 Хорошая удовольствие, чтобы участвовать в целибрации. И, да, мой talk is about wild coverings of Berkwich Cures. Я не надеюсь хорошей familiarity with Berkwich Spaces, так что я попробую, как-то, дать базу, без давления слишком деталей о Berkwich Spaces. Ок, теперь... Ок, теперь просто... А? А, я... Спасибо. Так что целибрации, я думаю, мы всегда работаем в геометричных ситуациях, так что мы фиксируем альжебракт, в полной реальной целибреции. И мы будем участвовать в Berkwich Cures, только в таком целибреции. А теперь фанетморфизм, геоналитичный Berkwich Cures, может быть... В принципе, вы можете тоже думать об эквивалентной языке, с этой языком формы модели, или рюкзитной. В конце концов, мы все изучаем, более-менее, то же самое. Объект от чуть-чуть различных тенголов. Так что это может быть довольно сложным. Я могу дать какие-то фотографии в сервисе, когда я попробую convincing you, но, на самом деле, не очень просто, хотя в данном данном случае. И они могут иметь очень... что-то я называю, топологичную ромфикацию локусу. Все эти нотения будут представлены позже. Итак, мы хотим дать какой-то комбинаторийной описания. Для обычных курсов стабильная редакция дает более-менее лучшую комбинаторию в детстве. Для морфизм вы увидите, что есть определенная стабильная редакция. Она не описывает морфизм в данном случае. Так что, мы хотим какую-то хорошую информацию, когда стабильная редакция дает. Минуты будут следоваться в варианте. Первая функция это, что я называю, различная функция. Просто, в любом случае, она комплюет в различных, в том, что в том, что в том, что в том, что в том, что есть файна-заморфизм, который просто поднялся экстенцию и обладает более-менее, в реакции романтистерий. Так что, это будет Eating Brook و selfish function, currently, I cannot explain what it is towards the end of the talk. We will introduce and I'll formally some results. Ok, this is just the goals and sources. Это к альтервертой? Нет. В данном случае интересный кейс. В принципе, это можно. Если рейс, то характеристика 0, то это не интересная, потому что все это так. Но если это кейс, то это both mixed and equal characteristic, это equally interesting and we will not distinguish them at all. Вы не поняли, но эти маномиальные объекты, они маномиальные, в которых, в чем? В коэффициенте? Да, в коэффициенте 0,1. Просто у вас есть функция 0,1-0,1, в которой есть маномиальные объекты с некоторыми коэффициентами. Вообще, коэффициенты будут от абсолютной цели кейса, но я не оформлюрую так. Окей. Теперь, другое функция, она училась в социальной работе моего пшд-студия, и форма порта, эта пара пейперов была о профильной функции. И также есть небольшая веревка 2 пейперов, которые мы сейчас обрежем, и, ну, эти пейперы тоже будут на IHS сайте, а также я поставил на моих сайтах. Окей. Теперь план. Ну, за половиной пейперов я will talk about more or less basic results, what is known to experts on Berkowitz geometry, but again, I would like to recall various things. So, first of all, we will discuss Berkowitz curves, then we will discuss morphisms of curves on what is more or less well known. I cannot say classical because Berkowitz introduced all this stuff in 90s. I used to say classical, but it's sort of classical for Berkowitz geometry. And then I will talk about different function and profile function in the last two quarters of lecture. Окей. Let's start with conventions. Probably, I will try to put some conventions here on board, because the only problem with slides is that you know, I can put very few information simultaneous on the board, so I will try to keep something. Here, ok, so evaluation always is taken real maybe I will put in such real evaluation, if it's not obviously, otherwise also we'll use notation k-circle, this is the ring of integers it's customary for rigid geometry and Berkowitz geometry, and the residue field will be denoted by tilde. Окей, so with notation with respect to valued fields now also we fix k, which is an algebraically closed ground field, complete ground field. And evaluation is not trivial? Evaluation can be trivial, when all results will be trivial, so it depends on you, if you prefer to allow or not, but in principle it's Окей. k-analytic curve X will be called nice, if it is smooth proper connected, in principle one can instead of smooth, only say, rick smooth and this boundary for simplicity I ignore this. In the papers one considers slightly broader notion of nice, so for us just smooth proper connected, and then it's just a notification of some algebraic curve, which is also nice, it is smooth proper connected. Окей, and f will denote a morphism we want to study so it is finite of nice curves. Окей, Good, now Go back one slide. If X equals X, is that what you need to say? If What is if, in particular? In particular it's a property of such a thing. If it's an it's not always an aleutification, it's a very curve, is it? Under these assumptions it is always it's written in particular what is the aleutification. Окей Now About points of I don't want to give general definition but maybe I assume a little bit with you often literally a little bit with region geometry. So Mercury spaces are defined similarly one takes affinoid algebras define some spectrum, which