 So I have given this talk before Christmas and which was one of the four very fast. And so basically these lectures now the next few lectures I would say the same thing again, more slowly, and hopefully more understandably with a bit more background. Okay, so And so we work with numbers and then you want to study the quality. She's on services. Okay, so maybe a show and say, what's what's the space. I don't give it besides condition. So, so for me, this will be essentially right to your scheme. Okay, so that means if I take the points of the more like space over the complex numbers. What last is called M M C with just the, the objects we want to know about and naturally means that if we have a family of objects like family of objects. What is the family was born into this family. What family is just that the fiber over every point is one such object one study. And usually one to make the family one requires some property like that. But anyway, this is just me. And let's look at some so the morphism obviously should correspond to the obvious map at your point. The object. Now, as we talk to you know, so let's look at first my favorite sample. Let's just do it in the case that s is a smooth surface. And then the same with points like this. This is the zero dimensional sub schemes of this length of point in the other scheme for the general point. Any point on s when these points come together we have a micro scheme structure, which is given. So, in a sense, it's given structure she has a scheme which by definition says, this is his final thought over sections of structure she is the same as some old points and this number is. This is the same. No point is that one point. This consists of two possibilities either we have two distinct points, or we have a point and double structure, which is the same. This is the same vector, and then the board section will be as follows would take itself and blow it up, and then we divide by the action of symmetrical. So in general, was shown by progress. This is one thing that offers of the industry points. This is also mentioned. So, would you have I haven't been here. Not the size of what one life space and sometimes you have something which is called a modernized space, which means that it also has what is called the universal family obeyed. This is the case here. So, we have a family, which is a kind of line. So this is a sub scheme of this that is that's correct. Just describe it as incident correspondence corresponds to their point in the sub scheme. This obviously has to test a project by a second point. And this, this map is what is called flat for families, I can view this as a family of zero dimensional substance, thermatize by itself. So this is the same image on the P sub scheme of the point for one into a substance. So this is this example. I'm really interested in topological invariance of one life cases. So let's look at the brain. This is the word for the moment. You stick my attention to the real dimension. For the final number. And points to the end. All the events at once. That's also something one. I don't want to start with what life spaces might spend on some parameters some numerical parameters like this number in here. And then one could go. More less places for different values. Somehow related to each other, but for both that one can find a generating function. So later means. This lecture. I mean, yeah. Maybe one doesn't know that. She is a generalization, which was the vector boundary. So one could say new sheet as a vector boundary. And but that's no. I mean, I expect to look at one last piece of sheets. I want to first say so that we have the situation there. I want to report. She thinks. It does. One of the. Yes. From our measure. To some extent, how far is it from the trivial. See to the end or it's all to the end. This will be one or if you take a line bundle. You can have a line bundle associated with the visor. Just one. If I take a direct. So, you also want to use for. I don't think so. I don't know if you can look at the. This is just a form of linear combinations. And so it's not quite bad. You have to be my. Provence. We treat an exact sequence of vector. As a sum. And so. It's also equivalent to. That's G. If we have an X. Another. She's. Now, if X is non-singular. These two are the same. The same mutation. And this is because. Every. On X. Has a finite resolution. This. And we can. We do have. So I don't know whether. You know, maybe the mix of. I don't know. I don't know. So now we want to look at. So the first. So. You know such. We need to restrict. What she wants to allow for one last place. So. Somehow. Seeing it. It's really impossible to have some things. If they have. And so. And here the condition is that. She's not allowed. Such. That's the same. And. Each zero. So. So this is. And. The. The. The. The. The. And so. When you. Yeah. The. The. The. The. The. The. The. So. Obviously this only makes sense. To either. Find what needs to be known. This is the same as saying whenever. It's actually because it was put the employee. Saying that. You can replace the zero. Just by the space of section. To the sub sheet. And then. I think. The is a model. That's a. Each media just will. Surface. Given rank. 20,000. And. Each table. The human rank. And the current. What to mention here. There's also no stable. Stable it is. stable of the modelized space, which pre-modelizes the space. So I didn't, besides for the exist, modelized space, usually