 Thank you. I like to saint organizers, they have done a marvelous job. And their generosity in time and attention to details and love for Gabber are something very impressive. I also like to thank Professor Gabber who's whom I have been in touch probably in the last 10 years. We had a common interest in purity results and this is joint work with Professor Gabber. We have two manuscripts. So, I'll try to combine. And I like to begin with an introduction. R is going to be a local regular ring, mixed characteristic, 0P, and dimension D, which we like to be at least 2. And we'll denote by x, the speck of R. We'll also need the puncture spectrum. The maximal ideal will be denoted by m sub R, the close point. And we'll also need the residue field, which is of characteristic P. So, I like to begin right away with the definition, what we are talking about. We say R for x is a p quasi healthy, if each visible group over y extends to x. And we have an analog notion for abelian schemes, p healthy, if each abelian scheme over y extends to x. Of course, in the A case extends as a visible group, and in the B case extends as an abelian scheme. I'd like to mention a few things, why do we care about sarsinks? In 91, in the book, a little bit, sure, thank you. Or maybe, oh, this doesn't go higher, OK, then probably this is the maximum. In the book of Professor Falkens and Chai, it was claimed, basically, that each R is both. And definitely this is not OK, so there has been a mistake. And there are at least two gaps and probably some smaller problems. This was pretty much after the work of Moret Bailey on an analog result for curves. So, in the intuition, if it works for curves, it should work for abelian schemes. It's quite tempting, but it turned out not to be the case. And second thing, that if M script is a good modular space, and let's say that x is p quasi healthy, then these two together should imply that the y-value points should be the same thing at x-value points in extension result. And in this way, one would like to prove the unigness of good modular spaces, such as integral canonical models of Shimura varieties over some DVR v, which, of course, is of mixed characteristic, but we want the index of ramification should be at the most p minus 1, because the most we can do about getting some udingness results. So, this is why we like to come back and work again on this area. It's something you could say classically. It goes back to 27 years, but because we have something which is not OK, we like to see what can be done. And, of course, the definition is much later, and the word healthy was kind of trying, was going on, sort of trying to model and ponder what's going on, and what R could still be worked out. Now, simple thing is, we could say, proposition 1. We can state it in two parts. If V is equal to 2, then R is p quasi healthy, even only if we have a statement of the following type. For each complex of the form of this type, of course, of finite, flat, commutative group schemes annihilated by a power of p. And I maybe would like to give a quick name to this to make it shorter, OK, over x. So whenever we have such a complex, star is a short exact sequence, even only if it's pulled back to the puncture spectrum, is a short exact sequence. And B, if we have d greater or equal to 3, then we are going to have basically only one implication. So what is behind the case d equal to 2? Yes? Is there an implication between the two notions of p healthy and p quasi healthy? That will be proposition 2, OK? Yes, thank you. So you see, when d equal to 2, if you have the category of this finite flat, I'm going to make it short over x. We can restrict it to y. So we have a natural restriction functor. This is very well known that it's going to be an isomorphism of categories and equivalence of categories. And therefore, for d equal to 2, this implication is going to follow easily from the very definition of visible groups where we have an inductive system and some short exact sequences are supposed to be satisfied to be in these short exact sequences. Now, for d greater or equal to 2, if you have a complex such that it's pulled back to y is a short exact sequence, but OK, so it's a short exact sequence. Oops, sorry, this has been inverted. And we have a CRM of Rhino from the book of Brine, Bertelo and Messig that we can embed g into the visible group on a billion scheme. And we can restrict it to y. And we can form this quotient. This is going to extend to an abelian scheme over x if and only if the complex is exact. In other words, h becomes a close up scheme or subgroup scheme. And of course, this means that star is going to be a short exact sequence. So this proposition kind of give us an idea what we like to do in order to find counter examples. We could try to look at situation where when you have the x, let's say in the FPPF topology of k and h mapping by restriction into this x1 of k over y, h over y, as long as this is not on top, then we can find the class here, which can be used to define first a short exact sequence over y, is going to extend to a complex over x, because we are in the case d equal to 2. So therefore, if this is not subjective, automatically we have a counter example. Counter example to the claim, because from our point of view would be examples basically. So that's what one can try to begin with to find counter examples. And I mentioned to you that this is a proposition 2 as well, that if r is p quasi healthy, then it is p, it is quasi healthy. Ups, boje, je, it's fine, it's fine. It is p healthy. Had the impressions that I forgot something. If it is, then it is p healthy. This