is richer than in region geometry and then glues them, gluing is a little bit subtle, like in region geometry so I want to skip the details but it's important to mention that points correspond to same evaluations to real same evaluations on affinoid algebras and to any point one can associate a complete residue field, a good invariant point unlike algebraic geometry where points are just evaluated and fields here are evaluated in complete fields, so a really a good invariant of point is completed residue field so for a point x in x, we are given H of x, completed residue field I think Valkovich consider more general affinoid algebras than than intact, so you draw the streak That's correct, but I decreased generality here from beginning, so we can assume everything is strict and I'm not going to use this to any detail, so affinoid algebras will not appear in this talk okay for any k-variety, so in principle since in our generality everything is algebraisable we can only think about an elitification of some stuff and it's maybe a slightly simpler way so for any algebraic k-variety Berkevich-Fontorel defines an elitification and a map from an elitification x to a variety curly x a fiber of a point z consists just of all real valuations on the residue field of z and for any such variation we can complete k of z with respect to this relation and we get completed residue field of a point so in particular for any closed point since k is algebraically closed by assumptions residue field is just k so we can put only one valuation compatible with valuation of k valuation of k so the fiber is just a single point and this is a classical point appearing in rigid geometry so we'll call it a rigid point usually now in particular if curly x is an algebraic integral k-curve when said theoretically a curve elitification of curly x consists of classical rigid points and everything else adjust valuations on k of curly x on field of rational functions okay now points of k-analytic curves okay so in principle points are divided to four classes this is more or less necessary five but okay let's try to do some other with this so type one adjust usual rigid points which are sort of classical ones everything else should correspond to non-trivial extension h of x of k is a non-trivial extension so when we are given a non-trivial extension of valued fields we can ask what extends if residue field extends or if group of values extends type two is the case when residue field extends so h of x tilde is strictly larger than k tilde and it turns out it requires an argument but it's not complicated in this case group of values is just the same as group of values of k I am using that k is algebraically closed in this claim and the residue field complete residue field of x actually is function field of a k-curve so it's translated as degree one this curve c of x so x type two and c of x is the residue curve curve of h of x tilde of k tilde okay now type three is the case when the group of values extends in such case one can show that residue field is trivial and the factor of groups of values is just z and fourth case is when we have a non-trivial extension but nothing extends neither group of values nor residue field this is the worst case, it's called type four now a small remark fortunately for us type four points will not be essential at all so I'll mainly ignore them throughout this talk which tries to prove stable reduction or resolution in high dimensions stable reduction in high dimensions this is really your enemy but once we have stable reduction for curves we are not real trouble okay now a fine line so let's consider example how these points look like for a fine line so take x is a fine line of k and fix a coordinate t then it turns out that any point is a very simple description just the norm corresponding to this x same evaluation when we shift coordinate by a is just a sort of monomial valuation is defined as maximum on monomials so it's a sort of generalization of classical Gauss valuation when r equal to 1 and a equal to 0 is Gauss valuation okay in particular if x is of type 1 this happens if and only if r is 0 so this is just classical point equal to something and otherwise x is the maximal point which satisfies the inequality absolute value of t minus a is less equal r because obviously any point which satisfies this should also satisfy less or equal in wet equality so x is the maximal point of subset given by absolute value of t minus a equal r so it's just a disk so any such x is maximal point of a disk so points of types 1, 2, 3 are just classical points and maximal points of disk what do you mean maximal point? the same evaluation it defines on polynomial in t is larger than larger equal than same evaluation of any other point contained in the disk okay now we can parameterize point by a so the question is what is the redundancy of such parameterization into sound redundancy is very simple the disks must coincide for points to coincide so non-archimedian disks coincide if and only if radius is the same r equals s and a and b are close enough any point is center of disk so this is and because of this we can draw a picture of x very easily for example for any point a we