 It's a pleasure to introduce Louisa Nunez-Batenkort. Yeah, and he will speak on Nash Blow-Ups and Prime Characteristics. Okay, well, thanks, Dale. I want to take this opportunity to thank the organizers for giving me the opportunity to speak today. It's a great pleasure to participate in a celebration of the work of Craig and Mel, to whom I and many others feel very, very, very grateful. Okay, so this is a young work. This is the talk is based on two papers. The first one is with Daniel Duarte, who is at Dunham, Mexico. And then the second is between the three of us, Daniel Duarte and Jack Jeffries, who is a University of Nebraska-Lincoln in the U.S. Okay, so let me start with a setting for today's talk. So for us, K is always going to be an algebraically closed field. Second, X is going to be an affine algebraic variety over K of dimension D. And a lot of times I want to think of X, and my mind is going to be just a speck of R, and then R then is going to be a finely generated K-algebra, which is a domain. Okay, so R is just a polynomial ring over a prime ideal. Or sometimes I'm going to take localizations of R and R, so I write in R. Okay, so let me introduce the definition of the main protagonist. So we consider X in the defined space. So if I take any smooth points, then here we're going to have a map for every X, little X, we're going to have that the time in the space is well-defined, and it has dimension D. And then, so I want this to leave, it has to be in the grass manian. Here, here. This map is well-defined as long as X is smooth. So the Nash Blob is defined. So what we're going to do is take this map, look at it too much, and then take the Sariski closure. So that is a right-byting. And this comes together with a map. So we have a map from here to X, which is given by the projection map here. So we take the projection map here, we just restrict it. So that's a particular map. Okay, and this was... Yeah. I mean, this was appearing in a paper by Nobile in 75, and then he created this Nash for hearing this question from him for over a period of years. But if we go back a few years, there is also a paper of Sample in 54 that describes the same object. Okay. So what we're doing here is replacing the singular points by limits of tangent spaces. And we hope that that improves the variety. So the main question here, which I'm going to call question number one, is the following. Can we resolve singularities via Nash Blobs? So in other words is... Okay, let's start with a variety. Okay, there exists a T, natural number, and a sequence of varieties, such that, well, here what we do is Xi plus one is equal to the Nash Blob of Xi. Well, X0 is equal to X, the original variety. And then we need maps here, which are the maps that we took here, such that this happens, this happens, and the last one is smooth. So this way, if we take this one and the composition of all the maps, we obtain a resolution of singularities. So that's the end goal. And it will be given a very nice algorithmic process to get a resolution of singularities. If the answer to this were yes. And we don't know if that is true. And for this question, just to be meaningful, we need that... What we expect is that Nash Blob improves the singularities. So the first things that we need is it changes the variety because if we just get the same, then nothing's going to happen. So for this, it's necessary that if X is singular, then the Nash Blob of X has to be different to X. That's what we need in order for this question to be meaningful. And this is a theorem of Nobile, same paper, 75. He proved that if the characteristic of the field is zero, then the Nash Blob of X happens... It's equality if and only if the variety is smooth. Or we obtain this one, which is... So he also proved, same paper, that if the characteristic of the field is zero, the dimension of the variety is one, so it's a curve, and then our question, the first question has a positive answer. So we can resolve curves using this. And then he also gave an example, same paper, in prime characteristic, X being the spec, we take F2 X, Y over X square minus a cube. So we take the cost in characteristic two, and then the Nash Blob of X is equal to X. So Nash Blob does nothing to this curve in prime characteristic. And this was one of the first papers, maybe the second paper in Nash Blob, and even when that comes this example, there is no point in thinking about this in prime characteristic. And then most of the work in 40-something years was trying to get results, at least in this, in characteristic zero. So let me mention a couple of results in characteristic zero. First, I want to modify question one, something related. So a related question that has been studied also in characteristic zero is the following. Can we solve singularities with Nash Blob's normalization? So specifically is that X is a T, natural number, and a sequence of varieties, such that, so we start with our variety X0, X1, da-da-da, XT, and I just realized that there is no zero, sorry, I just keep thinking about resolutions and psych diagrams. So we have this, such that, again, we want X0, we don't require to be X, but the normalization of X, and then XI plus one is equal to the normalization of the Nash Blob of XI, and the last one we want it to be as moot. So we don't take just Nash's normalization, normalization of Nash, normalization of Nash, normalization of Nash, and then if this is moot, again, we get in those maps a resolution of singularities. So that question for curves is also not meaningful because we know that normally icing is enough, so we don't have to take the... Well, the answer will be, yes, but it's not interesting. So 15 years after Nobile's paper, Espira Kopsky proved the following, I know, sorry, I want to do something else. A couple of years later, Gonzalez 77 proved that the answer, the second question, the second question, too, is positive