not there. Now I have a three minutes. So, this was this set the theme of now, but now we can kind of please ourselves on this and try so. So what we're interested in now is to complete explicitly what the other power series up. We want to know what they are. So here we see some of the forms physics says that all these things should be somewhat special in terms of model forms. This one. And so we want to try to do that. So, remember that this GSA equal to we have defined what this GSA is for any surface. For any people of just class in the second for more than you don't have this. You know, if I all right. That's what I classes to put up class in the second for more. We can just try to take some services, which are particularly nice. And some nine bundles on them, which are particularly nice and compute for those. So we're not enough examples. And then for good for those, the term of the process. We know what a two is, but it's good for permit. But what is the policy is that we have to do it. Yeah, they were more, because we don't have this. Yeah, that was one. Surface alpha. The alpha. In the promotion. And so it just means that we have an action to start and to start on the surface. And the line, but this should be a current line for this action. And actually, the services we just did. So it's always be either. That's not the history. And then the line, but this would be something. One or minus one. So maybe I will end with three pieces as my time out. So I can just say, then please. And that's how we presented some. But anyway, maybe I stopped here. Okay, thanks. Thank you.