have a line of points p a r for any other point b we have a line of points p b r and they meet precisely at the radius absolute value of a minus absolute value of b so the whole structure of type, points of type 1, 2, 3 is just a sort of tree which probably even if you have not studied York Visionary, maybe the picture of tree you saw because it's very easy to draw reviews you just see such a tree now a small remark not important for us, type 4 points correspond to embedded sequences of disks without intersection classical intersection so it may happen that we are given some intersection of disks without anything and when it is type 4 point I put it in LeBrakis this is not important ok now skeletons so let's say with a subgraph gamma of x by a subgraph we mean the following it's a connected subgraph whose vertices are only of types 1 and 2 maybe on this picture I should also explain what are the types so all these points are type 1 and type 2, 3 so the intersection points they are type 2 the interesting points disks of rational radios these are points where we can specialize to different points, to different directions and something like this is a type 3 so we want vertices of the graph to be only type 1 and type 2 and such a graph is called a skeleton if the complement of the set of vertices is a disjoint union of open disks and open annuli or maybe semi annuli by semi annulus I mean disk open disk with a punch, so disk minus a point because my vertices can be at type 1 points I also allow such a creation so a typical example take something here make a cut what you have between 2 cuts will be an open annulus can you also join 0, infinity and p1 through all the count no, ok I am trying to keep a little you want at least one type 2 point you are absolutely correct ok, I try to you want one type 2 point any skeleton gives a good combinatorial approximation of x because of a felling fact first of all x without whole gamma so for any annulus we have a central chord like here which connects the two infinite directions of the annulus so if we remove the central chord from the annulus we get disjoint union of disks and fact is that maybe I just I did not spell out we want the vertices to be as I explained and we want edges to be central chords of the semi annuli so if we remove the whole graph the whole skeleton from x we just get disjoint union of open disks so curve becomes extremely simple without the skeleton indicates the skeleton knows almost everything about the curve at least any combinatorial information any larger subgraph any larger guy which satisfies these conditions also is a skeleton only for curses just low dimensional phenomenon and one can compute for example genus of x as the sum of genus of all points plus the first betting number of x or one can only sum over vertices of the graph and add the first betting number of the graph it means that the graph contains all points of x with nonzero genus the graph contains all loops of the curve so graph knows a lot about the curves it follows immediately from this observation that x without gamma is disjoint union of open disks now by a genus of a point of a curve I mean the following if point is of type different from 2 when genus is 0 we cannot assign anything interesting but if it is associated to x a curve cx of a residue field and we can take genus of this curve what do you mean by h1 of x which kind of comorrhage is it h1 you can take usual with integral in which space as usual as usual topological space in fact I don't need it but okay as usual topological space okay now semi-stable reduction theorem asserts that any nice curve possesses a skeleton okay probably this is not a formulation you are used to but it's equivalent and this is so called skeletal formulation of semi-stable reduction theorem we do not really need its relation to formal covers to formal models I will just briefly indicate it so classical formulation is equivalent to skeletal one and it claims that x possesses a semi-stable formal model of a formal spectrum of k0 or since we are even in algebraic situation even of a spec of k0 we can take algebraic model and relation between two formulations is as follows we always have a reductional specialization map of closed fiber of formal model and then preimage of generic points of closed fiber of any generic point is just a single point of type 2 preimage of a node is an open analyst and preimage of a smooth point is an open disk so actually if you are given a semi-stable reduction and a skeleton and moreover you can naturally embed the impedance graph of the closed fiber of semi-stable model into Bjorkovich space and this set of vertices will be precisely parameterized by irreducible components of the closed fiber now local structure of nice curves actually it can be obtained from stable reduction but one can also go in other direction one can prove by direct analytic techniques local description and when it's not so difficult to glue it and to prove stable reduction by purely analytic method so first of all nice curve is a huge graph it's something like this but we should think about it as a sort of graph By the way I did not discuss the question what is the topology again it's not so