 We need to introduce Kaiichi Watanabe from Nihon University, and he'll talk about Gornstein property of normal tangent cones, of normal surface singularity. Please. Thank you very much. First, I'd like to thank the organizers to give a chance for me to talk here. And this is a joint work with Tomohiro Okuma and Kenichi Yoshida. And this is a part of the archive. And so the title of this workshop is Commutative Algebra and Advocate Geometry. And so our idea is to answer the problem in commutative algebra using algebraic geometry. So OK. So let A and B. So normal. Normal is very important. OK. And we assume maybe excellent of dimension 2. So I'll explain about why in this dimension 2. And so we assume this is actually closed, also included in A. And I always means in my talk that the integral is closed. And the main fact is that we can take the same thing, appeared already in Dale's talk. This means integral closure and g bar i. So they are quite natural things. And so our problem in the front is Gornstein. So maybe you would think this is a very simple question. So for example, take OK. So this is a local OR graded degree 0 in case. So very simple example. Assume that and if we assume for simplicity, we know the answer for all cases. But for simplicity, and we are relatively prime, then G bar of maximal idea is Gornstein. If and or if B equivalent to A, mod 1 to 1 mod A. So Gornstein is very limited. OK. So our tool is a resolution of singularity, F inverse. Inverse image we write. In this notation, we always assume that these are irreducible components. And so what is good in dimension 2 is we have, so we call cycle. Cycle means some integer combination, linear combination of these EIs. And there is, yeah, so for Z and Z prime. OK. So there is the intersection number. And at prime, and at prime, sorry. So the intersection number is very important for. And so we can express the singularity by a graph, like. So if A is, so this means the genus of the curve. So these are E0, E1. So this means the genus of the curve. So this one means E0 has genus 1. And others are genus 0, namely, rational curves. And this means these two curves are intersected. So these are not connected, and these are not intersected. So we can express singularities up to a certain stage. Of course, we cannot know all of these singularities. But so some, yeah, we can know some of these properties. And then so we come to, so we can express i by this way. Since i is determined by z, we write z here. And so anyway, in theoretically, we can express every inter-close idea by writing cycles. OK. And so we have to know important invariant, so we define. So this means dimension of H1. Of course, this is very important. And it is rational singularity. And define one more thing. So we define Ea. So for a cycle, this is canonical divisor. So also Pa of A is soup, the positive divisor, Pa of Z. And it is easy to show that this is always bounded by Pg. And so the important thing is that this is calculated by the graph, only by the graph. And so that means in some sense, this is a topological invariant. OK. So this is something I forgot to tell. So rational singularity is very important and assumed to be the simplest of singularities. And then we want to define next simplest. And A is an elliptic singularity, Pa is 1. So note that Pg of elliptic singularity can be arbitrary large. So this is very important, one important thing of this Pa. And then we want to define, so reduction numbers. Actually, we define normal reduction numbers. OK, so for the given i, so we can take minimal reduction. So which is characterized by q o x equal. And of course, this is generated by two elements. And so we define first an r of i, minimal r, i hat r plus 1. So it's minimal. So the same, but yeah, i r such that q for every n. So some people will wonder whether these are different. So we do have one example such that this nr is 1 and r bar is arbitrary large. But one more thing to say is actually we don't have only one example of such kind. So to find plenty of such i, so different, it will be very nice. And so nice thing to have, yeah. Interaction theory is that there is a theorem of Riemann-Loch. So if i equal high z, we can compute corrects by intersection numbers plus sorry, h1. So again, this is a very important invariant. And so we define if i, i, z, we define q of i, h1 of x minus z. So the cohomology of these fields are very important. And so we all define. So notion of Cauchy-Roux complex for q equal ap. And in some sense, we can lift these two upstairs. And we have, and by cohomology is a sequence, along with the sequence, we can also, we can tensor x minus z. So then it is easy to show that to the length of q. So because if we take h0 of this one, so there comes a, and i, z. And this is given by a, b, and minus b, a. So i, so i square. And so the image comes here is q times i, z. And so by exact sequence of cohomologies, the length is the same as the corner of here. And then we have q of n minus 1 z plus n plus 1 minus. Yeah, yeah, yeah, I saw this, yeah, yeah, h0 of this is, yeah. So, yeah, sorry, yeah, this should be i, sorry. And one more thing is the sequence q and i is not increasing. And if equal at some stage, then equal forever. So by this fact, so since we have starting point is PG, it is very easy to show. So our bar of i equal 1. If I don't leave h1 as q of id equal to PG. So because h1 of this is just PG, and then comes here. And we call such a PG idea. So what sequence are there that action number is not 1? I mean, h1 of these things are all same. We have, so this means the action number is 1. OK, so this trivially shows that, yeah, a is a rational singularity. We found only if every i inter-closed