 I'd like to introduce Professor Sato from Kyushu University in Japan, and he's going to talk about the general hyperplane section of canonical three-folds and positive characteristics. Thank you very much. Thank you very much for your introduction. Can you hear me? OK, so thank you very much for your introduction, and thank you very much for giving me to talk such a great place. Thank you very much. OK, so let me start my talk today. OK, so my talk is about a beltine-type cell hem in positive characteristics. So let me recall the very classical cell hem in algebraic geometry, which is called a beltine cell hem. OK, so maybe if you study algebraic geometry, then beltine cell hem is very, you may know this cell hem because this is very standard and basic tool in algebraic geometry, which says that let X be a projective variety of algebraically closed field. And since X is projective, it is embedded in projective space. OK, and let H be a general hyperplane section. So for example, if X is like this in some projective space, then OK, so we can take hyperplane in PN, which is isomorphic to PN minus 1. And we consider the intersection of this hyperplane and X. Then this is a hyperplane section, H, plane section. And OK, so general means we learn this hyperplane very generally. So if H is sufficiently general, then we can show that if X is smooth, so is the hyperplane section H. OK, so this is a very strong tool in algebraic geometry. Because the dimension of H is smaller than the dimension of X, then this is very convenient to consider induction dimension. So this is very good cell hem, and very well known. And there is also a variant of this cell hem, which is called a built-in type cell hem, built-in type cell hem, which says that for the similar setting, for the same setting, so if we consider the case when X is normal, then so is H. And moreover, so for example, if X is regular incodimention N, or satisfies sales ASN condition, or many similar properties, local properties, then such property can be, the general hyperplane section also have the same property. OK, so this is built-in type cell hem. So today we consider built-in type cell hem for log canonical singularities, so instead of normal or RN or SN. So today we consider the questions about singularities in MMP. So singularities in MMP is several classes of singularities which appear in MMP. For example, log canonical singularities, or KLT singularities, or there are several classes here. So I consider singularities in MMP. So let me recall the, what are they? Log canonical singularities, so KLT or log canonical singularities. I will recall these notions. OK, so these notions are, so first defined in, so sorry, classes of singularities, singularities defined in terms of resolution, resolution of singularities, of singularities. And second, so these classes are very important in minimal model program. OK, so minimal model program is a program which, very roughly speaking, so which obtain a very good projective variety from, given some projective variety. And the program is not closed in smooth variety. So even if we consider smooth projective variety, but if we run MMP, then it may admit some singularities cases. And such singularities appear for the output of minimal model program. And thirdly, these singularities behave very well in characteristic 0. So this is very, so for example, KLT is very good with some co-homology, balancing of co-homologies in characteristic 0. But so in positive characteristic, the situation is more subtle. And it may, there may be several pathologies in positive characteristic. Finally, so for example, in dimension 2 and characteristic 0, the KLT is nothing but quotient singularity. OK, so this is a very rough explanation of these singularities. OK, so OK, so then I will write the main question again. So the main question is this, but OK, so I will, I write this again. So which says that today's main question says that if X is projective normal variety, over algebraic I closed the field, with and H be a general hyperplane section in section. OK, so the question says that if X is KLT, so X has only KLT singularities, or X has only log canonical singularities, then so is, so this is the main question today. OK, so there are several non-results in characteristic 0. So for example, if the dimension of X is smaller than 3, smaller than or equal to 2, then this is OK. It's very easy to prove in this case. In higher dimensional, so we first remark that lead, lead proves that if characteristic of the base field is 0, then the main question is OK for both KLT cases and LC cases. But in positive characteristics, so there are not so many results. So the only one result was proved by Shunsuke Takagi and myself, which says that if dimension is 3 and characteristic is larger than 5, then KLT case of main question is OK. So this is non-results. So I don't explain this result, but I give some comment for this result. So we use X singularities to prove this result. So this is somewhat X singularity type theorem. But even in dimension 3, so there is no result about LC case. And OK, so I want you to prove the LC case. But OK, so I can recently prove the LC case, so it says that if characteristic is larger than 3 and the dimension of X is 3, then main question is true both for KLT case and LC case. OK, so this is today's main theorem. So there are two different from the previous one. So one different is so the LC case is contained in my main theorem, today's main theorem. And the second is that in this paper, so we need to assume characteristic is larger than 5. But this theorem also proves in characteristics. So this contains the characteristic 5 case is OK. A little procedure. OK, so it's a unique question, OK? So I will give a sketch of this theorem to prove. Basic idea is very standard. So we consider a