is essentially done in the book. All the ideas are there, because the argument aims to so much in reality, it's not that easy to follow. There's a more recent proof of mine from 2004, but entirely inside is much shorter, of course, because we just concentrated in what we need it. So definitely we have this implication. And next I'd like to start mentioning the first counter example, which is due to Reino, Professor August, and Professor Gabbard. Well, OK. So this thing is due to Reino, who mentioned it to Professor August, who mentioned a few details to Professor Gabbard, and then a letter of Professor Gabbard to Professor August. May I say, I'd like to say something about it because of Reino. I was at the Institute, and I was working on this question. I needed it for some reason, which isn't so important. And Mumford came to give a talk. So I asked Mumford if it was true, that is, that every billion scheme could be extended in this context. And Mumford, after the next talk, he came up to me and said, you ruined this talk because I spent the whole time thinking about this, but I couldn't solve it. So Reino came, and I asked Reino, and he made a counter example very quickly. Wow, OK. So this is OK. So the letter is from 92. So this must be a talk in 91, something like that. OK. So we take d equal to 2. We take r to be strictly Encelian. And we want each irreducible component of the closed locus of p to be of multiplicity, of multiplicity divisible by p minus 1. And we want at least two of them. Yes, so each, so there are at least two of them. And the letter even gives an explicit example. We can take the ring of bit vectors over nudge by closure of a finite field, formal power series into variables. And we divide by an equation. We take exactly two reducible components. And multiplicity has to be d0 by p minus 1. The smallest is p minus 1. Now, what is going to be in this case? H is going to be z mod pz. And k is going to be mu sub p. So I repeat that we are looking at a specific x1 map. So we are looking at mu p, z mod pz, because it's strictly Encelian. This is going to be 0. And it means by restriction. And we don't want to be subjective. So, therefore, this being 0, we want to have something here, which is non-zero. Now, such or exact sequences are splitting locally in the tau topology. So, therefore, this becomes an H0 et al over y of the home from mu p, z mod pz, because I put y there. I'm not going to repeat it here. Here, I mean, H1, z mod pz, x1, x1. OK, yes, x1, sure, thank you. So, yes, it's x1 as above. So we go to H1. This is easy to compute. We can consider the jota from y minus the close locals of p equal to 0, open into y. So, therefore, we're going to have jota lower shrik of z mod pz. I'll mention in a second that this condition are going to imply that we're going to have a primitive root of a unity of order p inside the ring. And so, we can rewrite it down like that. We also have a short exact sequence on the tau topos over y. And here, we have the irreducible components of v of p equal to 0. There are s of them, as we mentioned, is at least 2. And, therefore, we get z pz over these generic points in characteristic p. So this is short exact sequence. So we can use it to compute the H1 et al. It's mapping into H1 et al of y z mod pz, which, due to the purity, is going to be 0, OK? And, therefore, it's going to be isomorphic. With the co-kernel, we get the level of H0s, OK? And direct sum i going from 1 to s in, say, eta i z mod pz. And this is isomorphic with f sub p to the power s minus 1 and is different from 0. So this completes the counter example. So next, we like to move towards the classification for d equal to 2. It's a complete classification in the Henselian case. In the non-Henselian case, we have some partial results. It's not that easy to build up finite flat group schemes over, let's say, finite regular over z, z localized at p. So, therefore, passing down from the Henselian case to the general case, we need some more work, OK? So, I'd like to begin with the CRM, it's the CRM, OK? So, I'll need some notation for the completion. The completion is going to look as follows, divided by an element f, a regular element. And the CRM is in three parts. The first part is due to my self-hensink in 2010, which basically says that if f doesn't belong to the ideal generated by p, the x to the power p, y to the power p, and this product, then r is p equals to health. The b part, which is due to Professor Gabar and myself, this is one of the manuscripts we are talking, which is completing the Henselian case, but needs some little bit more work in the non-Henselian case. So, therefore, we are going to assume that r is Henselian, and f does belong to this ideal. So, in this case, we like to say something more. This has implication, as we can see, in different kind of homologies, and, therefore, it's kind of good to say some more details. So, in this case, there is going to exist a g, a finite flat over x, with its special fiber connected, so that we have a couple of more properties. The first property, because in the case of short exact sequences, we can just restrict to the epimorphism part. So, there is a homomorphism from g to mu p. We like to keep the same mu p, which is not epimorphism, but let's give a name to it, alpha. But when we restrict it to alpha, it does become an epimorphism. Second, the order of g is either p to the power 2 or p to the power 3. And the third thing, we can say something about h as well. h is, how can we define the h? Of course, it's going to be the extension to x of the kernel of alpha sub y. So, we can define it directly, but I'm keeping