important for intuition I think it's enough there is some fancy topology but let's ignore it okay any type one or type 4 point lies in an open disk because on the we can only get type 1 at vertices we can have only type 2 and 3 any type 3 lies in an open analysis and description of type 2 points is a little bit more fancy so it turns out that any point of type 2 has many directions and these are parameterized by points of the residue curve so in the picture I placed here every ramification looks like p1 over residue field because all genuses here are 0 but in general points can be of type 2 can be more complicated and they do not have to be locally embeddable in 2p1 so there might be many points which contain some genus and okay well now one more basic fact about curve so any curve possesses a canonical minimal metric so that logarithms of absolute values of functions are piecewise linear with integral slopes so under this condition there is a single minimal metric its existence can be proved by few but in essence you can use stable reduction and you can avoid using it so its not as difficult as stable reduction now I will prefer to work with exponential metric or radius metric so radius of an interval is actually exponential of the metric of the length of the interval with respect to metric then absolute values of functions will be piecewise monomial as we should think about them so there is some dominant on any subintervals there are dominant monomials so in essence we avoid playing with logarithms going back and forth with logarithms if we use radius exponential metric but how can this be let us say for the curve there are some cases where the semi-stable reduction is not unique for p1 if you want to I never said anything about me but let's see p1 so here is the metric the distance between points with the same center of radius s and t is just s over t whereas depending on which one is larger than one so it does not depend on any choice given a skeleton gamma inside y we denote by r gamma the inverse exponential distance from gamma this will be important in the last part of the talk so maybe I'll put it here so if we are given gamma inside y a skeleton when r gamma is the distance from gamma so actually it means equivalently we can remove gamma from y when everything breaks to disks we can normalize them to be unit disk and when the distance inverse exponential distance from point y to be skeleton is just the radius of y inside the disk normalize to be of so the same normalization I think was used by Francesco in his work on solutions of differential equations good now so far this is the picture of course now let's discuss morphisms so first of all I would like to introduce a multiplicity function of morphism which is a morphism so maybe I'll put it here and f from y to natural numbers so for type 1 points multiplicity is just usual ramification index for other points the maximal ideal is trivial but the field is not complete residue field is not directly closed it can extend so the degree of this extension is multiplicity of map at y this is a very simple fact where f is a local isomorphism at y if and only if a multiplicity is 1 so actually it's an interesting question for example to describe where f is not a local is not a local isomorphism I'll call such set ok let me first say about this I'll call such set topological ramification locus so we consider multiplicity loci of f to be all points where multiplicity is at least d this will be denoted nf а? no no no it depends at some stage it will be but a priori you don't have to so all y such that nf of y is largely equal to d ok so one of our answers to describe this also we'll say that f is topologically tame if multiplicity is invertible in residue field ok now fact for any interval inside y if we consider it parameterized with radius with radii parameterization when first of all the image of i under f is a graph so and this graph we have natural exponential metric with respect to this metric the restriction of f is a piecewise monomial map and the degrees of this map of slopes strictly speaking one has to say degree but I'll say slope in this example these are just multiplicities so if we know function nf if we know everything about metric structure of the map we know completely what are the slopes so we completely describe the metric aspects of ok now simultaneous semi-stable reduction so skeleton of f is a pair of skeletons on y and on x so such skeleton on y the vertices of skeleton on y contain this set contains the ramification locals so in fact the only reason why I insist to allow vertices of type 1 is that I want ramification points to be inside the skeleton ok now theorem any finite morphism between nice curves possesses a skeleton why? because we know that once you can find one skeleton you can find as many as you want just in legend you still get skeleton so you can easily play with x and y and find a pair which is compatible so the theorem is not essentially strong it can be deduced relatively easily the price will pay is that it does not give such a good description of morphisms as simultaneous reduction provides for curves ok so this is this argument how to prove deduce with theorem from stable