 plus the variety following way, so u is x minus d and there's a map, the toric variety x bar d bar, and this is a sequence of blow-ups. And each blow-up has the following form, so there's a center of the blow-up. d which is the intersection of a component of the boundary with a hyper-tourist defined by a character, and this character is determined in the following way. So we have a form, holomorphic symplectic form, sigma on x and a corresponding form, sigma bar on x bar, and what has to be true is that the residue of sigma bar is equal to d k i over k i up to some constant. So it's a very constrained situation where the blow-ups have to satisfy this property. So for instance, maybe I just make the remark straight away. If we only want, just recall, so if we only want log klarbiao, any co-dimension two center will do. The same thing contained in a unique component of the boundary. So you see the divergence in dimension bigger than two between the cluster case, which is roughly holomorphic symplectic case and the klarbiao case in this strong restriction on the sort of centers you're allowed to blow up. So what I want to do next is connect this with the usual definition of a cluster variety. So cluster algebras were introduced by Fomin and Zelovinsky in 2001 and then they were given a more geometric interpretation shortly afterwards by Fokon-Goncharov. I think this was in 02, so geometric. So what I'm paraphrasing here is roughly what Fokon-Goncharov said so that there's a variety U, which is a union of algebrate tori. This is a union of open sets. And these are related by transition functions of a very special kind. These are the composites of what's called a mutation. U from the torus, a birational map from the torus to another copy of the same torus and given in coordinates by the following formula. So Z1 maps to Z1, Z2 times 1 plus Z1. You have a coordinates staying fixed. So this is in some coordinates. So the picture is you start with... Oh, thank you. It's best to have the same dimension. Okay, so the picture is that you have a bunch of tori. These are labeled by the vertices of some graph and if two tori correspond to adjacent vertices, then the transition map between the two charts is given in some coordinates by a map of this form and then the variety U is obtained by taking the union of all these tori. Okay, and so how does this relate to our picture? So let's relate to our picture. Incidentally, the set of labels are called the seeds. Yeah, so of course, depending on exactly what sort of generality you want, so obviously it's very natural to put a constant in here and I think the original definition, it was exactly what I just wrote, but then people generalized. Well, let me try to explain how this relates to our picture and then maybe you'll see sort of the level of generality that's... That's correct. So I should say, so let's remind ourselves so, there's a parallel series of lectures by Corti and Kasprick and they've been talking about mutations and yes, indeed the two mutations are related, but I want to emphasize I'm talking about the cluster varieties, which are roughly the holomorphic symplectic case and they're talking about... Well, they're talking about far-nose varieties, but they're really talking about log-calabi out varieties in some sense because if I take a far-nose variety, I can just take a section of minus k to make it a log-calabi out and so they're really in this slightly more general context where they can blow up any co-dimension to center, meaning instead of writing a character here, or essentially a monomial, I can write a pretty general function. So that explains why my mutation is rather more restrained than theirs. Okay, so let me just say, so what's the picture here? So this corresponds to the center being z1 equals 0, I'm sorry, z2 equals 0, that's my boundary divisor, so this is c, intersected with, you know, my character is z1 and my lambda is minus 1. So let me sort of draw the picture for suitable choice of compactification. So again, remember, what I mean by this is that, you know, I have this guy x bar, but I really don't care which toric variety it is, I'm allowed to sort of blow up arbitrarily. I'm not changing the interior, just these toric blow-ups. So what's the picture? So then I have... So this boundary divisor is a component of my boundary, here's z, and there's also a sort of opposite boundary divisor. So in terms of the fan picture for the toric varieties, this divisor corresponds to a ray, and I've just inserted the opposite ray, in terms of the fan picture, so there's an opposite divisor, and I can also suitably refine the fan, again corresponding to a toric blow-up, so there's a P1 bundle structure on my toric variety. So this is a picture of x bar. Then what do I do? I blow up z. So this is the familiar picture to most algebraic geometries, geometers of what's called an elementary transformation. Let me sort of use the notation from the surface case. So if we're in a surface case, this is really just a P1 bundle over P1, possibly with some degenerate fibres at zero and infinity. This is a fibre, a copy of P1 with self-intersection zero. This becomes a union of two minus one curves, the exceptional curve, let me call it E, and the strict transform of this fibre. And then I can