the same notations. H is either a form. We cannot make it all the time z mod pz. In the previous case, it was possible because of zeta sub p being there. So, it's either a form of z mod pz or a bt1 of height 2 and dimension 1, whose fiber over k is supersingular, supersingular bt1. And, of course, if you combine a and b, what one gets, that in dimension 2, suppose r is henseljen, then r is p quasi healthy, if and only if f doesn't belong there, or sometimes it's convenient to do the reduction mod p of f. f bar doesn't belong to the ideal generator by x to the power p, y to the power p, x to the power p minus 1, y to the power p minus 1, which is an ideal of this ring of formal power series. Is not, thank you, sure. Yes, thank you. And f bar depends only on r up to automorphisms and also up to units, this way to multiply by units because we're dealing with an ideal. And we call it the coon element of r. And it can be defined in arbitrary dimension. And I repeat, it's unique up to units plus automorphisms. So this is a complete classification. And what is the essence of the b part? Part of the a part is also going to show up when we go to the proof of this proposition 3. OK? We like to do it more general. So let r script be a Henseljen loka ring of residue field of characteristic p. So suppose there is elements x, y in the maximal ideal of r script, elements a, b in the ring r, and element c, which is either 0 or a unit. That's kind of a major restriction. We denote the units by u. Sarset, we have an equation of the following form. So we can express p in terms of x and y, and these coefficients a, b, c as follows. So then what is the conclusion in this case? Zenselj, going to exist g script finite flat over x script, spek of r script. OK? Sarset, the following properties hold. First of all, g sub k is connected. And moreover, the analogs of i2 and 3 are going to hold. OK? First is that we are going to have an alpha. I'm not going to change the letter now from g script to muza p, which is not epi, but is going to be epi when restricted to x script minus the spectrum where x and y are both 0. OK? This is the first condition. Second condition is refers to the order. We're going to have the order of g equal with p to the power 2 if c is a unit. And of order p to the power 3 if c is equal to 0. And the third condition is going to say something about the kernel. Of course, the kernel is going to define only initially over this one, but it's going to extend. So we can speak about h script. And h script is either a form of zemot pz. Of course, it has to be the order p to the power 2. Or, again, bt1 of dimension 1 in h2. And the same properties that the fiber over the field, the residue field, we can denote it again by k, is going to be connected, and is going to be super singular. And there is also complement of it. If I change the pips minus 1 root of c exists in the ring, then h is going to be actually isomorphic with zemot pz. So we don't have to kind of go around for a form of zemot pz, but actually it's going to be zemot pz. Now, how one would prove such a proposition? So I'd like to concentrate on the case when r script is complete and regular. And moreover, we have x and y as part of a regular system of parameters. So we can write down r script something which involves formal power series in a finite number of variables. Beside x and y, we need z1, z2, zd minus 2. And we have to divide by a similar f. And that f is basically nothing else but the equation which we have here, which I could rewrite it down again. So next one we like to consider a frame. We can consider s script. We can choose a nice Frobenius lift of it. For instance, as a sigma of t is t to the power p, for all t running through x, y, z1 up to z, d minus 2, and extending Frobenius of the cone ring of k. And one builds up in this way with zinc and law calling frame. We have the ideal f. We have r script. And then we have z sigma. We also have a sigma dot. And we have sigma of f. So this is going to be a frame in the sense of law 2010 and previous works of zinc as well. Many frames can be used, but this is basically the one we're going to need here. And sigma dot, it goes from f to s script. And it's going to be given by some formula as follows. It's not going to be important for this talk to know much about these frames. And we're just going to use nil potent broj modules. This is the terminology zinc and myself have introduced on our world generalizing broj's conjecture and a kissing result on the classification of finite plate group schemes over this evaluation ring of mischaracteristic, complete and with a perfect residue field. We work for families where we had Eisenstein element in multiple variables. And the law was able to do the general case following quite close the work of zinc and myself. So now I'm going to introduce some matrices just to get the flavor what is involved. Subcase, when we have c equal to 0, we'll also need to introduce s script divide by p. So this is going to be. So we like to start with a matrix, which is 3 by 3, with coefficients in s, which has the following property. And with the determinant of a0 being equal with f bar to the power 2. With it, we can construct nil potent broj modules as follows. We view s to the power 3 as a free module over s of rank 3 and so on. This will be the Frobenius pullback. We choose the standard basis on both sides and we take the linear map, which defines a0. Here we're just going to choose the multiplication by f. And here is going to be the map, which will be given by this matrix. And here we have the p power, the Frobenius of the previous map we had as a vertical