reduction and one can also formulate it in language of formal models will not need it so just a size remark it's more or less equivalent to existence of a finite formal model ok with both curly y and curly x semi-stable now to which extent simultaneous skeleton trivializes morphism ok the answer is written here on the complement of skeleton our morphism is just disjoint union of finite italic covers of disks by disks but such covers can be complicated that's all the stories so they can be really complicated here you exclude the characteristic pre-radical maps because you want ramification locus to be content yeah yes in this theorem in the end in the realisation theorem we will allow I agree with you ok let's ignore this point, it's minor you can always split to purely-radical description of tamed morphism so any topologically tamed that is morphism which is topologically tamed at any point italic cover of disks by disks actually is trivial so and also for annuli any topologically tamed italic cover of annulus by annulus is kumer so it's given by a simple formula and it follows that if f is topologically tamed then the morphism splits outside of the skeleton any action can be only on the skeleton and it's constant along any edge on any edge we have just the same power e, the same kumer map because any edge is just a skeleton annulus so this gives very good description of combinatorial description of tamed italic covers so in particular actually what we get we get a map of graphs with multiplicities and it satisfies natural conditions I'll run through first two because they're really obvious you have some local constancy of multiplicities so let me not comment on these two but just sort of a little bit more subtle condition is that we also have a local Riemann Hurwitz condition for this map of graphs so we are given a map of graphs probably maybe I'll start here map of graph gamma y to gamma x I know something like this and for a vertex here its image we can relate with genus of v and genus of u using the ramification along the edges by usual local Riemann Hurwitz and not surprisingly this thing follows from Riemann Hurwitz formula for reduction curves so it's not stronger than usual Riemann Hurwitz and in fact these three conditions where the only numerical conditions the map of graphs should satisfy there are lifting results that if you give a morphism of graph which satisfies this, that and that then you can lift it to be a skeleton of a morphism of burkwish curves ok now problems with the wild case ok, as I said the tal cover of this by this can be complicated and the extension of residue fields can be purely inseparable if it's purely inseparable we can say nothing interesting about the map of reduction curves so Riemann Hurwitz just completely breaks down third, even if it is separable the local terms of E is now larger than E and E-1 and it involves different in general so in case of wild ramification even if extension is separable we have trouble ok and two more examples the non-splitting set can be huge for example and here I would like to draw a couple of pictures for example let's consider a map T goes to T to the P over let's say C L it's C P over there but let me take here just general L and then maps looks something like this and it turns out that there is a metric neighborhood of the chord zero infinity the map is not split and it splits outside so this size is absolute value let me write it just immediately now it turns out that this radius R equal to 1 if L is not P if we are in the same case but it is number which is equal to absolute value of P 2 1 over P-1 if L equal to P both facts can be explained very easily you should consider radius of convergence of P-th root of 1 plus T this is 1 if L different from P and this is this is absolute value of P to P over P-1 if it is if L equals to P so just you try to solve this equation in H of X and you cannot solve in this radius but you can solve when you are outside so we see that in general the non-splitting set in non-time case it can be a huge chunk of a graph ok and the situation becomes even more complicated if if we consider for example C2 mixed characteristic 0 2 and we consider double cover of P-1 F lambda from E to P-1 of K in such case we have 4 interesting points infinity lambda 0 1 and it turns out that if absolute value of lambda is large and absolute value of 1 over 16 the number which is written here becomes absolute value of 2 squared so if this thing of exponential radius 1 4 does not touch this locus then the situation is relatively simple we know that we have 2 disjoint here we can extract root of T-lambda here we cannot extract root of T-1 everything else is split and ok and above this interval which connects the 2 non-splitting regions we have 2 pre-images so what we get here we have something like what I draw so what we get is a bad reduction a bad reduction curve and one can even compute the length of the loop in terms of absolute value of lambda so this happens if absolute value of lambda is large when 1 over absolute value of 16 but if ok I don't have time to explain what happens if we are on the border but it's easy to see if these 2 guys touch if just they meet at a point when