 We welcome Dr. Joab Lan from the University of St. Andrews. He's going to talk about tropical green varieties. Thank you. All right, well, thank you very much for the introduction and for setting up this wonderful summer school. And also for humoring me when I wanted to switch my time several times, so I really appreciate that. Right, so I'm going to tell you about tropical green varieties. We heard a lot of nice talks about the usual algebraic green varieties and I plan to give a slightly different perspective on that topic. And let me remind you that is Angela told us earlier today. If I have a double cover of algebraic curves, there is a natural way of getting a an ability and variety out of that called the pre variety. And if the genus of the care downstairs is G, then this pre variety as a group and as a topological space is identified with a G minus one dimensional tours. So what I want to do in this talk is tell you about an analogous construction in tropical geometry, where instead of a double cover of curves. This time we have a double cover of metric graphs, so metric graph is just a graph where the edges are given some links, so the whole thing become a metric space. So if we have such a double cover we will have what we call a tropical green variety. And this is identified with a real tours of dimension G minus one. Okay, so I assume most people here are not well versed in tropical geometry so I want to kind of very quickly give the general idea of this theory. I don't have any intention of going to too many details right now. So one of the things that I wanted to share this philosophy is that I have some problem in algebraic geometry. And I use the process called tropicalization to reduce the problem to combinatorics. Now when I use the word reduce it kind of sounds as if combinatorics is easy and that's that's definitely not the case. But in a way, the tropical tropical geometry lets us kind of get rid of stuff that are less relevant to the problem and focus on the combinatorial aspect that were already present in the problem. It kind of streamlines the whole combinatorial approach to problems in algebraic geometry. So suppose I start from an algebraic variety and I perform this process called tropicalization. What I end up getting is a polyhedral complex a collection of polytopes attached along their faces. This tropicalization process is pretty, pretty deep and I'm, I'm not going to get into it I'm going to block black box the whole thing during this talk. But for instance, it's something that preserves dimension. So if I started from an algebraic surface with complex dimension to the tropicalization is going to be some sort of a two dimensional political complex. So a collection of polygons attached along their faces. First we're mostly interested in algebraic curves. And when I start to an algebraic curve, what I end up getting is a one dimensional polyhedral complex, which is, which is just a metric graph. If things go well is not always the case, but ideally, if my curve had genus three, then the tropicalization will also have genus three in the sense that it has three independent cycles. Okay. As we learned earlier today and yesterday. What we need to define the pre variety is by looking at the divisors. And so we need to have a notion of the divisors also on the tropical side. And this is a very natural thing. We have a divisor on algebraic curve with just some finite combination of points with positive or negative coefficients. And when I tropicalize I just give you a finite combination of points on the metric graph. So the tropicalization is a point wise map and every point goes to a point. So divisors go to this finite combination of points. And such a combination of points is we either call it a tropical divisor or we call the chip configuration. In the sense we think of them as playing chips that we place on points of the graph. Now, there's a very natural question that comes up. If I started from two equivalent divisors. And I tropicalize both of them. What can I say about the relation between their tropicalization. In other words, what is the appropriate equivalence relation between divisors on the tropical side. And this is given by something called the chief firing game. So this was introduced chief firing games actually existed before tropical geometry but the person who connected chief firing games with algebraic geometry is Matt Baker. 15 years ago 12 years ago. And basically if I start from two equivalent divisors on the algebraic curves, their tropicalization will be related but what what is known as cheap firing. And the idea is that that if I want to go from one divisor on the graph to an equivalent divisor. I'm allowed to continuously move the chips, as long as I maintain a condition that I call the zero momentum along cycles, meaning that if I look at the momentum of all the chips that are moving at any point that moment the total momentum will be zero along each cycle. So let's see some quick examples of how that works. Let's say I have this metric graph. By the way if anybody has a question feel free to stop me and ask. I have this metric graph and I have a divisor on this graph consisting of two chips, one chip on every one of these points. And I want