 Thank you very much for coming to the second lecture of this mini-course. Here is the plan for today. First, I'll explain the theory of Tamkin's metrization and the Kondasevich-Soborman essential skeleton. Second, I'll introduce skeletal curves, which is a key notion in the theory. Third, I'll explain where do skeletal curves come from in practice, natural sources of skeletal curves. And fourth, I will introduce naive counts of skeletal curves. And finally, I will give a proof of the symmetry theorem by skeletal curves as an application of the theory of skeletal curves. We will see other applications of skeletal curves in the next lectures. Okay, so let's start with the first part, Tamkin's metrization and the Kondasevich-Soborman essential skeleton. The idea is the following. By a college, non-alchymidian analytic spaces have very complicated underlying topological spaces. For example, the analytic p1 is an infinite tree, containing infinitely many vertices and infinitely many branches. And the Baikovic analytic elliptic curve is infinitely many trees attached to a circle. It is impossible to visualize Baikovic analytic spaces in higher dimensions. But they contain very nice piecewise linear subsets called skeletons. In general, skeletons are not unique. They depend on the choice of formal models. But if we are given a volume from omega on the analytic space, then we can define a unique skeleton, sk of omega, associated to the volume from omega. Thus, for Kalabiya variety, where we have a unique volume form up to scaling, we have a canonical skeleton called the essential skeleton. For example, the circle inside this elliptic curve is the essential skeleton of the elliptic curve. Here is the history of essential skeleton. In 2000, Kondasevich and Soboman constructed an essential skeleton inside non-archimedian analytic Kalabiya space, X over C double parenthesis T of maximum degeneration. Here, C double parenthesis T denotes the field of formal Laurent series. Their method is the following. First, they define a weight function psi on divisorial points, X div inside X using semi-stable models of X. And then they define the essential skeleton sk of X inside X to be the closure of the minimum locus of psi. After that, in year 2012, Mustafa Nikas extended the weight function psi to the whole analytic space, so it is no longer necessary to take a closure for defining the essential skeleton. Then in 2017 and 2018, Brown-Mason and Mori-Mason-Stevensson, they extended the weight function and the essential skeleton to pairs. And in 2014, not really in chronological order, Michael Temkin made a vast generalization. He bypasses completely the use of semi-stable models. In this way, he is able to extend the theory of weight function and essential skeleton to any non-archimedian base field, not necessarily of characteristic zero or discrete valuation. And moreover, his theory works in the relative situation for any analytic space X over another analytic space S. His method is the following. First, he provides the shift of scalar differentials on X, omega X with the maximal seminorm called scalar seminorm, which is the maximal seminorm making the differential D from the shift of functions on X to the shift of differentials on X a non-expensive map. He calls this maximal seminorm scalar seminorm. Then this gives rise to a seminorm on the canonical bundle kx by taking top exterior power of the shift of scalar differentials. Now, if we have a volume from omega, and if we apply this seminorm to omega, we obtain a real-valued function. And the Temkin proved that this real-valued function is equal up to a constant to the Kondasevich-Sobomar-Mustatonicus weight function in the situations where the weight functions are well defined. And the essential skeleton in Temkin's language is just the maximal locus of this scalar seminorm of this volume from omega. This is roughly his method. And since we will need Temkin's formulation to establish some properties of essential skeletons for some of our proofs, here let me give more details of Temkin's construction. So first let us define some seminorm at the level of rings. Definition given a seminorm of the ring B and a homomorphism of rings, phi from A to B, we equip omega B over A, the module of relative scalar differentials with the scalar seminorm by the following formula. For any element x in this module of relative scalar differentials, we define its scalar seminorm to be follows. First, we write x as sum of C i d B i, where C i and B i lie in B. C i and B i are elements of B. And we take the maximum over i of the norm of C i times the norm of B i. And then we take inf of this maximum over all possible ways of writing x as sum of C i d B i. So this gives the definition of scalar seminorm at the level of rings. And Temkin proves that gives a canonical characterization of this scalar seminorm defined by the explicit formula. He proved that this scalar seminorm is the maximum seminorm that makes the differential D from B to the module of relative scalar differentials a non-expensive a-homomorphism. Now let's consider the global geometric situation. Given f, a morphism of k analytic spaces where k is any non-archimedian base field, we apply this above definition at the level of rings, we obtain immediately a pre-shift of scalar seminorms on avinoid domains. Then via shiftification, we obtain so-called scalar seminorm on this shift of relative scalar differentials. And similarly, we have a canonical characterization as in the lemma above. Temkin shows that this norm defined from shiftification is simply the maximum seminorm on this shift of relative scalar differentials making the map D from the shift of functions to the shift of differentials a non-expensive map. And now if we take top exterior power and arbitrary tensor product, we obtain the scalar seminorm on pulmonary volume forms. There is a small technical point is that in fact, we have to consider so-called geometric scalar seminorm after passing to algebraic closure in order to get better properties. Now here is a theorem of Temkin. For any pulmonary volume form omega, we can take its scalar seminorm and we obtain a real-valued function on x. Temkin's theorem says that this scalar seminorm of omega is an upper semi-continuous function. The theorem says that this real-valued function seminorm of omega is an upper semi-continuous function. Now we make the following definition. We define the skeleton of x associated to any pulmonary volume form omega to be simply the maximum locus of the scalar seminorm of omega. It's possibly empty if the maximum doesn't exist. And we denote this skeleton associated to omega by sk of omega, considered as a subset of x. So this definition of skeleton depends on the choice of some