this will not be what I mentioned, a fine-modelized space. So it doesn't necessarily mean it was a family, but it sometimes has. In general, it's what's called a coarse-modelized space, into which one can, what I said here, objects, such that the modelized space exists, should have no non-trivial automorphisms. One can show quite easily, from this definition, the exercise that stable sheets are simple. So that means that the automorphisms are only not flying by constants. So this is one indication. But you know, why precisely? Same is stable, I mean, it's a long proof. So one has to somehow, you know, you can somehow parameterize what you want to do. So this is somehow a GIT problem. So you can have a suitable box scheme, if you know what that is. I mean, I think you know, but I mean, I don't know the audience, which parameterizes such sheets together with some vector space of sections, say. And then to get the modelized space, you have to be able to take a quotient with a set of GIT. And the notion of stable and semi-stable that I introduced here is precisely the notion of stable stability and semi-stability that one needs. That, you know, is imposed by GIT. Maybe just as an example, seeing an example, although it's not obvious, mainly if I look at the hybrid scheme, it is the modelized space keys of rank one equal to n, identified for comma or g over s with integers. This is by identifying with its idea sheet. So the idea sheet of a zero-dimensional sub-stream of length n, you know, it's an idea sheet, so a sub-sheet of O. So the Q, the GITs double-dual, this will be O. So the first term is zero, and it follows from the definition, works it out that the second term will actually be the number n. So how many points is different? So this gives some kind of set-theoretic correspondence to the non-trivial fact that the proof is actually isomorphic. So this modelized space is also something which is called the expected dimension, which is a strange thing, which however we do. Let me assume for simplicity that the first-party number is zero as per model test. It will be quite similar. It has many components of varying dimensions. But it has something, but it has an expected dimension right now. This is two minus, obviously this means the second set beyond the surface, C1, right by C1, we will evaluate if there's a number, so this is a certain number. So what does this mean? We have assigned this particular number to the modelized space. Again, it's the expected dimension in spite of the fact that the actual dimension of the modelized space might be different. So what do we mean by that? So it's somehow the dimension that M should have. And so, for instance, one can run an issue map. Mainly at every point in the modelized space, we have some vector space and then there's the run issue map to indicate which I could call the construction space. So you have this at every point in M, at every point in P in M. So the picture such that first is different. So the expected dimension I called the E of M. So that is the E of M because of the difference of these numbers, M minus K, and a neighborhood, an open neighborhood of our point P in M. So if it's a problem, it's the inverse image of zero. One could say that, and K are not space, so it depends on the point in the modelized space, but the difference is always the same. This is the virtual image. This is fixed. And so one could say that locally in the electric topology, the modelized space is obtained as a zero set of K equations in the M dimension of the prime space where the difference of these two is this. So if these equations were transversal, the dimension would be because the M minus K, so that's the virtual dimension. So if somehow the work was good, this thing would be smooth, would be expected. Now we will come back to that. We can see in particular it follows the dimension of every component of M. The thing that you should look at was if they are transversal, I want to maybe, which is somehow something you want to interpret on it. In this case, it takes S, and I assume first homologies assume that this adjusts the column of two forms on S. So there should be non-table form of two forms on S. Usually it will be two such as such as five and I mean more surfaces that will not be two for something like two. Now we choose a line bundle and we have, we can look and we choose also C1, so such that the modernized space now on the direction of space two C1, C2 consists only of spaces. So if you look at the condition, this actually becomes a certain condition of space. There must be certain things that we were able to define. For instance, this is the case, but it is not modernized. And then in that case, formula for the order numbers of these modernized spaces, which come from physics. I will maybe so which means this is just something like this for alpha then this is always meant to be I'd say the pairing it is commonly class the fundamental class of S according to the way we do it is the number. And now I will just make a simplifying something to make the formula nicer. Maybe later I will say the general formula of today's assume in the canonical linear system. So it means to assume there is a system here on the super which is the zero set of the polynomial this would, for instance, be the case if it is 0 for minimal and in that case, we have the formula which says 994 so we put the data we have already seen with the data function then create expression 8 times this I