to find other divisors that are equivalent to it. As I said I'm allowed to move the chips continuously so let's say I move them like this at the same speed. In this case if I look at the total momentum of things that are moving along the cycle, the total momentum is zero because they're moving in opposite directions. Well, well no one was moving along the cycle so the total momentum of chips is zero here. So as long as I move those chips at equal speeds in opposite directions, I get a family a one dimensional family of equivalent divisors. Now using this idea I can actually go from this device on the left to this device on the right. I can move these two chips here until they both reach the vertex assuming they started at equal distance from the vertex. Moving along bridges is always allowed because there are no cycles there so the condition is satisfied. Once they reach this vertex they can move in opposite directions until I reach this divisor here. On the other hand if I started from a less symmetric divisor like this one. There's no way I can reach the divisor on the right, because if I try to move the chips to the vertex only one of them will reach the vertex. And there's no way that I can move it here but then only have one chip here and there's no way it can move along these edges here. So so the street fire relations somehow is very much dependent on the position of the chips and the geometry of the metric graphs. Okay so now that we have the notion of divisors and I have the notion of equivalent solution between divisors. I can define the tropical Jacobian in a way very much analogous to the way that the Jacobian is defining algebraic geometry. I take all the divisors of degrees zero the divisors were the coefficients sum to zero. And I mode out by the equivalence relation given by chief firing. Now this is very naturally an algebraic group I can just add and subtract divisors from each other, and that actually respects the chief firing relation. In the show and I'm talking in Jarkov. If I started from a graph of genius. Sorry, here it's supposed to be gamma. The gamma hygienist G, then the Jacobian is going to be a real torus of dimension g RG mode CG. So just one quick example to see why this is true. For instance, if my entire graph is just one cycle. Any divisor of degree one is equivalent to just a choice of one point on this graph. And so, every point on the graph gives me a divisor of degree one. And one can show also that every point gives me a non equivalent divisor. So the set of divisors of degree one is identified with basically this entire circle. And the divisor degree one are also identified with divisors of degree zero. And so the Jacobian of a cycle is itself, just like you expect from algebraic geometry where the Jacobian of an elliptic curve is isomorphic to itself. If you go to higher genus more complicated graph it becomes much harder to see this isomorphism, but it still holds. And then we have where we only have a Jacobian. We can start to think about a tropical version of the cream variety. So now instead of double cover of break curves we will take a double cover of metric graphs. And what I mean by double cover well there are two types of double cover I'm only going to show one type here so these two are all both of the same type. So double cover I just mean that it is a topological double cover, every point has, every point has two pre images. And the map is locally in isometry. So here, this edge and this edge are both the same length as this, this edge, this edge and this edge both map to this edge or the same length. And here, the picture is a bit misleading but here I mean that these two edges are both the same length is this edge. This is another example of a double cover here. As I was saying we need to allow not just these these free double covers where everything is an isometry. There's also a second kind of these called harmonic double cover. But that's a little more complicated to define I will not go into it right now. I should say that this is somewhat related to the two types of singularities that Angela mentioned in her talk earlier today. So now when we have a double cover. If I have any divisor upstairs. I can very naturally just push the divisor downstairs just take the point wise map that takes the divisor from upstairs to downstairs. And that gives us an induced map on Jacobian that is called the normal. And I think that this that the map that pushes divisors down is also it respects the equivalence relation the cheap fire emulation. And so we actually get a map on divisor classes. So a few years ago, Dave Jensen and I, we studied this map. And we showed that the kernel of this map has either one or two connected components. So in fact is one or two connected components correspond to two connected components. Two connected components when it is nice free double cover that I explained, and one connected component is if it's a harmonic double cover, which, as I said, we're going to not think about today. And moreover, each of these connected components is identified with a real towards the dimension G minus one. I should say that from the way I presented here it just looked like everything works exactly the same as an algebraic geometry. But