pulmonary volume form. And now let's introduce the definition of essential skeleton, which is just a union of all such skeletons over all possible volume form. So here it is. Definition essential skeleton. Let k be a non-alchimedian field of characteristic 0 and let x be any smooth k-barite. We define the essential skeleton of x denoted as sk of x to be simply the union of skeletons associated to all omega, where over all log pulmonary volume form omega. And by definition, a log pulmonary volume form is just a section of this line bundle, which is some arbitrary tensor product of the logarithmic canonical bundle. And here we take any SNC compactification x in y, and so y is any SNC compactification of x, and d is the complement of x. And one can show that this space of sections is independent of the SNC compactification we choose. So we can just choose any SNC compactification, consider all pulmonary volume forms as sections of any tensor powers of the logarithmic canonical bundle, take the associated skeleton, and then take a union, and this is by definition the essential skeleton of x. Since we have taken union over all volume forms, it's just canonically associated to x. Let's introduce a notation for later use. When a compactification x in y is fixed, it's usually quite natural to consider the closure of the essential skeleton of x inside the identification of y. And we denote this closure by sk bar x, and sometimes we call it the closed essential skeleton. So that makes sense if we have a compactification fixed. Let's give some examples of essential skeletons. First example, we take x to be the algebraic torus. In this case, the essential skeleton of x is homomorphic to Rn, and it lives in the identification of the algebraic torus. One can show that the essential skeleton of x is in fact a birational invariant. With respect to volume forms, of course. So if u is a log-Kalabiya variety containing azarisky open torus tm, m being the co-character lattice as in the previous talk, then the essential skeleton of u is just equal to the essential skeleton of the torus, and it's homomorphic to mr, the lattice m tensored with r. So it's just Rn. So for our log-Kalabiya, the essential skeleton is very simple, just Euclidean space. Second example, we take x to be p1 minus some closed points. In this case, the essential skeleton of x is equal to the convex hull of these points. So recall that the analytic p1 is an infinite tree with infinitely many vertices and infinitely many branches. And we take out some closed points from this tree. The closed points, they are points on the boundary of this disc. Then the claim is that the essential skeleton of the punctured p1 is equal to the convex hull of these points. So here we take out four points, four closed points, and then the essential skeleton is the convex hull of these four points, which is this red subtree inside this infinite tree. Example three, we take x to be an elliptic curve with bed reduction, whose electrification is infinitely many trees attached to a circle. In this case, the essential skeleton is just the circle inside this analytic space. Then we have a two-dimensional analog of this example three, where we take x to be a k3 surface with maximal degeneration. And in this case, the essential skeleton is homomorphic to s2, two-dimensional sphere inside the n-identification of x. The final example we want to give is the following. We take x to be m0n, the modular space of p1 with n marked points. Then we show that the essential skeleton of x is homomorphic to chop 0n, the modular space of rational tropical curves with n legs. So we show this by considering the classical DeLene-Bernford compactification, m0n bar of x, consisting of stable n-pointed rational curves. And then we show that it gives rise to a minimal compactification. And we further deduce that the essential skeleton is just the usual, the skeleton associated to the compactification. And that skeleton was previously studied in the work of Abramovic, Caporoso, and Pain. So that's, for the moment, that's what I want to explain for the theory of Tamkin's matrization and the essential skeleton. Now let's turn to the next section. We will introduce the notion of skeletal curves, which is a key notion in the theory. The idea is the following. Let's consider an analytic curve C in the log-calabial variety U-analytic. We have our log-calabial variety U-analytic, and the verification of our log-calabial variety. And inside we have this blue essential skeleton, piecewise linear subset embedded in this analytic space, this blue essential skeleton. And we consider some analytic, this red analytic curve C inside our calabial. If the dimension of U is greater or equal to 2, then by dimensional reason, the curve C never meets the essential skeleton. Because the points in the essential skeleton are valuations on the generic point of the variety U. And the points in the curve, the curve C is a one-dimensional subspace. The points of the curve C, they are at most of dimension one, while the points in this essential skeleton, the points in this essential skeleton is of top dimension. So this curve C has no chance to meet this essential skeleton, just because of dimension reason. But we can let the curve C touch the essential skeleton of U, if we allow the curve C to be defined over a big non-alchimedian field extension, k prime of k. And here is the surprise. And soon as some k point of the curve C touches the essential skeleton of U, then the whole skeleton of the curve C must lie in the essential skeleton of U. So we observed that in general by dimensional reason, there is no chance for a curve C to touch this green essential skeleton. But if we allow the curve C to be defined over a big enough non-alchimedian field extension, then as soon as some k point of C touches this essential skeleton SKU, then the whole skeleton of C will lie in the skeleton, the essential skeleton of U. Now let us give the precise statement. We fix some log-calabia variety U, the orange U over k, some volume form omega on U, U in Y, here Y, some SNC compactification, and let D be the divisors at infinity. We denote by D essential inside D, the union of essential divisors. By essential divisor, we mean divisors where the volume form omega has a pole. So here in the picture, these dark blue curves denote essential divisors while this light blue, light blue curve is a non-essential divisor. And now we will consider some curve C in Y, this red curve that touches some points of the boundary divisor. So as we said, we must, if we want the curve to touch the skeleton, we must pass to a big enough field, base field extension. So let k and k prime be a non-archimedean field extension, and we choose C, a rational nodal curve