found this our series look like if I have the statement is the order number of the this is a the formula which says that the order number of the modelized space depends only on the holomorphic stress and there is a closed formula which does it for all places at the same time for all surfaces in terms of just this is rather formula there is a few problems with it so there are some issues I mean it's not clear to me whether it's actually true so maybe we have to be sure what we mean by it so we actually I'm convinced it's true in a certain sense but this is only if we interpret what we mean by the holomorphic and the second one that is not an issue but I can just say if you look at the actual paper there are more terms there are some more terms but these are not supposed to compute the order number of the modelized space but they are supposed to describe the variance of other modelized spaces some modelized spaces so there are other terms what these are I want to first deal with the first issue so what is wrong how do we interpret this order number something that I there's one thing I forgot to mention when we talk about but maybe I can say more precisely what I said or actually the non-singular to project the price and we have to set the parameter and we can associate the number to it so so we get intersections which are in particular the order number this way the order number the operation on the parameter the operation the number of singularities and so now we want to do this in our situation with the modelized space but we want to do it in such a way as if the modelized space was non-singular but we know it's singular somehow it's supposed to be behave like it's non-singular of dimension the virtual dimension or the expected of two things we need that we also these things like this we only do it's non-singular so that's it the long dimension we want to define the variance was non-singular so found this in terms of what is called the perfect structure so let me call it to let the scheme with an embedding obviously from how this is the case where it's projective space we look at the ideal she the ideal she why if you look at how long you have the standard exact sequence by square branch branch and the corner is the difference sequence and so you could say the dual of the tangent so you could say that the tangent space is somehow what prevents us from going in the direction inside m which we have this furnished picture at m should be something like we have the sequence tells us somehow the tangent space of this we can see somehow the potential space the tangent space here and you can somehow see and the other thing that we assume so this is the situation that had now we assume we have one thing more indeed that this thing can be described in terms of vector bonds that's two vector bonds e-1 goes to zero by a method d this situation here so we have with a more complex e-1 d with a more complex which is such that everything can be used and such that the following holds at this level one formality level is an isomorphism which is subjected to more precisely by taking p induced map from the cokernel of p here to the cokernel of p second is that the same induced map by p on the cokernel of p cokernel downstairs here the fuel of the cokernel is the tangent space we have vector bundles which from which we can recover all the tangent spaces and such that in the cokernel we have contained the obstructions somehow we can describe all tangent spaces and all obstructions in terms of just two vector bonds so this thing is for perfect obstruction sometimes one says one perfect obstruction but then in this case and then in this case we can I should maybe say with respect to dimension which is the exact dimension to the difference of the range of the two vector bonds so I should have said this now in this case there is something which is nice in the homologous vector dimension you see it's a bit more it's also behaves in super ways like the fundamental class that you have with you now and it's also what is called structure sheet it's actually not a sheet it's just an element but which again has some properties which behave like the sub structure sheet of the smooth variety you all the time I don't see what there are in the position of the other number and it's looking nice I mean as we are there to see the Euler number it's just one more minute I mean I will read that especially but then so this is somehow like this replaces the potential so it must be the dual so this is therefore just the class E0- bundles on M the tangent bundle is no longer the vector one it's a complex and then the virtual Euler number I mean you just write down the top index 0 for the Euler number you see it's integrated with a virtual fundamental class I forgot to modelize this but what we have done is we just have proposed that with this new definition of the tangent bundle we have this top index this is the virtual definition of Euler number so it's rather more complicated but then it's actually reasonably computable and then it will be nice anyway so this is the definition I mean in some sense it's not quite the definition but I have to explain to you how this works for our amp so what is the abstraction here for the modelized space of sheets but anyway a few more yeah that's the number that's the number I guess we can not make a movie but maybe you can do a recap of the last one on the what on the last page from your amp okay okay I mean we can yeah okay