 So, these appear, and then it's, yeah, of course, R, and sorry, okay, excellent, oh. It's a meaning, since everything is E, A. And then, end by tensoring, and one more thing. So, consider the map A to A, R, A, so this is a generically flat. That means, so, localizing by one thing, so adding, so, A inverse, we can assume, also, yeah, yeah, R, A is flat over A, okay. And so, we call this R, A, a model, model of R. And then, to take any maximal ideal of A. Then, of course, A over mu is a field, finite regenerated over Z. That means, this is a finite field. And, of course, characteristic is some P. Going to consider some property, so P, which is preserved under a basic extension field. I don't say explicitly a rational, a pure, et cetera, is preserved under a basic extension field. But, that's, and so, this, yeah, okay, so, the property P has P does not depend so on, so, so, individual at only P. So, choosing some A, so, we, for some mu, that makes models very simple, okay. Okay, so, and we have two, so, we say, so, for some P, open P type, except for finite, R has dense, sorry, I forgot if P type, you can, for infinite, okay, so, for example, so, this R has, yeah, R P is a rational, except for P equal to E5. So, this is, this R is a rational type. It is same thing as regular type. So, but not R, okay, R equals, so, so, we have seen that R P is, if I'm going to leave, yeah, P is equivalent to 1 model P. So, this is R is dense P type, but not open P type. So, yeah, it's very easy to understand. Okay, so, okay, first, yeah, so, it is shown by, so, maybe, this, so, R is defined by character zero. So, R, if I don't leave, R is rational singularity. And, so, this is a very recent result by Mar and Swede, and going, Tucker will talk about, so, it's necessary for, for some P, R P is a rational. Okay, so, so, one way is by Smith, yes, and so, if, okay, so, we talked about, so, should rational ring, and so, yeah. And then, you can choose, so, so, as said, should rational for rings of character zero is equivalent to, say, rational singularity. And so, so, if we have the kernel, and so, if we support this kernel, this is not rational. So, that means, support is, has, yeah, lower dimension, and then, so, yeah, R is open to rational type. And, and the converse is, converse is by Harlan and Mehta, Surivas, algebraic geometry of higher grade. So, yeah, it is too much for, to treat here, and perhaps I must confess that I haven't, yeah, understood. The proof, myself. So, but it, so, very, in some sense, strange that, so, if R is rational singularity, then it is necessarily open FPR type. And then, okay, of course, single one is less, yeah. And, of course, needless to say that single, so, the fact that single P is sufficient, then we can, in some sense, computable. Computable is a very important thing, I think. If, if we need infinitely many P, that, that is not computable, but single P, yeah, certainly computable, yeah, by computer, okay. And then, so, we come to section five. So, first, we must, yeah, define the notion of discrepancy for, yeah, take, so, this is normal. Yeah, it is sufficient to assume normal, but, yeah, for simplicity. And, so, these are some irreducibility by the co-dimension one variety. And, so, we call it, E is exceptional. So, we consider dimension A equal D. So, of course, E has dimension D minus one, but if dimension becomes strictly less than D minus one. And, for such E, and so, we assume that A is a cuboidal sign. If, so, I forgot to say that A is normal, canonical model. So, for some times, principle. So, in general, we can take index one cover, so that, yeah, K is R. Anyway, we have, then, yeah, a general point, D of A. So, let X1 to XZ, regular parameter system of O, X, Z. And, also, yeah, T, so, E is defined, T equals zero, or Y, Y equals zero, near W, Z. And, then, so, we can, so, we can find some, actually, we take differential form, but, yeah. And, then, we can put, sorry, I erased the long place. So, sorry, again, general point. So, zero. And, so, X1, XB, regular SOP, sorry. And, then, put, yeah. Okay. This is your unit. Okay. And, then, define XE equals R of S. Ah, sorry. This, we should put R here, because, so, this is generator of R's. And, so, we define, terminal singularity for every A of XE, yes, positive, to log terminal singularity, minus one, okay. So, canonical singularity, non-negative, and log canonical singularity, at most, minus one. And, so, it is easy to do by blowing up that if less than minus one for some key, then, we can find the infimum of AX. E for X, and we move X and E equal minus infinity. So, so, the minimal is called log discrepancy. This takes, yeah, either minus infinity or some things are, yeah, this is called, okay. Actually, I'm not myself very familiar to these guys, sorry for. It is, so, for example, in this situation, dimension two is very special. If A is terminal, then A is regular. Log terminal, ah, sorry, canonical, then, rational and golden standard. So, so, so-called famous. Log terminal is equivalent to, so, quotient singularity. The field is complex numbers. Quotient singularity means, so, so, of course, G is in SL, then, we come to this case. From dimension three, these things are totally different, and, so, yeah, for terminal singularities in dimension three was, yeah, classified by Shigatou Mori, and that was a very important result in our model theory. Okay. So, so, let me talk about, so, example of, so, this is a project variety and smooth project variety and ample impartial. So, canonical shape is, we have a, so, X, X is obtained by growing up, maximal ideal, and then, so, A is extension asset, and then, we have A, X, E, A plus one. So, for example, a, a famous, a third, and this is square Y, and there is, so, four generators, and I, I planned to calculate, in this case, yeah, this is, yeah, blow up resolution is, then, we can compute that. Growing up is, let's say, X and minus one over three, and, so, all, it's a very famous that all normal logic bring, so, same grouping, same grouping, normal same grouping, regular log terminal. Of course, this should be larger than minus one, but it is, actually, I contained a computation in my regimen, but unfortunately, the computation is mistaken. I, I made some confusion, and so, sometimes I'll make some revised version. Okay, so, now, so, my main point of today is the next one, so, so, A is in character zero, and if A is the regular type, sorry, of, yeah, this is dense is enough, but, actually, it turns out to be open, so, the same thing to say open or dense, and in this case, definitely dense. So, the essential part, the following then, is, so, this is very easy for finite, then, obtained by, this is one thing, and another thing is, in part, this is here. So, this is a very trivial fact. Okay, so, the theorem, for simplicity, yeah, we assume, so, X spec, and, assume that A is F here, of course, this is normal, and, so, then, there is some, this, A, and there should be something. So, this is splitting, and then, so, splitting means, and then, this defines, so, so, put L to be, so, the quotient single field. So, this defines, five defines, and then, so, since we are in algebraic situation, so, X, we can consider, this is, we can define, and so, since we have I to the inclusion, and, we should have, so, we want to say that, yeah, A is, if otherwise, should exist some E of AX, this is minus one, since, now, we have, by definition, so, this is by, formula, one over Q, and since this is less than minus one, so, this number will be bigger than Q, so, consider the generic point, so, so, at the generic point, of course, you know, the localing is a discrete variation ring, and, so, and then, so, if, if something has, is in, in M, Q, so, this value means, this thing is contained in M, so, multiplied by, and then, since this is principle, so, and, so, image, of base of N, so, this contained as, yeah, contradicts to the, this is a splitting, this is identity, so, so, the same thing will be, yeah, part one is in the same manner, so, because, there is some extra, and, okay, so, I, I thought of this proof, when I was in, over, in 1996, and, so, I, I was very glad that the next morning, Yanos Koral said to me, so, you did a very good job, thank you very much. So, yeah, I forgot to say that this is, yeah, contained in my paper with Hara, so, H-A-W-A, so, sorry.