over k. We consider F, a k prime analytic map from the base change, from the base change of C to the base change of Y, such that the pre-image of F, the pre-image by F of the divisor D is equal to the pre-image by F of the essential part. In other words, the curve C meets only essential divisors at infinity. And furthermore, we ask that the pre-image of the essential divisors is some linear combination of k points, P i in C, such a curve, which mainly lies in the interior U, and when it hits the boundary divisor, it hits only the essential part at some k rational points with some multiplicities. So F is a k prime analytic map between the base changes, and we consider the composition of F with this natural projection map given from the base change. So we have made a base change, and we have the natural projection of base change, and we consider the composition which we denote by F y. Now the claim is that if F y of x lies in the essential skeleton of U for some k point x, then F y of the essential skeleton of the base change of the punctured curve C naught is just C minus the marked points. So then F y of the skeleton of the punctured curve will lie totally in the essential skeleton of U. In other words, the whole skeleton of the curve lies in this essential skeleton of U. And recall from the example that we mentioned above, the essential skeleton of such a punctured curve is just equal to the convex hole of all the marked points in the analytic space. So this is a precise statement, and we call such F skeletal curves. Here is an example of a skeletal curve. We take U to be the algebraic torus, and we have seen from the examples above that the essential skeleton of the algebraic torus is just R n, this blue plane, essential skeleton homomorphic to R n, and we take our curve C to be just a p1. So it's an infinite tree, and we choose four marked points in p1, 1, 2, 3, 4, four marked points in p1. Then the essential skeleton of the punctured curve C naught, C minus the four marked points, is just the convex hole of these four marked points, which is this red sub tree inside this infinite tree. And now we consider a map from this p1 to the algebraic torus. So as we said, in general, this map, the image of this p1 has no chance to meet this blue essential skeleton, just because of dimensional reason. But if we pass to a big enough base field extension, then it might happen. And the theorem says that if some k point of the curve C hits the blue essential skeleton, then the whole skeleton, this red sub tree, the whole skeleton of the curve will lie in the essential skeleton of U. The major advantage of skeletal curves is that they have canonical tropicalization. Since the map fy maps the skeleton of the curve into the essential skeleton of U, we can just restrict this map fy to the essential to the skeleton of the curve, and then we get some tropical object from some finite tree, some tree, which we denote by gamma, to this polyhedral object. And this restriction is independent of any choice of retraction map from the identification of U to the essential skeleton of U. So in general, for general curve, this image of the skeleton of the curve will does not lie in the essential skeleton of U. Therefore, to get anything tropical, we must further compose with a retraction from the identification of U to the essential skeleton of U. But this retraction is not canonical. For example, different minimal compactification U in y gives different retraction maps. So then for general curve, different retraction maps gives different tropicalizations. But for skeletal curves, the compactification does not matter. We always have a canonical tropicalization. And we call this restriction the spine associated to the skeletal curve. So in the example above, the spine, the associated spine is simply the map from this red sub tree to the blue to the blue plane, this red curve. And this is canonical independent of any choice of retraction. Now let me explain the idea of the proof of the skeletal curve theory. Let's first recall the statement. We have some narachymedian field extension k prime of k and a rational curve nodal rational curve C over k. And we consider a k prime analytic map of the base change of C to the base change of y such that the curve hits the boundary divides meets only essential boundary divisors at some k points. And we can see the composition of F with the projection map of from the base change. The claim is that if f y of x, if f y sends some k point of the curve to the essential skeleton of u, then f y sends the skeleton of the base change of the punctured curve, which is just C minus all the marked points to the essential skeleton of u. In other words, the whole skeleton of the curve lies in the essential skeleton of u. Here is the idea of the proof. So for the proof, we put the map f above into a family. And we consider the skeleton of the family and also the skeleton of the base. We want to relate various skeletons together. In order to put the map into a family, very naturally we consider a home scheme consisting of all maps from the curve to y analytic. And then we consider the subspace of the home scheme h consisting of all maps f from C to y analytic of the same curve class and the same intersection pattern with D as the given one. We have the following diagram. So h is some space of maps. Over h we have the universal curve, which is just a product. Since it's just a space of maps, the domain curve doesn't change. So it's just a product, c times h. We have two projections, p c to c, p h to h. Then we have the universal map from the universal curve to y, which we denote by e. And we consider also the map phi from the universal curve to c times y. Whose first factor is projection to c and the second factor is given by the universal map. By the deformation theory of curves, we can show that the map phi is a tile over some dense Zariski open subset of the target. It's generically a tile. Furthermore, using deformation theory of curves by computing the tangent spaces of h, we show that the volume from omega on u in y, the volume from omega on u gives rise to a volume from omega h on h. So it induces a natural volume from omega h. Then we do an explicit computation. One can see that the pullback of omega, omega is here, the pullback of omega by e and the pullback of omega h, omega h is on h by the projection map p h. They agree on p h horizontal tangent spaces of the universal curve. So they may not completely agree, but they agree on horizontal tangent spaces. This implies that for any one form alpha on the punctured curve c, if we pull back alpha by the projection map p c, and we wedge the pullback of omega by e, this is equal to the pullback of alpha by p c, and a wedge the pullback of omega