actually it's not always the case, and all the techniques end up being different and it's kind of a miracle that all of the results and definitions somehow mirror exactly what happens on the algebraic side. In any case, once we know that the kernel of the norm map is basically a real torus, we can define the tropical premap. Either the kernel had one or two connected components, we will take the connected component that contained the identity element. Right. So we have this tropical prem variety. And now there are some very natural questions to ask about that. First of all, in tropical geometry, there is also a notion of polarization. So it's natural to ask what we can say about this polarization. And also, since we, the reason we're all here is to study algebraic geometry. The question is, how does a tropical prem related to the algebraic prem. So this was studied in joint work with Martin Woolish. And I should say that parts of this result was also refined in later work, joint work with Dimitri Zakharov. So the tropical prem is what is known as the principally polarized tropical ability. And second, if I start from some algebraic double cover and I tropicalize it and I get the double cover on metric graphs. And identification as tropical principally polarized ability varieties between the tropicalization of the algebraic prem and the prem of the tropicalization. So this is exactly what you'd hope exactly what you'd expect. And now when we have this, we can start using the tropical prem in order to study problems in algebraic geometry. So what I wanted to do in the last part of my talk, I want to explain how we can use this to prove new results in algebraic geometry. So these new results are in real notice theory. And first I want to say a few words about what you know the theory is. So broadly speaking, I have some algebraic curve of genius G. And we're looking for divisors on that curve of degree D and rank R. And we're looking for line bundles of degree D that have R plus one independent global sections. And the set of those divisors is called W or D. And the main question one asks is what is the dimension of this. You can also ask other questions of course whether or not it's smooth and other questions relating to a geometry, but the basic question is what is the dimension. So this was studied extensively and it's one of the main motivation for a lot of modern algebraic geometry. And the real notice here and the main result in this area is that if my curve is general in the modulate space of curves, we know exactly the dimension of this locus, the dimension is going to be G minus R plus one G minus D plus R. So this formula is kind of well laid out because we can see that G is the dimension of the Jacobian. And then the set we're interested in has this code I mentioned. So somehow there are these many independent conditions for W or D. And this is attributed to one set one direction to camp and other direction to be fit in Harris, but there are a lot of other people who are involved in improving these results. Lazarus failed full done. And Eisen but I think they say a lot of the major figures in algebraic geometry in the 60s 70s 80s are involved in this result. Okay. Okay, so so that was true for generic curve. The difference when the curve is not generic. Well, this is very recent using tropical geometry. So, so one way of stratifying the modulate space of curves is to is by the gonality of the curve, the lowest degree of a map to P one. So there's an open dense set where there is a certain expected gonality. And as the gonality goes down get lower dimensional sets in the modulate space of curves. And so if you look at the figure and then Dave Jensen and Groove Ranganathan, we're able to find formulas for the dimension of the W or D for characters are generic within the K Bono locus. I'm not running it down because the formula is a little complicated and there are different formulas depending on all kinds of parameters. So I came here today to talk about pink varieties. And there is an variation of the usual, you know, the theory that actually talks about print varieties. And right so now, instead of looking for divisors inside the Jacobian, we're going to look for divisors inside the pink variety. We want to ask whether their devices of Frank are in the print variety. We just could ask the question this naively this way. The question doesn't actually make sense, because the print variety consists of divisors of degrees zero so clearly are is greater than zero. I'm not going to find any divisor Frank are, but we actually take a certain variation of the print variety which I will call print K, which means it's going to be the pre image by the norm map of the canonical divisor of C. So the usual print variety you can think of it as a component of the pre image of the zero divisor right it's everything that goes to zero it's a kernel. So here we're going to replace it by the primitive the canonical divisor. And I do this I get something which is isomorphic to the print variety so it's just a translation of the print variety. It's actually the same object just said now. These are divisors of degree to G minus two. And it makes sense to ask for the rank of these divisors. Right. Now, if I have