left arrow. It's easy to check that the identity we have there builds up this commutative diagram. And basically, the theory of law generalizing the work of Zink and myself allows us to pass from such a diagram to an alpha, which is nothing else, but the one we want from g script from up. In this case, g is going to have order p to the power 3. Now, the question is how one builds such a matrix, which will have all the properties. How one builds such a matrix a0? One can work co-variantly or counter-variantly. And therefore, to make it exactly as in the case of law, we are going to first define another matrix and then kind of take a joint in the transpose. And one needs to choose alpha 1, alpha 2, alpha 3 so that the determinant of a, it's exactly a bar. And there are several ways to do that. One relatively nice is to take alpha 3 equal to 1. Alpha 1 is going to be minus b times 1 plus x to the power p minus 1, y to the power p minus 1, everything to the power minus 1. And alpha 2 is going to be a bar times 1 plus alpha 1, y to the power p minus 1, where in the local case, x and y are in the maximal ideal. So, therefore, all these inverses are going to exist. And now one takes b, the joint of a, and a0 is going to be the transpose of b. One has nice identity, it's a b times b a. The last law of the matrix is bar x. It's top c, this is related to the second term. I hope this is clear, it has to be three terms. So, this is going to be nothing else, but a bar times the identity matrix 3 by 3 and the same thing a0, b0, and b0, a0. And, of course, determinant of a0 is a determinant of b, and determinant of b becomes easily computable as f bar to the power 3 divided by determinant of a, so, therefore, becomes the second power. And the fact that we have such a sink implies that the co-carnals of these two maps are annihilated by f bar. And, of course, the determinant of a0 is equal to f, the same thing holds for the map defined by a0. And, also, if one look at a mod x, y, what we get, we're going to get zeros on the first two rows, and therefore, sorry, the first two columns, and therefore it's going to be of rank one, which is going to imply that the joint, also, is going to be zero, mod x and y. So, therefore, when we take the transpose a0, we're definitely going to get something which is nilpotent. Now, we have also the subcase, where c is a unit, and the same story is going to be, except that we can make it 2 by 2 matrices, which is very good. Every single is going to be the same, multiplication by f bar. Here is going to be the matrix a1, and here we're going to have just x, y. And, of course, x to the power p, y to the power p. And I can also write down who is a1 in this case. A bar, here I didn't put, I mean, I put a bar here, a bar and b bar means reduction. I mean, would be basically leaves to s, which modulo r are going to be those elements. So, a bar b bar. And here, you see, we already left it here. When we wrote it down f in this way, we implicitly left it and then left, we denoted it in the same way. But when we take a modulo p, we are going to denoted by bar. So, let's complete the matrix. Plus c bar, y to the power p minus 1, a bar, y, and b bar, x. And here, b bar, y plus c, x to the power p minus 1. And in this case, the determinant of a1 is going to be c bar times f bar, but c is a unit. So, everything is going to be annihilated, the co-kernel of what we get here is going to be annihilated by f bar, because this is a unit. OK, so this is basically how one proves this case. And the general case is pretty much standard. One considers some, you can push this one up first, universal rings, which are going to be regular for zlokalizat p, so for which the approximation, artyn approximation is going to be true. So, therefore, from the completion, we can go down to the Hanselization. And because we have universal rings, the general case of a Hanselian is going to be pulled back of the Hanselization or localizations of the universal rings. So, of course, the universal rings are kind of obvious. I'm just going to mention, for instance, when c is equal to zero, we can put zp, we can put two variables and divide by f where c is basically zero, so we get rid of it. It's going to look, of course, we have to put also ab. Let me not forget that part. And something similar, when c is involved, this would be a universal ring, and all our scripts are going to be basically pulled back of Hanselization or localizations of this ring. So, that's how the proof goes for dimension 2. So, now let's move to higher dimension 4. Let's understand what is the trouble for dimension 3. This is a second gap, you could say, in the book I have mentioned before. So, let's consider the simplest case where r is w of k. It doesn't matter if it is algebraically closed, perfect so that I can keep writing w instead of c. We can put two variables. So, it's of dimension 3. And, of course, we can consider portions of it where we divide by powers of y, r divided by y to the power n. Of course, it's regular only for n equal to 1, but what we define p quasi healthy would make sense in any case we have something which is a lockering, and what is the fact here, which is surprising, is p quasi healthy if and only if n is equal to 1. So, let's try to understand a little bit what is the problem. We can even kind of consider counter examples, which are coming from elliptic curves. Let's use x as an example. So, let's try to understand counter examples, which are coming from elliptic curves. Let's use x as a variable. Of course, back of it, sorry. We can get the map from speck of r tilde n, where we invert x, and at the level