h by p h. It's just because they agree on the horizontal tangent space, the two forms. So if we wedge anything vertical, we get equality. We denote this by equality star. And the second, for any k rational point, k point x in c, which is not a marked point p i, we consider the evaluation map at x, e v x, which is a map from this space of maps h to u, just evaluating at x to u and l. Then since such x gives a horizontal section gives a horizontal section of this projection p h, we see from since these two forms agree on the horizontal tangent spaces, and if we pull back using the horizontal section, we see immediately that the pullback of omega by e v x is just equal to the volume from omega h. And that implies that the pre-image of the essential skeleton of u by e v x is just equal to the essential skeleton of h associated to the volume from omega h. This is because again, by the deformation theory of curves, we can one can see that the evaluation map e v x is generically a tile, and that implies that pullback of skeleton is equal to skeleton of pullback. So here pre-image of skeleton by the etanus of e v x, pre-image of skeleton is just a skeleton of pullback. So this omega h is pullback, and we denote this equality by double star. Now let's pick one fiber of our family. So we choose any point f in h, h is the space of maps, we choose any point f, we denote by c f the fiber of the universal curve at f. So it recalls that the universal curve is just a product. So the fiber at f is just some base change of c. And f h is the space of maps, f is a point in the space of maps. So f gives a map from the fiber c f to y analytic, which is just the restriction of the universal map e from the universal curve to from the universal curve to y analytic. There should be no c here. And it's natural to denote this map f because it's really given by f. Assume, now assume that f x lies in the essential skeleton of u for some k rational point x. Since f x is just evaluation of f at x, so this equality double star implies that f lies in the skeleton associated to the volume from omega h. Because we assume f x lies in the skeleton of u and f x is just evaluation of x at f. So by this equality, we know that evaluation e v x of f lies here means that e v x of f lies in skeleton of u means that f lies in the pre image of the skeleton of u by e v x, which means that f lies in the skeleton associated to omega h. So we get a very nice characterization of f now just from our hypothesis. And recall our goal is to show that f of the skeleton of the punctured c0 the punctured fiber c0 f lies in the essential skeleton of u. So in order to show that let's compute this pre image, pre image by phi of the product of the skeleton of c0 times the skeleton of u. We want to show this, we compute this product, we will use the map phi. And by definition the skeleton essential skeleton of c0 is just a union of skeletons associated to all possible log volume forms on c0. Here taking union over log volume forms or puluri log puluri forms they are the same. But first equality is by definition of essential skeleton. Next using Temkin's theory of matrization, Temkin's matrization theory one can show that skeleton of product is equal to product of skeleton. So here we have product of skeleton and it's equal to skeleton of product with respect to the wedge of the volume forms. And the next recall that phi is generically etal by deformation theory. This implies that pullback of skeleton is equal to skeleton of pullback. So here we have pre image of skeleton by some etal map and this is equal to skeleton of the pullback of the form by this map. Now recall by definition phi has two factors. First factor is the projection to c. Second factor is the universal map e. So by definition of phi, by definition of phi, this is just equal to the skeleton of pullback of alpha by pc wedge pullback of omega by e. And now we apply our explicit computation, this equality of forms on horizontal vector spaces. We apply our explicit computation, we deduce that this is equal to skeleton of this wedge product. So we replace this wedge, this pullback of omega by e by this pullback of omega h by ph. And then to summarize this by definition again is just the essential skeleton of c0 of the punctured curve times the essential skeleton of omega h. And we observe that by Tamkin's matrization theory, a point z lies in the essential in the skeleton of a product x times y if and only if z projects to a point y in the skeleton of y of big y and the z lies in the skeleton of the fiber. x y. So a point lies in skeletal product if and only if it projects to skeleton of the base and moreover it lies in skeleton of the fiber. Therefore, since f lies in the skeleton associated to the form omega h, so we think h as the base here, therefore for any x in the skeleton of the fiber, the skeleton of the fiber, the punctured curve at f, this computation, the equality above between this one and this one shows that x just lives in the pre-image of the product of skeleton because by what we just said, f already lives, so we look at this line, f already lives in the skeleton of the base. Now if we choose any point in the skeleton of the fiber, then this point actually lies in the skeleton of this total space and that is just equal to the pre-image of this product of skeleton. So this shows that x lies in the pre-image by phi of this product of skeleton and we deduce that just to recall the definition of phi, we deduce that f of the skeleton of the punctured curve at f lies in the essential skeleton of u. In other words, the skeleton of the curve maps to the essential skeleton to the essential skeleton of u. So proof complete. Remark, by adding extra k points to our curve c as marked points, the above argument has a stronger and perhaps more surprising result. We can show that the convex hull of all k rational points inside the fiber cf maps to the closed skeleton, sku, the closed essential skeleton, which is just the closure of the essential skeleton in this fixed compactification by analytic. So not only the skeleton of the curve maps to the skeleton of the target localabia, but the convex hull of all k points will lie there. So that is all I want to say for the proof of the skeletal curve theorem. If you did not follow every line of the proof, no worries. And now we will move to the next topic. So the question is, the skeletal curves seem so nice. They have canonical tropicalization and we will be using them for many purposes. So the natural question is, where do they come from in practice? And in the next section, we will talk about the natural sources of skeletal curves. Let's first make five minutes break before moving on to the next section. So the skeletal curves, they seem so