for toric surfaces. So with the toric surface, this is going to work to get a resolution of singularities. But he also requires in characteristics here. So, and let me just mention two ideas of the proof. So first, he defines a logarithmic Jacobian ideal. So we think in a fine toric variety, as the speck of a semi-group ring. So this one is going to be a monomial ideal, which is defined by linear relations, the rators of the semi-group. So he proved that in this case, the Nash blob is equal to the blob of this logarithmic Jacobian of X. So now he proved that we get the resolution of singularities looking at the combinatorial data that's attached to this. Okay. Okay, and then later, in 1990, Espira Kopsky, building on work of Hironaka, he showed that the answer to question two is positive for toric surfaces in characteristics here. So what about prime characteristic? So we're going to solve the question one is not meaningful in prime characteristic by the example of Novelia. In a young work with Daniel, we show that if we take a normal variety in prime characteristic, then the Nash blob of X is isomorphic to X if and only if X is smooth. So question two is meaningful if we just normalize, which is not a big price to pay. A few ideas of the proof or steps of the proof to see what fails here that doesn't fail here. Okay, first is we take R to be the terms of a point, so we fix a point, the variety, and we need to prove that this point is regular. So if the Nash blob of X is isomorphic to X, it was proved, but this year, that the model of scalar differentials is what it has to be plus possibly some torsion. So then the X is partial derivatives or derivatives, linear derivations and elements X1, Xd inside R such that the i of Xj is equal to the Kronecker delta, and in characteristic zero, this finishes the proof because then one can go to a result of Sariski and then complete and then see it and then R has to be another ring. I joined one variable for any of these that we have. So at some point, he just recovers a power series ring. And what's happening is that this torsion doesn't exist in characteristic zero. In prime characteristic, in this example, we have that this is equal to R plus something that does have torsion. So torsion does exist there, and that's the problem. So how we solve this, well, we cannot use this criterion for smoothness or play much with this, but we have constant theorem in prime characteristic and it's what we use. So using these derivatives, we built a map, fc, that goes from Rd to P to the d to R1 over P we define this map globally and then we prove that fc localized at Q is an isomorphism for every prime of height one. This one is... Thank you, thank you, Nick. Yeah, it's just a free map of rank P to the... Thank you, thank you, Nick. Okay? And then this implies that fc is an iso, R1 over P is free. So then what we start doing is, well, now that question two or like Nash Blob is meaningful for normal varieties in prime characteristic, we can look at all the words that has been down in the 40 plus years and then try to prove it for prime characteristic. And the first thing that we wanted to do is prove this theorem by Mark, and we couldn't. There are some parts in the proof that our characteristic zero is absolutely... Well, the way we were trying to prove it was absolutely necessary. One of them was this Rassolowski that I mentioned and there are other parts that we couldn't... Yes, we couldn't. Look at this. So we were not successful with this and then we say, well, let's try this one. That's a good start. And this is the new result in which Jack joined us for this project following if X is a toric variety prime characteristic, question two has a positive answer. So we can resolve same as Gonzalez-Springberg. So now let me say a few key ideas of the proof. So what we wanted to do at the beginning is, this Jacobian ideal makes sense in prime characteristic. I mean, we can just look at the monomials that we have and then try to do that. But then it turns out that that Jacobian ideal does not define the Nash blob. So we cannot use this one. So then we start looking for a characteristic P version of that Jacobian ideal. So we define the characteristic P version of the logarithmic Jacobian ideal. What we did is, here is looking for when some relations happen or do not happen. So if an equation becomes or does not become zero. So we just look at that mod P. This is something very natural. And it turns out that this is true. That the log P Jacobian, it is the Nash blob is a blob of the logarithmic characteristic P logarithmic Jacobian ideal. And actually this is true in general, not only for surfaces. The next step is when we do really need surfaces. Then we have to prove the combinatorial data. Sharp P logarithmic Jacobian is the same the data of the characteristic zero logarithmic Jacobian. And then we're done because we just follow the proof of Gonzalez-Springberg. Once we have that, the ideals are different but the combinatorial data is the same and that's what he used to prove the resolution of singularities. So once we have these two, we're done. Sorry, I couldn't hear you. Oh yeah, surface, sorry. Thank you. These two are true in general. This needs surface. And in fact, we have an example of dimension three variety where this is not true. They have different combinatorial data. And that's everything I wanted to mention today. Thank you. Are there any questions? No, just took the same definition characteristic zero and then instead of put equal to zero put congruent model of P to zero. The Jacobian, yeah. It is not reducing the monomial ideal from characteristic zero to characteristic P. It's just reducing the relations on the exponents which gives a priori different ideals. For P big enough, it's gonna be the same but in low characteristics are different. As we have an example where it gave different combinatorial data.