a curve that is general in Moduli, and this this time the model is the hood stays. Then there is a classical result that tells me exactly what the dimension is by classical I mean from the AIDS. And the old filters showed that the dimension of this locus is g minus one minus r plus one choose to. And so again g minus one is the dimension of the print variety. And this is just the core dimension inside it. Okay, now we can naturally asked what happens if I is not general in Moduli. And just like for the usual Bruno theory, what happens if my double cover is in the K-Gonal locus. And so here by K-Gonal locus I mean that the curve downstairs the curve C is a K-Gonal curve. So this was joint work with Martin Ulrich and then also with Stephen Creech, and we're able to find an upper bound for the dimension of this locus. When our downstairs curve is generic is K-Gonal. And the formula is that the dimension is bounded by g minus one minus L plus one choose to minus L r minus L, where L is K over two rounded upwards. So this is an upper bound for now, I should say that the same methods also lead to an upper bound in the really generic case. And in that case, the upper bound coincides with what is not. So hopefully that should suggest that maybe this upper bound is the right upper bound. Okay, so what's the idea here what the strategy that we use. So first of all, for problems of this flavor in algebraic geometry. Using upper semi-continuity properties. You actually need to only find one example. One case where your statement of the theorem holds. So this is what people often call it's a problem of searching for hay in a haystack, right, because we're claiming that some property holds almost for every curve. And all we need to just find one curve that satisfies that those should be easy as easy as finding hay in a haystack but somehow coming up with a generic curve is not an easy problem. Right, so so we know that the cream construction commutes with tropicalization. From that we know that actually suffices to find a single tropical example that satisfies our condition once we find a tropical example that will tell us that there is an example which tells us that the theorem is correct. And besides because now we can just choose our favorite graph and and compute, or the double cover of graphs and compute the cream real not a lot of that computer dimension of that and hope that this will be something interesting. And we went for certain graph that is known as the chain of loops or rather the bottom graph is the chain of loops. And which we call the folder chain of loops. Yeah. So we started the brilliant theory of this double cover. And we're able to show that our formula actually holds on the nose for for this graph so it's not on the tropical side it's not actually an upper bound it's a precise dimension. But that implies an upper bound on the algebraic side. And as I was saying what suggests that this might be a good graph to look at is that when we apply the same ideas. Oh, I should say what makes this cave on all is that the length of the upper loop is k times the length of the lower loop for each one of these loops. But if you choose generic edge length, then you recover the formula by Beckham and voters, which makes us hope that this is the right formula. So, all of these tropical prim stuff are all very recent. Dave and I define a tropical team in 2016. So there are many, many other things to discover either on the tropical side and also the connection with the algebraic side. So hopefully this is just just the beginning of a much wider story. And hopefully there will be new interesting results in coming years. Yeah, so thank you all for coming and I hope you enjoyed my talk. Thanks. Yo, very nice talk. Is there any question on the mark. I have one curiosity. So, yes, I have already heard some some conferences or some seminars on this tropicalization of curves. And I know that it is possible to get information on linear systems on the course, but these techniques. I can hear you very well. You said it's only possible to get information. Yes, on, yes, on linear systems or on on classical algebraic curves by tropicalizing. For instance, I have seen a paper on the monality sequence of a curve by these tropical techniques. Do you think that these techniques would help also to, to attack some problems like tomorrow to this morning problem, given, for instance, a curve of genus three to find explicitly some cover on the leaf ticker or something like this. Um, yeah, I think they should be possible to use tropical geometry for other purposes. It's usually these things are not immediate to basically kind of need to develop new theory for every type of problem that you encounter. Yeah, but I think sorry. Could this be related with the results somehow that you found or not. That's a good question. Yeah, I'm not sure. Do you consider the cover of K-Gonal curves? And so does your construction can then go back to construct covers of algebraic curves of small genus? Yes, I'm not sure which problem you're exactly referring to. There is definitely work about covers of curves and trying to lift tropical covers into algebraic covers. And so the answer is probably yes, it's possible but not obvious. Any questions?