nice, but where do they come from in practice? And that's what I will explain in the next part of this lecture. So let's explain where do skeletal curves come from. Recall from the Frobenius structure conjecture that we are interested in counting rational curves in Y with prescribed intersections with the boundary D. So we have Y, some SNC compactification of our log-Arabial U. And we are interested in counting this kind of red curves whose intersection numbers with the boundary divisors are fixed. Or if we can also phrase it in terms of the interior, in other words, we are interested in puncturing the rational curves in U with prescribed asymptotics at the punctures. Anyway, U is what we ultimately care about. So let's fix some notations for convenience. We have a tuple, both p, consisting of pj, where pj are integer points in the skeleton. So in my last lecture, I give an explicit formula for this SKUZ, which is just zero disjoint union with the positive integer multiples of essential divisorial valuation. And now, in this lecture, I explained the theory of essential skeleton, and they are just integer points inside the essential skeleton. So we fix this tuple in order to prescribe intersections of our red curve with the boundary D. Some pj can be zero, and we call such j internal marked points. For example, we can have an internal marked point, p4, internal because they maps to the interior. And for nonzero pj, we call such j boundary because these marked points are supposed to go to the boundary, and we write pj as in this explicit form, multiples of some divisorial valuation, mj times nuj, and the divisorial valuation is just some divisor at infinity. So we can always assume that nuj is given by some component of our boundary D after making some blow up. Now let's consider the modular space, the modular stack, mu, both p beta, consisting of n-pointed rational stable maps from some nodal rational curve c with marked points pj to y of class beta, such that each boundary marked point pj meets the interior of the divisor dj with tangency order mj and no other intersections with D. So exactly the sort of modular stack we consider in the Frobenius structure conjecture. And if we pick an internal marked point pi, then we can evaluate at this internal marked point and we obtain something in u. And we can also take the domain and take the stabilisation of the domain, we obtain a point in the Dlingman for the stack of n-pointed rational curves. So recall that the domain of a stable map may not be stable, thus we need to take a further stabilisation in order to get a stable curve. And we put them together, we have the natural map phi i, very analogous to the map phi we considered in the proof of the skeletal curve theorem. And now we have the theorem source of skeletal curves, which says that phi i over the skeleton inside the target has finite fibres. And moreover, the fibres, they consist of skeletal curves, which just means that the pre-image of phi i by phi i of this skeleton inside the product consists of skeletal curves. So that's the way we produce skeletal curves in practice. And just a small point, here we consider closure of the skeleton, so it's a bit stronger than we just consider skeleton. And that is important in the theory because we also want to consider degenerate domains. And in the proof of associativity, for example, it's, and also just in the classical theory of Gromov-Witton invariance, it's useful sometimes to degenerate stable maps and to break them apart. So that's why we also consider the closure of skeleton, which will contain these degenerate curves. So that's the way we produce skeletal curves in practice. And the proof is the following. For finiteness, we again use the deformation theory. We can show that for any fixed modulus of domain, the fiber of phi i at mu is finite at al over some zarisky dense open subset of the log-calabi yaw. And skeletalness follows from the skeletal curves here. So here finiteness allows us to count curves naively without using virtual fundamental classes. And let me explain now how do we count them naively using this finiteness result. So let me explain now naive counts of skeletal curves. The above theorem source of skeletal curves suggests a simple definition of naive counts associated to spines in the essential skeleton of yaw, which we explain now. And the study of properties of such counts is the main technical foundation of our theory. So recall we have our natural map phi i going from the modular space of stable maps to the modular space of stable curves by taking domain modulus and to our log-calabi yaw by taking evaluation of some internal market point. Now any stable map inside the pre-image by phi i of the skeleton in the target is skeletal by the above theorem. So we have a canonically defined spine, which is just we take restriction of f to the skeleton of our curve and this maps to the skeleton of yaw by the skeletal curve theorem. So here we take the closure of the skeleton, it doesn't change much, just more convenient to work with, because otherwise it's just infinite curves in like in Rn. If we take closure, it's just more convenient for notation. We can say where infinite point goes. So that's a very minor point. And conversely, given any abstract spine h from some graph, some tree to the skeleton of yaw and some curve class beta, we want to count all skeletal curves of class beta giving rise to this spine h. So this is our goal now. We want to define the count nh beta, which is supposed to be the number of skeletal curves with the spine h and the curve class beta. So first the question, what is an abstract spine in the essential skeleton of yaw? First, observe that the essential skeleton of yaw has an intrinsic conical piecewise integral linear structure. The idea is the following. So if we take any S&C compactification of yaw, we obtain a simplicial cone complex structure on the essential skeleton. And now such structures given by two different S&C compactifications, they are just related by some piecewise integral linear map. So therefore, this essential skeleton has some intrinsic piecewise integral linear structure. And thus it makes sense to define a spine in the essential skeleton to be a piecewise integral affine map h from some nodometric tree to the essential skeleton. Here is a picture. So now we consider such nodometric tree gamma. This is our essential skeleton and we consider a spine inside. And we denote by vj the set of one valent vertices of gamma. Let us first consider the case of extended spine. In other words, let's assume that all the vj's are infinite vertices. And we denote by pj the weight of vectors at every vj. In other words, just the derivatives. So these purple vectors are pj. And we denote the