instead of this you'll just get one point which is an ordinary reduction point now but the interesting thing is when absolute value of lambda is less than 1 over 16 this is equivalent to the fact that absolute value of j is less than 1 because there is scaling factor between j and lambda and this is equivalent to j tilde equals 0 which is the curve reduction curve is super singular super singular reduction and in this case just by hands one can compute the following situation lambda infinity 0,1 equals 0, infinity the same locus 0,1 but at the touch point there is first of all there is some direction I think something like square root of lambda in direction of square root of lambda maybe not completely certain where you'll get a this distance or distance from this point to the two axis is smaller than absolute value of 4 so this point is located much closer than you would expect you have meeting of two things and suddenly they cancel one another ok so this picture is definitely a little bit strange when you see it for the first time now I'll try to explain where it comes from and give some logic to ok so the different ok just motivation these last two remarks actually especially the last one says that different might be an interesting invariant to look at and especially because different measures the wildness of extensions if residue field extension is inseparable we have wild extension so it's natural to check how wild it is ok so we'll indeed look at different now the definition which I'll use in this talk is very different of separable extension is absolute value of annihilator of omega of integers of L over integers of k and before author says that this is incorrect definition let me make two remarks so first of all we use multiplicative language small remark so usual different will be minus logarith of what is written here like exponential distance and state of distance so we use multiplicative language and important remark is that this definition is the right one only because in our situation omega L0 over k0 is of almost rank 1 it's a subquotient of L0 so in general you should use a smarter definition which involves fitting ideals or something like that so it can be it's doable but I prefer a simple definition because one just proves by hands since we are sort of one dimensional situation we are only studying curves of algebraically closed field in our case one can show that this model is a subquotient of L0 this definition works also one can consider log different it's defined similarly but using logarithmic differentials if k is discreetly valued when these two are related by uniformizers and in general there are no uniformizers and they are just equal so classical Riemann Hurwitz formula for morphism of algebraic curves is written using difference so I just put it here but it's in Robin's book it's very classical ok and now the different function so now naturally I will denote by hyperbolic the set of points of types 2, 3 and if you wish 4 but again 4 is not important for us and to generically tile f from y to x we assign different function that is to any point y we just assign different of hy h of f of y now small remark, side remark we work with not discreet valuations unless we consider trivial valued case which is not interesting otherwise valuations are not discreetly valued and then between different and logarithmic different in fact what I define behaves better if you interpret it as a logarithmic function but we do not have to distinguish because just they coincide but maybe ok now this guy measures wireless of f in particular is just constantly equal to 1 in topologically tamed case and it easily explains all phenomena or so far I want first of all to give you intuition why this is helpful after that I will formulate and stuff like that ok so first of all here it turns out that different in mixed characteristic the difference is always larger equal than absolute value of p for extension of degree p so there is a minimum shift on the on the very bad locus on the curve which connects with two ramification points it's very natural and the different we measure different on y so this is y2x so here and also so this is absolute value of p and it decreases it increases in all directions with constant slope p-1 so on the distance absolute value of p to 1 over p-1 the difference becomes 1 and this is precisely the boundary of the wireless ramification locus when different reaches 1 wireless ramification stops so it completely explains what goes on here and it turns out that here also one can describe the picture a little bit more funny way again it's absolute value of p here and here so just slope 0 it goes with slope 1 in all directions outside of here so it goes here and here with slope 1 but from here you have slope 3 and this leads to the super singular point where you have more ramification everywhere else you have just slope 1 so from this picture you can completely describe the wireless ramification locus this will not be a metric neighborhood now because the difference is changing but it will be sort of conic neighborhood ok now general facts so, first of all it's piecewise monomial on intervals probably this is due to little bit bomber we