of the rings, we like to map x into x plus x inverse times y, or y to the power n minus 1, just to kind of basically the last one, which is non-zero. We have here even in a billion scheme, but we can concentrate on this visible group. It pulls back to a visible group here. If you look at the puncture spectrum, it's going to be the union of what we have over there, the localization, and we're going to have something coming from speck, some locating o, y divided by y to the power n, and this o is a dvr of characteristic zero. So, therefore, by triviality, because we have nilpotent, what we get here is a visible group, and, therefore, what we get over o automatically extends to this part which has nilpotent elements, and, therefore, the visible group, which we initially get here from the pullback, is going to extend to all of it, but it's very easy to see that there's no way, because of this x inverse, there's no way it's going to map into r tilde n. There's no way to factorize it. So, therefore, where the mistake in the book shows up, r tilde one is p quasi healthy, but when we like to leave to r tilde two, the argument of, okay, torsor, so visible group doesn't work out, and the reason doesn't work out, okay, zero dimension two, but all the deformation theories we have learned from visible groups require a power of p to be zero, and, therefore, when we divide by some power of p, no matter which power we choose, we get dimension one. So, there's no way we can, outside something of dimension one, outside of the punctured spectrum, we cannot extend it. So, this is the problem, and, therefore, many claims in the literature that there are quasi healthy schemes in dimension three, or, oops, no, this one I need, and this one Lord, thank you, or the other way. No, you're fine, you're correct. So, I have enough time to at least say what's happening in dimension three, or higher, and... Did you define quasi healthy only for regular rings, though? Well, I mentioned that it can't be defined for any local ring, and I also use it here when I made the last fact, stated above, I meant it in the same definition. Okay. But as you see, as soon as you have nipot in elements, there are no much hopes to get quasi healthy, so you can define it, but it's not going to be something very useful. Okay. So, now let's move to dimension three, or higher, what is the main result for d greater or equal to three? We have zero m2. Okay. So, of course, suppose d greater or equal to three, and similarly, we can consider the completion, and the completion, we can write it down, x1 up to xd, divided by some element. We can denote it by f again. Okay. So, suppose d and we write, this is actually not an assumption, but n we write it in this way. If f doesn't belong to the ideal generated by p, the p is powers, and 2p minus 2 power of the ideal generated by x1, xd, then r is p quasi healthy. This includes all the formal power series over a DVR of index of amplification at most p minus 1. So, we recover everything, we need to get a new proof of integral canonical models of Shimura varieties, the first proof being by Zink and myself, which found a way to go around this difficulty for d greater equal to 3. So, I'd like to say something about what are the new ideas involved in proving this. There are basically three of them. It's a result of Rhino, which I don't know if has been used since 74 before, and also we aim to get good sections. Of course, the good sections are required for d equal to 3, so that we can use the results for d equal to 2. And so there's a very nice constancy theorem, and also we have quite a lot of more or less standard standard tools. There are plenty of them. For instance, we can work out in the faithfully flat topology, and we can assume that we're going to have an algebraically closed field k, which is also uncountable. We really need it in the argument, and okay, so x1, xd over f. It becomes our R, we don't need to completely because we can use faithfully flat descent. And also we can actually reduce to k is d equal to 3, because okay, the difficulty I explained to you for d equal to 3 are not anymore, the grotendik messing deformation theories take care of the passage from dimension 3 to dimension 4, and so on. And this is where we need k equal with k bar to be uncountable. So now I have two, three more minutes, I would like to say a little bit more first of all more general statement than this. If R is as in CRM2, then a principal group over the field of fractions of R extends to R if and only if it extends to all dvrs, which are, of course, loka rings of R. It extends to all dvrs of x. So this would be basically what would say kind of a strong p p healthy sort of because I define p quasi healthy, but this would be basically p healthy. This is why the quasi shows up. We like to kind of move towards the global results and one reduces. Now what is nice about this condition here, that this is sorry, here this is stable under generalizations and there are very nice tools of proving this, which I don't think they are dead standard, but anyway, and because of that the two statements are equivalent. So now just maybe because it's such an honor to remember Rhino, I'd like to briefly keep your attention for one or two more minutes just to say where at least Rhino si orem pops up and which one because he has so many. So what is the idea that one considers the blow up of x along the close point, which is the maximal ideal. This works for all d. So we have a projective space sitting inside is going to have a generic point and it's a dvr, the lockering and of course how one starts with the d over y a feasible group