whole all the pj, we put them as a tuple both p. So here we have five one valent vertices v1, v2, v3, v4, v5. And v5 shoots up vertically, which means that the leg v5 is mapped to a point. The map h can be constant on the whole leg. And in this case, this p5 is zero the derivative. So recall that we said that the essential skeleton of m0n is homomorphic to the modular space of tropical curves, rational tropical curves with n legs. And in fact, this holds also after taking closure. So the closed essential skeleton of m0n is actually a homomorphic to the modular space of stable extended nodal rational tropical curves with n legs. That's gamma. The northern. Sorry. Yeah. Oh, I have a question on this. So the trap bar is just a naive closure of the tropicalization. The trap bar is a compactification of the modular space of tropical curves. So you're all internal legs of infinite lengths. The legs, yes. I allow some edges to have infinite lengths. Yeah. Because I don't know. Like the, I think the Jonathan Weiss and the melody Chen, they have, they define the trap bar, like which is, yeah, I don't know. Is it, is it this coinciding with the Jonathan Weiss and the melody Chen's definition of the trap bar? So this trap bar 0n, I think it was first considered in the paper by Abramovic's Caproso and the pain. Okay. Powered tropicalization of modular space of stable curves, probably. And with the show, we show that here the essential skeleton is just the essential, is just the skeleton given by the classical Dulin Man for the compactification. And then we apply a result in the paper of Abramovic, Caproso and the pain, which identifies the skeleton associated to the Dulin Man for the compactification with this modular space of extended tropical curves. Okay. Thanks. Yeah. So they are really natural objects when we consider compactification. Yeah. So we have our nodometric tree. And it's just a point in this modular space of tropical curves. So by this homomorphism, we obtain a point in the skeleton of M0n. And recall, we have our natural map phi i from the modular space of analytical curves to the modular space of domain times our local abial. Inside, we have a product of skeletons. And then we have a point gamma in the skeleton of the first factor. And we also have the point Hvi. So in this picture, it's just this point H of V5. We also have this point in the skeleton of U. So the pair together gives a point in the target. And now we just take the preimage by phi i of this point in the target. And by the skeletal curve theorem, the preimage is just a finite set and consists of only skeletal curves. But now we have a finite set. And not all curves inside this finite set are good. So we further restrict to a subset F i H beta consisting of stable maps whose spine is equal to H. So this subset, this set phi i inverse, it just says that our curve has the correct domain and the internal mark point i pi maps to the correct place. That's all. It doesn't say anything about the spine. That's why we consider a subset with the right spine. And then the count n i H beta that we want, that was our goal, we want to define. We just let it be the length of this subset considered as a zero-dimensional analytic space because probably we have some new potents or multiplicities. If we pass to a big enough infinite, if we pass to a big enough algebraic closure, if we pass to an algebraic closure, then it's enough to take just the cardinality of this set. So we define the count n i H beta to be this length. And n i H beta just means the number of skeletal curves associated to the spine H curve class beta and by evaluating at the ice marked point. So intuitively, this number counts these purple rational curves, close the rational curves with the given spine, given red spine. And more generally, we consider also non-extended spines. Sometimes we call it truncated spines. In other words, we allow some one-way lent vertices, vj to be finite. So the idea is to use toric tail condition to define the counts associated to truncated spines as in the first lecture. Here is the picture. We have skeleton of our log-cala-bio. And we consider a truncated spine. So here, the vertices v1, v3, v4, they are finite vertices. And v2 and v5, they remain infinite vertices. And in order to count the skeletal curves associated to such spines, we recall that we have a torus inside u with co-character lattice m. And this implies that the essential skeleton of u is equal to the essential skeleton of the torus and is homomorphic just to m tensor with r, rn. And now we can extend the truncated spine, this truncated spine h together with curve classes. And we obtain an extended spine h hat and an extended curve class beta hat. So I wrote since regarding curve classes in blue just to mean that you can ignore it if you are not familiar with the theory. They are more auxiliary. So we can just focus on the spine. So we apply the constructions above. We apply the constructions above to this extended spine h hat and extended curve class beta hat. We obtain a finite set, fi h hat beta hat as above consisting of closed curves with spine h hat. And now we consider subset, a further subset satisfying the toric tail condition. We ask each punctured tail disc to lie inside our torus. So then we are ready finally to define our count associated to such a truncated spine to be simply the length of this subset considered as a zero dimensional analytic space. So intuitively this number just counts this kind of open curves with given spine. And by open we mean curves with boundaries. So that's the definition of our naive counts. And we have the following theorem concerning this number, this counting number. So assume the spine h is in the general position. More precisely we assume h is transferred to walls inside the skeleton of u. So I will introduce the notion of walls in the next lecture. Here let's just imagine that h is in some general position. And then the count hi ni h beta, meaning the number of skeletal curves associated to the spine h and the curve class beta by evaluating at i mark point is independent of the choice of the internal mark point i and the nor of the choice of the torus inside. So remark the independence on i used to be called the symmetry zero and had a tricky proof by a deformation in variance. Now we have a much more conceptual proof via skeletal curves and let me sketch below. So that shows another application of skeletal curves. We can get a conceptual understanding of this independence of the choice of the mark point where we evaluate. So if there are any questions you can ask. Otherwise I'll just go to the proof of the symmetry theorem. Yeah, so let me explain the symmetry theorem via skeletal curves. So the symmetry theorem is just the