deformation for tackle this problem. So let T be a projective space. So there exists a projective space T under the closed sub-scheme that of the product X times T, such that for any T, the fiber P2 is the projection to T is hyperplane section, hyperplane section. And actually, so this gives a bijection between the closed point of T and the hyperplane section, the set of hyperplane sections of X. This is bijective. This is somewhat well-known, well-known and very standard argument. And we consider, so for proofs of a routine type theorem, this hyperplane section is nothing but the closed fiber of this flat family. So we need to check the properties for this flat family. So let us consider the generic fiber, which is the sub-scheme of the base change of X to the function field of T, which is generic fiber. And we also consider the geometric generic fiber, which is the base change of the generic fiber to algebraically closed field, which is the closed sub-variety of times KT bar, base change of X. And I also remarked that this is three-dimensional, so this is two-dimensional, so two-dimensional variety over some fields. Then the idea is that we can show that, first, since X is KT, I explain only KT case now because LC case is very similar. So now, let us assume, since I assume X is KT, so we can show that the generic fiber is KT and the generic fiber is KT in price, the geometric generic fiber is also KT, and this in price, general closed fiber is KT. And this is what we want, this is what we want. OK, so this is a general strategy, but OK, so what is the difficulty? OK, so this part is not so difficult because this is not so difficult because natural morphisms, sorry, geometric fiber, the first projection, Zeta to X, is regular, so this morphism is a very good morphism, so the property, KT property, ascends to Zeta, OK? And this is also OK because this is standard argument. The difficulty is this part, OK? So in positive characteristics, so this operation, so the base change of variety over non-perfect field to algebraic closure is very, sometimes it is very difficult to treat. And this is very surprising, but I think this is very surprising implication, but in this situation, so actually something like this implication holds. So this is my next main theorem, so I will write that. OK, so let X be a two-dimensional geometrically normal variety over some field of characteristic P, which is larger than 3, OK? And let us consider closed point. We assume first, so the residue field of small x is separable over the base field. And secondly, so two-dimensional variety, X is KLT around the closed point A, OK? Or in LC case, so in log canonical case, we assume that X is log canonical and so some assumption. So we need some additional assumption, but I omit here today. OK, so then the base change X to algebraic closure is also KLT for LC. So this is my theorem B. And OK, so here we apply the theorem B to prove this implication. Any question? So OK, so the last five minutes, I will explain about theorem B. So because this is two-dimensional, because this is three-dimensional. I also say that this assumption is also holding for this. This is also non-trivial, but all assumptions satisfied in this case, OK? So I will explain theorem B, but to explain this theorem, so I think so why this, I need to explain why this theorem is surprising. So you have to explain it. We consider the count-type example in Characteristic 3. So let us consider Characteristic 3, which is not covered in this theorem. So this theorem holds only in Characteristic larger than 3. But if we assume Characteristic 3, OK, so let us take some element A. And L be the purely separable extension. So let us consider the variety surface, which is defined by xq minus yAyq plus y10, which is a hyperplane of A3k. Then we can show that x is geometrically normal and KLT. The base change of x to this purely separable extension, xL, is not KLT. I didn't define the definition of KLT today, because the definition is in terms of log resolution, resolution of singularities of x. So we need to explain this example. We will consider resolution of x, x and xL. OK, so we first consider the resolution of x. So x has one singular point, which is the origin. This is the singular point. So we blow up this point. Then we obtain some surface, another surface. And there is exceptional curve, which is isomorphic to P1k. Exceptional curve. And also there is a singular point, unique singular point, which is I write this P. But the residue field of P is actually L, which is a purely separable extension of k. So we finally blow up P. Then we obtain such a variety. So the exceptional curve is P1L, because this is L rational point. And this actually gives the resolution of singularities. So x2 is smooth. So this is the resolution of singularities of x. But so we consider the base change of this morphism to L. Then what happens? So this part is very bad in this case. Because of time, I can't explain more. So this is actually not smooth. So that is so in this case, the base change of resolution of x is not resolution of xL, base change of resolution of x, resolution of xL. And because KLT or LC is defined in terms of resolution of singularities, so this is very large abstraction. And actually, so in this example, so we can say such pathologies. So this is what happens in Characteristic 3. But I give some comments. So but if Characteristic is larger than 3, then such bad example does not exist, does not exist. So the base change of resolution is again resolution if in this assumption. So this is somewhat surprising, but this is proved by the classification of dual graphs of KLT singularities or LC singularities. OK, so I'm sorry to, I can't speak more. So thank you very much. I have any question for Professor Sato. Then I would like to thank him again and for others.