proved also for type 4 we improved a little bit the idea of proving this is very simple on intervals divide interval to pieces which are embeddable in annuli between annuli you just write down a series and you compute it extends to a piecewise monomial function that is on any interval it behaves like a piecewise monomial function also to type 1 points and one can describe the limit behavior at type 1 points so it turns out that it becomes constant if you have tamer ramification for example like here in positive characteristic you may have wild ramification type 1 points and different vanishes and you can describe the slope and the main property you have a balancing condition at any type 2 points and this is the most important because in order to understand this situation we would like to understand how it happens where there are points where we have slopes 0, 0 and everything else is 1 and so on what is the combinatorics of this picture the answer is here it's just a sort of Riemann-Hurwitz form or an analog of Riemann-Hurwitz form so this is precisely the genus part of Riemann-Hurwitz and local contributions are also sort of something very similar and in particular almost all guys here are 0 so almost all slopes equal to in separability degree of extension of raise to fuels minus 1 so almost all local terms vanish ok now what about the proof I will really really give a very brief indication of the proof so proof of balancing maybe I have to make one remark it seems that this formula is a close relative from my discussion with Ahmet very probably it's close relative of formula for vanishing cycles of Cato and probably this formula was described by Rino for covers of curves over DVRs they seem to be independent we work here not over DVR and so on so formally probably there are no implications but it must have relative but again I do not have any formal claim here but our proof is very simple it takes about 2-3 pages idea is that delta f is a family of difference so just if by definition of different so we consider a lattice in omega x we consider a sort of lattice of integral differential just minimal OX0 model which contains differential of OX0 and then consider omega y integral over public of omega x integral this is a torsion shift of K circle models whose stocks almost cyclic and precisely measures with different and then ok, then choose some element of K0 of absolute value equal to different and compare reductions of omega y integral and public of omega x integral but shifted by this number and these two reductions produce non-zero meromorphic map between over reduction between omega c sorry, it should be public of omega c y c x to omega c y and the balancing condition just boils down to computing degrees of this shift y poles and zeros the degree of this shift is precisely the left hand side of balancing condition and local terms will give you ok so I just should say here that if extension is inseparable in general the natural map from here to here is zero so there is no interesting map but if such a picture appears as reduction of map between analytic curves then we can produce an interesting meromorphic map and it is responsible for slopes of difference ok, now minimal skeletons so we say that a branch at the point of type 2 is trivial with respect to different if the slope is the expected one if you remember almost all slopes are equal to some number and when they do not contribute to local terms to balancing condition so in such case we say that this direction is trivial from point of type 2 if we are given a skeleton of x and we want to enlarge it and put inside a skeleton so we fix something on x to avoid situation like p1 where we can choose different skeleton so we just fix some skeleton of x and then we ask what can be the minimal skeleton of x and y simultaneously which contains given skeleton of x and the answer is given by a theorem if we have skeleton and we take its preimage gamma y, a priori gamma y does not have to be a skeleton but it turns out that we do get a skeleton of f if and only if ramification locus of f is contained in the vertices of gamma y and for any point y all branches are pointing outside of gamma at delta f trivial so this thing actually produces you at least in theory and algorithm we need to give a cover how to find stable model of what you have above for example here you start with obvious for candidates for bad points, ramification points so we should include them we should take the convex hull but we see that there is a strange point on the skeleton where the different behaves not normally so we must include this direction when we include all this stuff non-trivial different we get skeleton of both ok ok and also different controls we set of if a degree is p when the different as I explained here for example and on that example different completely controls the non-splitian locus but if a degree for example is p2 or pq when it may happen that there is a huge locus where a degree is p and where a degree is pq and so on cannot be controlled by a single invariant so the question about all non-splitian locus is still open different cannot control it in general and this leads to the last part of my talk so let's say that a