say, well, this is a symmetric picture, let's blow down the other divisor. So here's my new toric model, the prime isomorphic to z, of course, and this is x bar prime. And so what I'm claiming is that this map on the tori, this is just exactly this mutation. So it's an elementary transformation of P1 bundles. And so, again, to say the same thing in different language, if I take the union, let's call this T prime maybe. So the union of T and T prime is this intermediate variety. What shall we call it? Maybe, I don't know, x1 or something. Let's call this, yeah. So, x hat maybe. Okay, so this guy in the middle, you know, again, so he's a log-calabi-ow, let's write u hat again for his interior. So this is u hat, the interior of this middle guy with a co-dimension two subset removed. Namely, this subset here, let's call it w. And, you know, here, so maybe just to emphasize z, w and z prime are all identified. So this union of two tori in the atlas description of fucking Goncharov just corresponds to two toric models, and two tori for those two toric models which are related by elementary transformation of P1 bundles. So I didn't hear the last part. Literally equal, yeah. So this map is an isomorphism. So the dictionary, what this gives is that, so, you know, in the cluster story, they have seeds, they correspond to toric models, they have mutations, they correspond to elementary transformations between these. Okay, and so now we can start drawing some combinatorial pictures to sort of show the effect of these mutations. So we can view mutations using the fan of the toric models x and x prime. So let me just show by one example, I think you'll get the picture. What do we have here? So let's do an example. So as everybody knows, if I do an elementary transformation on P1 cross P1, I get the surface F1 for the picture. Here's my copy of P1 cross P1. Here's my z, just a point in dimension two. But again, let me label boundary divisors by self-intersections. Do one blow up, and then we blow down the strict transform. So that's just the same picture I drew over there in this particular case. But now if I try to draw the toric fans, so what's the toric fan of P1 cross P1? It's just the fan given by the positive, the quadrants in R2. And the fan for F1, well, we have to make a choice of coordinates, but one choice is like this. That's the fan for F1. So this mutation, mu, has an analog at the level of fans. It's called the tropicalization of mu. I'll write mu with a subscript t, so it's called tropicalization. And how is it defined? Well, I can say, in terms of these two half planes, the left and right half plane, on this half plane it's the identity. And... Oops, I'm sorry, I've got the fan wrong. Sorry about that. I should, you know, draw these rays so they correspond to the boundary devices as in this orientation. So this fan should be like this. This is the contractable curve of self intersection minus one. Sorry about that. So here it's a shear, the vector v. This is the vector v, which corresponds to the component of the boundary containing z. The boundary component containing z. So I know that there's been a some discussion of toric varieties, but let me say again, if you've got a toric fan, a fan corresponding to a toric variety, the rays of the fan correspond to the boundary devices. This ray here corresponds to this boundary divisor along which I'm blowing up. I'm going to shear in that direction on this half plane to produce this fan here. And the general statement will be there'll be a similar picture. We have a transformation defined by a shear which gives the action of this birational transformation on the boundary devices of your compactification. Okay, and I want to say again, so this is an instance of the mutations that have been discussed in the parallel course by Alan Alessio. And this is the picture that was discussed today where there was a shear transformation corresponding to a mutation. Unfortunately, there's a clash of notation. Al was talking about mutations in N, and N was monomials, but here this is the mutation in N, the lattice of one parameter subgroups, which corresponds to that under mirror symmetry. But it's the same picture that was discussed in the previous lecture. Okay, so now how about an example? So let's talk about the first example of this kind of thing. Variety X will be the blow-up of four points in P2 in general position. So that's the del petso surface of degree five. There's a unique isomorphism type. So this surface has 10 minus one curves. Those are the exceptional devices. There are four exceptional devices and the strict transforms of the lines joining two of the points. And they have dual graph of the following type, so-called Peterson graph. Hopefully there are no Satanists in the audience. So this graph, so there are 10 vertices corresponding to the curves, and I'm just joining two vertices by an edge if the corresponding curves intersect. So we'll now select or choose a cycle of five minus one curves. So that's our divisor, D inside X. So that's going to be an antichronical divisor. So this is a log Calabiow. Okay, so let's try to write down a Turing model. We've got these boundary minus one curves. And what this graph tells you is that, well, first of all, there is an interior minus one curve