independence of our count on the choice of the place, the point where we evaluate. I mean if you think why this is true it's not really obvious because we evaluate at an internal marked point i and we want to show that it doesn't depend on the choice of this internal marked point. So maybe we want to move if we have two different places where we evaluate maybe we want to move from one place to another. But the trouble is that when we move from one place to one place to another at some point we will across some walls and the spine is no longer transverse. So this kind of deformation in variance no longer holds if we move across walls. In general, we will have some wall crossing formula if we go through move across some walls. And here the way we want to show via skeletal curves is that we can actually move through the walls if it is a skeletal curve. So let me give more details. So the idea is to move from one place to another in the skeletal curve setting and in that setting we can go through the walls without some complicated wall crossing formula. Yeah, so let's recall the setting from the proof of the skeletal curve theorem. We have a home scheme parameterizing maps from domain curve C to the target Y and we consider a subspace consisting of maps with a given intersection pattern with the boundary and also some given class data. And we also had this natural maps. We have universal curve to projections. Universal curve is just a product and we have the natural map phi. First factor is just projection to C. Second factor is the universal map. And on Y we have volume from omega and on H by deformation theory we produced a volume from omega H. For any point F in H we denote CF the fiber of the universal curve at H and we denote the map, induced map again by F because that's what F means. So recall from the proof of the skeletal curve theorem, F being skeletal is equivalent to F lies in the skeleton of H associated to the volume form and phi X lies in the skeleton of the target if and only if F lies in the skeleton of H and X lies in the skeleton of the fiber. So that is what we have shown. The main point in the proof of the skeletal curve theorem. If you're confused about the rest, just yes. No, you can't formula what F lies in the skeleton. F is a map. F is a map but F is also a point of the space of map. Ah yes, sure. Sorry. Yeah, so F is a map but it's also a point in the space of maps. Yeah, so here we really showed that F as a map is skeletal if and only if F as a point lies in the skeleton. So that's what theorem says. Yeah, so now we assume F to be skeletal. In other words, we assume the point associated to the map lies in the skeleton. So we have a canonically associated spine which is just given by restriction H restricted to the skeleton of the fiber which is the same as the skeleton of the curve. So it maps to the skeleton of U. So that's all what we have done in the proof of the skeletal curve theorem. And now let delta denote the graph of the spine H. And here we make a claim. Assume that the spine H is in general position. In other words, assume it is transverse to walls. Then the skeleton of the fiber, Cf inside the pre-image by phi of delta is a connected component. So recall from this equivalence or recall from just from the fact that the whole skeleton of the curve lies in the skeleton of U. The skeleton of this fiber just lies in the pre-image. And we claim that this subset is a connected component. So I drew a picture for your understanding. Recall that our natural map phi goes from the universal curve C times H to C times Y. And we have the graph of the spine delta inside this target C times Y. And we have phi going from C times H. So this is C times H. H is the base, the space of maps, and every fiber and this total space is product C times H. Every fiber is curve C. So if we take pre-image by phi of delta, by the finiteness of phi, we obtain some graph inside the product C times H. We obtain some graph. So this fiber, the skeleton of this fiber Cf, it lives inside the pre-image because this goes to the skeleton as we have a skeletal curve. But we also have some other pieces. And the claim says that this fiber is actually a connected component. They do not, it doesn't touch with other fibers. So it's not difficult to see that, to show the claim. First, by the equivalence star, we see that this fiber, the skeleton of fiber is equal to the fiber of the pre-image to the fiber of the pre-image over F. So this implies that since it's a fiber, and the fiber is always closed. So this implies that the inclusion is closed. And we are left to prove that the inclusion is open. And we suppose the contrary, we pick some, so suppose the contrary, we pick a germ of a path, like this green germ, starting from the fiber, the skeleton, this skeleton of a fiber, we pick a germ of path zero epsilon, going to the pre-image by phi of delta, starting from this fiber and then goes out. And we can, so since the image of by phi of this alpha lies in this product, we can write it, write alpha as two components, QTFT. Maybe I should say that since, so since we have shown that the pre-image of phi, so alpha is a germ of path in the pre-image by phi, but we have shown that the pre-image of phi is just the product of skeleton of C and the skeleton of H. So we can write alpha as two components, QTFT. QTFT is some point on the curve and FT is some points in the modular space of maps. And we denote, since everything is a skeletal here, we denote by HT the span of FT. And now observe that the condition that alpha lies in the pre-image by phi of delta or phi alpha lies in delta and the delta being the graph of H0, this just implies that HT of QT is equal to H0 of QT. So we have QT fixed, a fixed point on our curve and it implies that for this small deformation of our map H, the image of this point doesn't move. And then by the continuity of tropicalization from FT to HT and the rigidity of transverse spines, this I will give more details in the next lecture, we deduce immediately that this equality must imply that HT is constant. In other words, there is no way to perturb HT, no way to perturb the spine while keeping this equality. So intuitively it's very simple, we have a spine and we have a fixed point QT and we fix the image of that point. Then if this spine is transverse towards, we cannot move this spine, it's just fixed at that point. In other words, this HT is constant. And if HT is constant, it means that QT FT lives in the pre-image of this fixed point, Q0 H0 Q0 for NET. And that is a contradiction to the quasi-finiteness of the map phi. So I said that by deformation theory, generically over the target, phi is finite at all. So in particular it's quasi-finite. But here we just produced infinitely