is ideal. So it is a very important thing to characterize singularity by the ideal theory. And so, yeah, this part is, of course, clear. So the compass uses some, yeah, result of a deal. And so we define, so define, inspired by the advice of Malidina. This is a loss. Yeah, we call i an elliptic idea if r bar equal to 2. So yes. And so the important fact is if a is an elliptic singularity, then for every i inter-closed, yeah, i is either PG ideal or an elliptic. So of course, this means that elliptic and r bar of a is 2. This is our new result that, so, r bar of a. We haven't defined maybe, so r bar a is max of r bar i, so far i. It is so easy to show by the exact sequence. This is at most PG of a. So that actually is a better bound. So this is PA. It is very interesting to know that. So such, as I said before, this invariant is a topological invariant for a. And such ideal theoretic invariant is described by such topological invariant. So we, yeah. And it is, so we are very happy if we will have the converse. But unfortunately, this is not the case. So then it is easy to calculate PA. And it is not easy to calculate, but we could, yeah. Of course, so this shows a is not elliptic, but yet r bar is 2. And then we come to the main topic of this talk. Also maybe, yes, these two, yeah. So this shows that r bar is bounded by 2 plus 1, but yeah, not elliptic. And so maybe one more example. So take fd, where fd is a smooth homogeneous degree d. Homogeneous is important. Then we know that r bar of a is r bar of the maximum ideal. And I hope, yeah, d minus, is this right? On the other hand, so PA is PA is genus of collage a. So of course, it will be very different. If d is d goes bigger. OK, so in the case of if d goes 5, so r bar is like, yeah. Looks like this is correct. 6. OK. And now we come to the main topic. So if first with i is a pj ideal, so i is g bar i is collage a, is equivalent to say, i is good defined by goto ei so i miss goto ei very much. So this is by definition, so the collage is just 1 half of. So in our example, this is just minus d square. And for every a, we have plenty of good ideas. And so that is proved in our area paper. And so in this case, the i for which g bar i is going to die is quite plenty. But if we consider the case, if r bar i is not pj ideal, the situation is quite different. So we have one theorem, g bar i is constant, and r bar i is r. Then g bar is also Gorenstein, and r bar is just 2. We want to give some classification of ideals of which g bar i is Gorenstein. So this shows, so in some sense, it is sufficient to consider elliptic ideas. But so elliptic ideas, it is very, so these are, of course, our joint work. This means always joint work, not only by myself. If r bar i is elliptic, g bar, then g bar is Gorenstein. If I don't leave, chi of z is 0. So this is the same as, yeah. Chi z is described by such a cycle. So first thing to say is that it is quite simple. So since z, sorry, i equal i sub z. OK, sorry. Thank you. So since, yeah, by Hudeck, first g bar i is Koyama Kore. So we don't need to worry about Koyama Koreness. And so by the way, we don't know so many examples of g bar i, but not Koyama Kore. We know that the example we gave that nr and r bar is different plays also an example of not Koyama Kore. But unfortunately, that's only examples we know. But maybe some people here know more count examples. It seems not known enough. So take minimally less than q. And then we can take degree 1. So this is, of course, a homogeneous parameter. So we put g bar i divided by this a star, b star. So since r bar is b0 has only 1, only b0 plus b1 plus b2 and baster. So since we want, of course, b is Goranstein 2, we can assume. So a length of b0 should be length of b2. OK. And of course, this is Goran's bar i. And as a rule, Riemann local theorem says, as I said, g a minus. And this is computed by i2 bar over qi. So this is equal to, yeah, oh, sorry, yeah, 2, 2, i. Minus twice. But since r bar i equal 2, we have the PGA is strictly larger than qi. But after that, all are the same. OK. So we can write this is qi. And if this is equal, we can cancel this. And this part should be 0. This ends the proof. But so this results very surprising fact. Let me give some example. So consider the case of erythmiac singularities. Consider. So for example 1. And in this case, Pg is given by n, maybe this is. And we have no with r bar is 2 g bar i Goranstein. On the other hand, of course, we have still plenty of good ideas, so no erythic ideal with g bar i Goranstein. And so I need to check carefully. OK, yes. This is again, oh, sorry, I don't. Exponent is x square plus. Yes, 6n plus 6. OK, sorry, yeah, yeah, 3n plus 3. And in this case, yeah, Pg of a is n plus 1. And in this case, we do have examples. So if r bar is 2 and g bar Goranstein, then i equal x, y, z, n plus 1. No, oh, yeah, yeah, OK. I hope this is correct. So anyway, so I'm, yeah, at most. So the number of i and g bar i Goranstein is bounded by Pg. I must confess that I am not very confident with this. I'm asking whether this is Pg plus 1 or n plus 2. Sorry, but you can consult the archive and yeah. And so the finite theorem is still true for ideas with more PA. So further examples, so it seems there are some finiteness of such ideas. But we are challenging about that. OK, thank you very much. Thank you very much. Are there any questions? Yeah, given a, so ideas, the ideal, no isomorphism. Yeah, so we have, so in higher genus, we do have examples of family of such ideas. So for example, some linear series plus m square. In some cases, such ideas do appear. But in this case, this is only finite. So this is proved by the theory of cycles, elliptic singularities. And thanks again, K.H.