meeting each boundary divisor, but they intersect in some complicated way in the interior which is encoded by this star in the middle of the Peterson graph. So I won't draw that, otherwise we'll get a headache. But what you can at least see is that two adjacent minus one curves here, a disjoint. Adjacent 3M minus one curves are disjoint. Adjacent meaning, so for instance these two here, that meet adjacent boundary divisors. So let's blow those down. I'll get a surface like this. I won't bother drawing the interior curves anymore. So this is now zero, zero, minus one, minus one. That's the Turing variety for this fat, D bar. And one way to say it is it's the blow up of P1 cross P1 in a point. Okay, so that's the Turing model. Let's see some adjacent seeds, adjacent Turing models. Let's look at a mutation. Okay, so I have this Turing model, let's call it X bar. So let's select say this center here to blow up. And notice actually that we already have a P1 vibration here. So this zero curve gives a vibration, this minus one, minus one combination is a singular fiber. So there is already a vibration. So take the fiber through Z. Again, we have this picture, we blow up. I've got these interior minus one curves. Leave the other center alone. Blow down to get to the other Turing model, which looks suspiciously like the first, but its sort of orientation has changed. And again, at the level of the fans, what happened, we had this picture, patient, what's the corresponding tropical picture? We're just shearing in the direction of this boundary divisor. So it's the same picture I have before, an extra ray. So maybe it's good in this picture to mark the rays of the fan corresponding to the blow up. So here's the, and what you find is if you keep going, there are exactly five Tori, corresponding to Toric models, union of five Tori, and the exchange graph, in other words, the graph with vertices corresponding to Tori and edges corresponding to mutations is just a pentagon. Okay, so that's a very nice combinatorial description. Let's go back to Algebraic Geometry and try to say something new from the point of view of Algebraic Geometry. So in Algebraic Geometry, what do you do with a Del Pesso surface? Well, you like to embed it in projective space via its anti-canonical linear system. Let's do that. This embeds in P5, and I'll call the coordinates x1 up to x5 and t. I'm going to choose this embedding, so I'm going to choose coordinates in a specific way here. So we choose coordinates, such that, well, first of all, the divisor d, that was a section of minus k, is just given by the hyperplane infinity, t equals zero, and x itself, or let's talk about u, actually, the affine piece has the following equations. I'll use indices mod 5 for brevity. So there are five equations. This is in A5, and I'm writing little xi for the affine coordinate big xi over t. The x1 up to x5, these are what are called the cluster variables. Oh, sorry, before I say that, let me say what's the connection between this and these tori? So the tori, ti, is just given by, there's just a locus where xi and xi plus 1 are both non-zero, so that's a torus with these two variables as coordinates. So that's a little exercise to check. So these are x1 up to x5, what are called the cluster variables, and the clusters are just the pairs corresponding to cluster tori. So one thing I wanted to mention is, okay, what's the basis of the cluster algebra, the global functions on the variety, the cluster algebra, global functions, u, that this has a canonical basis, okay, it's given by the cluster monomials. So what are they? They're just monomials with positive exponents in the cluster variables of each seat. So this has a basis, it's a complex vector space by this set. And so in the notation of Feminin-Zelovinsky, this is called the a2 cluster algebra. You can find it in Feminin-Zelovinsky's paper from 2001. Before I start erasing, are there any questions about this example? Okay, so let me give one more example then, which shows that things can be more complicated. So I wanted to give an example that was related to what's been discussed in Cortian-Casperich's lectures. So let's talk about the mirror to P2. No, no, so I is... So it goes from one to five, but I'm using... I'm just using... When I wrote down the equations, I used indices mod five. Yeah. It is... So it does... I mean, it is actually related to some of this work. So we actually used that construction in some of our work on cluster algebras. It's somewhat different in that the Hertzberg construction, if you remember, it's an analytic construction. You have to restrict to an analytic open subset of the torus when you take the quotient. So you're not actually getting an affine variety at the end. You're getting a stein manifold. And so there's... The ring of functions on that manifold is not... It's not a finitely generated C algebra. Maybe I can discuss with you later. But yeah. Okay. So the second example, the mirror of P2 together with a smooth elliptic curve. So I haven't told you about mirror symmetry yet. So you'll just have to... If you want, you can just take this as the definition. So what is it? Well, let's first start with a singular pair. Let's call it X prime, D prime. This is the following cubic surface in P3. So it's given