closed subset of curve is gamma radial with respect to a skeleton gamma if there exists a function on gamma such that s consists of all points whose exponential distance from gamma is bounded by r ok, maybe it's better to draw a picture to explain it just by single picture so let's assume this is some edge inside gamma and there is some function r on gamma from e to r and the set s is required to contain all points whose distance to gamma is controlled by this function on gamma so it's sort of something like radial set so in all our examples the locus was such a radial set with respect to different ok so theorem, there exists a skeleton of f such that its white part radializes all sets all multiplicity sets all together moreover if we found one such skeleton we'll do it's job and moreover in three cases you can choose any skeleton whichever you want these are cases when degree is p when f is tame and f is a Galois or even you can take normal curve which is composition of Radical and Galois example ok, if degree is p when actually the different trivial decreases gives you the radius of the set mfp ok how to prove it it's really very simple I call it splitting method it's used a lot in theory of valued fields so if you're given a problem about valued field you often want to solve it as follows for tame extensions for degree p extensions and do it by hands when extend to compositions and you get Galois extension because any Galois can be split to wild and tame and wild can be split to degree p and finally use some descent to get with no normal case and the realization theorem is proved just by this method but sort of globalized a little bit globalized one uses that category of italic covers of a germ orph at the point actually is the same as category of italic covers of spec of h of x so you can split locally you can split morphism like extensions of valued fields ok and finally ok, maybe I'll just since I'm starting to be out of time in a minute I think I'll just the last thing very quickly without giving details, spelling out details the last question is once we have a realization theorem we would like to know what the radii are we solve it for degree p this is different so is it a natural invariant of extension of completed residue fields and the answer is as follows just consider this radii r1, r2 and so on it's a better invariant because it's not compatible with compositions I give you composition of two functions it's difficult to express radii for composition through radii but you can rearrange it in a clever way ok, it will take two minutes to explain so just believe me you can rearrange it just as a sort of function from zero one that I call profile function and it's just equivalently original with radii and then the theorem says that if y2x is generic when for any point y of type 2 the profile function which precisely measures all this radii where splitting where we have locus of degree p to the n and then p to the n minus 1 and so on it's precisely the Herbrandt function of extension so last two comments even to formulate this theorem one has to extend Herbrandt's theorem to non-discrete setting actually it was not existing for non-discrete you just take proof of ser from from local fields and write something about almost monogenous extensions it's doable and when you can formulate the theorem and the proof is again straight forward by use of splitting method and for simple extension of degree p you just compare the two theories because both are controlled by different and the last is that this radii a priori defined only type 2 points but obviously they form some piecewise monomial family so there is a result about this ok thank you for your attention thank you very much questions just so you mentioned the fact that this is the difference so along the edges it is piecewise but is it so now what about it cannot be continuous in Birkwich topology for very simple reason the locus I describe you metric neighborhood it's not closed and not open in Birkwich topology it's closed in metric topology but metric topology is much stronger Birkwich topology your curve is locally compact so it's not ok at Birkwich conference I would stress this point that it is semi continuous for the Birkwich topology by easy it's always less than or equal to what you think yeah yeah yeah yeah semi continuity you have but that's all and it's continuous for metric topology it follows from this radialization theorem and it is definitely not continuous in some cases ok like what you said how does your high ramification behave in towers airbron function in this case again because you are almost monogenous you are sort of rank 1 composition of airbron when you have tower airbron function is a composition but for profile function which again it has very clear geometric interpretation profile function you just consider an arbitrary interval here from gamma to type 1 point you take its image this is 0,1 ok 0,1 and you have a restriction of f gives you some map this is the profile function if you are radial of the choice of this interval so for this definition it's obviously compatible with compositions for airbron function you should prove but it's like in classical theory of high ramification