many, we just produced a germ in the pre-image by phi of some point. And that's a contradiction, so that completes the proof of the claim. Yeah, and the claim produces us this nice connected component. And so let's just, I just explained the proof of the claim, but let us recapitulate what is the statement of the claim. So we have our natural map phi from C times H to C times Yn. And we have a skeletal curve, we have a skeletal curve F from C to Yn. And we assume that the associated spine is transverse. Then the claim says that the skeleton of the fiber CF inside the pre-image by phi of the graph of H is a connected component. And now observe the following. First, observe that the first factor of phi decides exactly where we evaluate for the second factor. The second factor of phi is a universal map. And the first factor the first factor of phi is the projection to C. So the first factor determines where we are evaluating for the second map. And furthermore, observe that if we take some of degree of our map phi restricted to this skeleton fiber, and here the degree makes sense exactly by the claim. Because we know that the map phi is finite at tau, generically over the target. So the degree makes sense. But if we restrict to a subset, the degree may no longer make sense. And here it still makes sense because we restrict this finite at tau map to some connected component. And then the degree is still well defined because the map remains to be finite at tau over some thickening of this connected component, some neighborhood. So the degree makes sense. And we take the degree and we take a sum of such degrees over all skeletal curves whose associated span is equal to h. And that is exactly the following count, nw hw beta, where we count the number of skeletal curves associated to the span hw, which is just the span h. But we add an internal marked point at w, meaning that we add some internal leg at w, which is contracted the leg. And then we consider the count of skeletal curves associated to this augmented span and the curve class beta by evaluating at the added marked point, w. And the left-hand side is equal to the right-hand side by the definition of this count. So now we can conclude the symmetry theorem for transverse span, the count. Now we see that the count nh beta is independent of the choice of the internal marked point. Because here we see that the left hand doesn't depend on the choice of w. And the right-hand side is the count of skeletal curves where we evaluate at this point, w, and w is allowed to move everywhere. So the count is invariant when we move w anywhere along the span. And this shows the symmetry theorem. Furthermore, we can show that adding or removing internal marked points does not affect the counts at all. So this is an illustration of how we use skeletal curves for establishing important properties of our counts. And we will see further examples of that in later parts of the lectures. So here for the symmetry property, symmetry theorem, actually we can have different proofs without passing through skeletal curves. But for other properties, we must use skeletal curves. And here it's nice to see that using skeletal curves, we really have the freedom of moving the point w everywhere. If the curve is not skeletal, there's no way to cross a wall while keeping the invariance. As a proof of the symmetry theorem, we don't move across the wall if we don't use the skeletal curve. So that's what I want to explain today. And for the next lecture, I will talk about deformation invariance, which is, and also many other properties of the counts that finally leads to the proof of the associativity of the mirror algebra. And for deformation invariance, as we said, usually it only holds outside walls. When we cross a wall, we are supposed to have wall crossing formula. We no longer expect deformation invariance. But for skeletal curves, actually, there is some trickier deformation invariance that somehow similar to this situation about moving around this marked point across walls. For skeletal curves, we can actually move across walls a little bit, as long as it's sufficiently transverse, but not really transverse. For non skeletal curves, it must be transverse in order to have deformation invariance. But for skeletal curves, we can relax a little bit the transversality condition. And that's actually important in the proof of associativity and also in the proof of wall crossing formula. Because in associativity, I mean the definition of structure constants, if you remember from the last lecture, the place we evaluate, we ask the point to go to Q. And the Q is, although it's a very generic point at the level of analytic geometry, it's a very special point at the level of tropical geometry. So all the spines that appear in the definition of structure constants, as in the previous lecture, they are all very special. They are not transverse at all. So in order, and of course, we can make them transverse if we don't ask the marked point to go to Q, but to go to some place, to go to some point sufficiently close to Q. But then we will have the choice of asking it to go to either the left of the wall or the right of the wall. Or if there are many more walls, then we have even many more choices of chambers. But in general, we have the choice of asking it to go to the left or go to the right. And it's not clear at all whether the structure constants for the marked point going to the left is equal to the structure constants for the marked point going to the right. And this going from left to right across in the wall, we have to use the theory of skeletal curves again. So I will explain more about that in the next lecture, the next amount. Thank you very much for your attention. Okay, thank you very much. And maybe you have time, maybe don't have time for the questions. Actually, I have a very simple question. You have this variety H, which has the same dimension as Y. It also has logarithmic volume form here. Yes. Yeah. But is it? Yeah, so it means that you can start to reproduce from some local abbey or another local abbey or in a sense. Yes. Yes. And does this H contain the torus? Again, if you assume that Y contains the torus. This H. Yeah. H is a cover of the torus. Probably itself is not a torus. Ah, it could be more. H is really the modular space. Yeah, I see. And also, we don't really have a good compactification of H. Ah, so it's not local abbey or more. It's a chromified cover, yeah. It's not clear whether it's local abbey or not, because we only consider the essential skeleton of H associated to this particular volume form. Yes. Yes. Yeah. Maybe there are other volume forms. Yeah. Oh, maybe this volume form has zeros. Yeah. It can have zeros or yes. Yeah. Okay. Okay. So thank you.