by the equation X, Y, Z equals T cubed inside P3. Or if you prefer another way to say it, you can take P2 and quotient by the action of the third roots of unity acting in the following way. So X0, X1, X2 maps to X0, Omega, X1, Omega squared, X2. Either way, what does this guy look like? Well, there's a triangle of lines at infinity. This is sort of X, Y, Z equals zero. That's going to be my divisor D prime. I guess I should say rather it's defined by the equation T equals zero, which is given by these lines. And the nodes of those lines, sort of a little analysis of the equation. Homogeneous equations are now passing to local coordinates. These are going to be what are called A2 singularities. So local equation will be a local equation, X, Y. So this is not a smooth surface, but it's easy to resolve. So let's resolve. What do you get? So the resolution of an A2 singularity, that's another little exercise, is just a chain of two minus two curves. So we have this picture. These lines are just the strict transforms of the components of D prime. A short calculation again will show that these have self-intersection minus two as well. So this will be our new divisor D, and we have a pair X, D. So this is a log-calabi out. So again, so here we call this map Pi. D is the full inverse image of D prime. And this is a very nice variety. It's actually the elliptic vibration over P1, elliptic vibration. Well, the fiber over infinity, being this reducible fiber D, this is a cycle of nine minus two curves, or in Kadaira's notation, this is an I9 singular fiber. So the picture is got three rational nodal fibers, and let's call this map W. That's the Landau-Ginsberg potential in the terminology of the talks by Alan Alessio. Okay, so let's see. So this is a log-calabi out with maximal boundaries, so it has a toric model. What is it? Well, in fact, if I sort of keep the picture as before, these minus two curves, which were the strict transforms of the components of D, have an exceptional minus one curve attached to them. Those are disjoint minus one curves. These are, in terms of this rational elliptic vibration, each minus one curve will be a section of that elliptic vibration. And the toric model is just given by blowing these down. Well, let's see. So I actually didn't think about this, but they're definitely there. I've been off the top of my head. Perhaps this is a good exercise for the reader. But yeah, maybe it's not hard to see using the equation of X prime. I won't spoil your fun. Oh, sorry. I can answer why there are minus one curves. So remember, minus one curve is a smooth rational curve with self-intersection minus one. Equivalently, by the junction formula, it's a... Okay, sorry. Yeah, in this model. And the answer is I didn't prepare before my lecture, but I'll leave it as an exercise to the reader. Yeah. Oh, is there another question? Is everybody happy? Okay. So now, I claim this is a toric variety. And, you know, again, sort of taking a cue from the other lecture series, a good way to see this is to draw the polytope for P2. This is sort of the polytope for P2 with polarization given by O3. So that's just the ordinary triangle, but we've scaled it up by a factor of three. And now if I just draw the fan given by all the integral points, that's the fan of this... Fan of this... The fan of X is given by this... So the corners are the minus one curves and every other curve has self-intersection minus two. So let's just do one mutation. So what happens if I start mutating? So again, let's mark with an X the points where I blow up. So for instance, if I mutate at this... this vertex here corresponding to blowing up along this minus one curve, this is going to be my Z, you can see that there's not a P1 vibration yet because I don't have the opposite vector in the fan. So what I first need to do, I take my X bar and I'm going to replace it by a new compactification. Let's call it X bar one. I'm just going to blow up so that this... I add the opposite vector to my fan. Maybe I'll just sort of draw it dotted here. To add that vector, that will be a blow-up of precisely this point here. So this is just an auxiliary change of our compactification so that we can see what goes on with the mutation. Let me draw that in. So now this becomes minus one, minus three, minus three. I'm really going through the chalk today. Is there anything longer than about an inch? All right. So there we are. Minus one, minus one. And now what's this? Minus two, minus two. There's some complicated thing like this. This is a vibration, a curve of self intersection number zero. And now I can do the elementary transformation. Blow up here, blow down there. Right, so I won't draw it. So now I'll get sort of X bar one prime just by saying picture X set. Now this will be a zero and this will be a minus three. Oh, I'm sorry. This is minus one. I guess maybe I don't need to draw this whole thing. Yeah, so the only thing that's changing is I blow up here so that becomes minus two. We got here minus one, minus one, minus three, minus three. And now what did I do here? I'm actually bothered now because that shouldn't be a zero curve. That should be negative. Something is wrong. No, I guess this is okay. Okay, so now the statement is so what's the exchange graph in this case? That was