and one we like to use the blow up somehow to show the d extends to x and the first thing is to show the d extends to p. This is at any point. Sorry. Yes, in z, of course. Thank you so much. So we have this feasible group and the first thing we like to extend is to p. How we do like to do p dominates r but in between there are many rings. Let's write it the level of the spectra. OK. And now we can choose p1 to be of dimension 2 so that pullback of y would be sort of y prime. So therefore all the truncations of d are going to first extend to p prime and thus also to p by pullback and now the result is if we have a feasible group over the field of fraction of dvr of mixed characteristic and if all truncation extend to finite flag group schemes over the discrevaluation ring then even the feasible group extends. No, no, you don't need anything. This is in the paper of 74 of professor Reino and I don't know if it has been used before if anyone is aware of any other application of this old result it was a complement where he did all those extensions when a feasible group over the field of fraction extends to p to the dvr and at the end he has this very nice complement as application of minimal and kind of maximal extensions of some given finite flag group schemes. So this is first of all that's how we win all the dvr of z and the section has the role of finding a good way to divide by one variable which is going to avoid all the sort of bad points about which we do not know what will happen and we can apply dimension 2 and I would like to say something more about the constant c theorem which is a very nice result but that's what the time permits thank you for today. Questions? So basically yes sure you see once we have a PDVL group which extends to p we are going to concentrate I repeat for d equal to 3 and 4 is going to exist as a band inside the projective space which now becomes a projective plane finite set so that we have finite flag commutative group scheme over z without sn because it extends also in dimension 2 so therefore what we are losing just to finding number of points now we are going and we can make them nicely one included in the other what we are talking about as infinite the union and the notion of Barca de Te group is open so therefore we can also find t1 t2 and so on tn which are going to be finite finite union of curves and we get t infinite for their union everything is sitting inside and the section refers that we like to divide the ring r by an equation of the form t1x1 plus t2x2 plus t3x3 we can assume that x1, x2, x3 are coming from regular parameters with t1, t2, t3 belonging to the ring of the vectors at least want to be a unit regular parameter so we get something of dimension 2 and in the blow up we get the strict transform of this up t which produces a p1 so this becomes a p2 so what we want we want p1 of k to avoid this countable set of points and we also want the generic point should avoid t infinite and this is possible as long as we go to uncountable k and this is relatively standard argument to kind of avoid these things it's not the difficult and now when you pull back the visible group over y we have a yt here which is nothing but the punctured disk the result on dimension 2 we can assure that x up t is like what we have in dimension 2 p quasi healthy because this condition will behave with suitable sections you have, if you look here 2p-2 would be exactly p-1 plus p-1 so we are in the right range and we have then a visible group over yt and it's going to pull back to z t it's going to be the pull backs of all the finite group schemes we get over z-sn and that's basically how one can conclude that actually this is a finite set this becomes a finite set because over p-1 p-1k t which depends on t we get a visible group so therefore this curve is going to avoid all the t infinite and now you are in p-2 if a curve avoids p-1 has to be the empty curve basically so therefore this becomes a finite set and that's basically where the constant is your m is going to follow that if you have a curve in a projective space and we have this finite basotite groups which pull back to something which become constant over a curve then automatically are going to extend to the whole projective space and it's going to be constant it's a very kind of nice result using the paper of hironaka mura on rational formal functions and goes back to 68 and that's a very delicate work and that's basically what goes next we're going to get something constant about p-2k and then we are going to do the completion of p-2k in size z all of them are going to be constant and then of course we end up also using the algebraization c or m of grot and d and so on but those are more standard and clearly this is what is the section we want to avoid the set as infinite and the generic point should avoid t infinite and then you win the game basically more questions? sorry you're not on your seat so I didn't see you this result of it's not in the ppt it is it is no that's why I mentioned the same year 74 yes it is there and I have not seen it applied yes other results of course yes so it has been a lot of work all together with ratios margin come 60 pages to fix one line from the book so I hope you enjoy something which is classical it is very nice to have new theories but sometime we have to go back and fix old theories so what to do so it seems that Alfred does not have a question on the paper of joint between himself so it is about your work so definitely we have worked out all details it is not like work in progress or something just starting no more questions that is time to speak again thank you