just one mutation. So of course if you saw the previous example you might be hoping for some nice finite graph that doesn't happen in this case. So in fact it's an infinite tree with every vertex of degree three. One can describe the combinatorics. So the combinatorics of the associated fans is given by the Markov equation. But anyway, so the picture is, this is this equation which you may have seen in connection with derived categories and et cetera. But anyway the point is that there are infinitely many tori. At each point there are free mutations I can do and that never closes up in the maximum possible way. So somehow you get this infinite tree with no relations at all and so just countably many tori in your atlas. So that's kind of another flavor of example that's actually more typical than the first example. So I should say something about that as well. So the Landau-Ginsberg potential, so if you remember there's some Laurent polynomials in the talk of Alessio and Al. So where are they? So we have these cluster tori T inside our U. U being this X minus D, that's the guy that has the Landau-Ginsberg potential mapping to C. Just remove the point to infinity. And so this W restricted to T is then a regular function on a torus. So it's a Laurent polynomial. So this is the polynomial described by, I'm sorry, Al. So depending on which torus you choose you'll get a different formula and the two formulas will be related by this mutation. I realize Greg this isn't answering your question maybe. You were asking for it described in terms of close four. Oh yeah, that will be true. So on the initial torus it'll be the usual formula. So on the initial torus it looked like this and then they'll be obtained by mutation from that guy. So I've got 10 minutes left. I just wanted to tell you now, suppose you want to go and do your own computations. How do you translate this picture into combinatorics? It's very easy, but I was talking with some guys just before the lecture. The problem with this, this is what Atea says, you have to pretend to sell your soul but not actually sell your soul. You need to work algebraically and combinatorially but think geometrically. But anyway, let's start the process. So what's happening here? I've got this XD described as a blow-up of a toric variety. So what's the data? I'll use the notation that was introduced in the previous lecture. So I've got my torus, T is U bar, that's a copy of C star to the N. But if I don't choose coordinates, so there are two abelian groups, so N, that's the first homology of the torus. Or what's the same thing, it's the one parameter subgroups. That's a copy of Z to the N. And then there's M, those are the characters, or I guess I could say also H upper one. And that's N dual, the dual group. So that's just the usual toric notation. Now what do we do? We have a bunch of boundary divisors. So a boundary divisor C, I said this before, that's just going to correspond to a primitive vector, V in N, so the ray generated by V corresponds to the boundary divisor. We have the character chi, so remember our center Z was C intersect chi equals lambda for some lambda, so chi, well that's just an element N, it's a character. And again, we may as well assume this is primitive, so this is just assuming that Z is irreducible. Not primitive, this is just a disjoint union of several connected components. And we've got our two form sigma bar on X bar. And as I said before, that's just got constant coefficients, so that corresponds to an element in wedge two of the characters tensed with C. Okay, and there was this condition that if you take your form sigma bar and you take the residue along C, you get d log chi up to a constant. And again, in terms of combinatorics, what does that mean? I think of my sigma bar as a skew form on N, so taking the residue along C just corresponds to inserting V in the first argument, and then this should be equal to the linear function on N given by nu times N. Okay, so now I need to tell you what did this mutation look like. So we gave an example before, and the general case is no harder. So we drew the tropicalization of mu so that was a map defined on the ambient spaces of the fans. So it's going from N, R, and lives in the space of one parameter subgroups tensed with the real numbers. And we had this picture, so what do you do? So I've got these two pieces of data. I've got a vector V and a functional M related by this formula. So M is determined by V up to scale. So again, V gives me the direction of the shear, and I guess maybe I should have drawn a higher-dimensional example. Somehow maybe I'll make this a three-dimensional example. This hyperplane, yes, I'll put it like this. So here, shear by V. This hyperplane is M perp, and I guess I'm just to fix ideas. It doesn't matter, the choice is irrelevant. Let's say this is the hash space M bigger than zero, and here it's the identity. And so, of course, you can write a formula the same formula that Al wrote last lecture, but in the specific case of cluster varieties. So it's actually an integral map defined by U goes to U. I'm not sure if my U agrees with it, but this is the formula, but that's just the algebraic expression of that picture. Okay, well, I think this is a good place to stop, so maybe I'll continue tomorrow.