 So, the title is here, Algorithmic Resolution, in fact it will be, what I'll talk about is an algorithm of resolution of singularities of logarithmic varieties, this is a sort of algorithmic identification of classical algorithm, so half of my talk will be about classical and half will be about the modification, let me start with introduction, I'll formulate one of many results and also with some motivation introduction, so first of all we always fix characteristic to be zero, the only case where classical is known, so we also only work here, what is over Q, we always work over Q, in addition by Varkey, I'll denote the category of integral or maybe locally, maybe the joint union, it does not really matter, integral varieties, if you care your characteristic zero, so for simplicity I'll only discuss this case and by Varkey log, so varieties of finite type of K, which are local integral, and Varkey log the same guys with FS log structure over K and trivial log structure, so classical resolution deals with the resolution of these things, canonical, factorial and so on, as I'll formulate in algorithmic we'll deal with this, okay now let me define maybe first a theorem classical, theorem one, which is classical, factorial resolution, it says the following, for any Z in Varkey such variety there exists a modification, it is proper vibrational map, Z res to Z, I'll denote this map F of Z, such that Z res is smooth, so this is indeed a resolution in a classical sense, proper vibrational, and the source is smooth, and in addition this is about funturality, F of Z is a funtorial, or compatible funtorial, for all smooth maps Y to Z, which is F of Y is just pullback of F of Z to Y, so funturality maybe just to mention few names, I don't know try it on the board, but this was proved without funtoriality just existence for any Z separately, by Hironak in 64, when it was done canonically, but is at least compatible with all toporphisms, canonical was done by sort of I think in the beginning of 90s by Gerstle-Millman and Bill Major, and later Lodaksek in 2000 actually showed that in fact this is even better, this is funtorial false smooth morphisms, there are many different descriptions of this algorithm, but it's always the same algorithm, so far, yeah and in fact the algorithm I'm talking about it also applies to the case when Lodaksek is trivial, so it also recovers this and it gives up in you, so I think I remember from some references sometimes it goes that the algorithms in the 90s like there was Bill Major first and it was not exactly the same that is there are different steps in some way, they are absolutely minor, but this is set up in neutral parts, you can do them in a little bit in different order or you can be not sure what it is okay and you can do it twice or twice instead of once and this is more or less with only difference, so to large extent it's the same algorithm, it's the same in gene, the same invariance and it was also insane as I think in Bill Major paper, yeah okay let's let's discuss it in questions, yeah okay and now theorem one law, logarithmic modification, this is by joint project with Abramovich and Lodaksek, everything I'm talking about here is in the framework of this joint project and it says we're folding, for any z in logarithmic there exists a modification z res to z, I'll denote it f logarithmic of z, a logarithmic algorithm such that the origin is log smooth and equals f of z product with y over z product in the fs category, so this is not just pullback, it is fs pullback we are working here, for any log smooth y goes to z and this log smooth functionality is much stronger than classical one, for example we can extract roots of exceptional devices with this we can cover branches covers branches of exceptional devices and it's still a kumar cover, it's log smooth, so we get from the divided bzb, so the main strength of this result is that it goes for log smooth, okay and what do you mean by the modification? it's just on the level of modification, it's no restitution on log structure, just modification on level of, but I will, it's just modification on level of under 1, right, but your condition of locally integral is not stable under the general local models of log smooth things because you you can have something which becomes at least for normal thing usually is the same but for just locally integral, if you take tens of zp, zq, it's not necessarily, so for even just extension of the field of scars doesn't preserve locally integral so it's, I don't know if it preserves, it presents a problem in the, it is definitely not a problem, in fact our theorem is a little bit more general, I just, I don't want to go to equidimensional and so on, just let me keep it yeah I allow for myself to cheat a little bit at two stages, yeah and the main cheating in this theorem is not what authors is saying, the main cheating is that in fact 0s is what we call a toroidal orbital, that is it is a dilemma for a stack with finite diagonalizable, so to get this stronger funtoriality we are forced to go to much more wide category and as far as we understand there is no way to get full funtoriality without passing those stacks, at least we have no idea how to do it and we suspect it's probably it's impossible, okay now one more remark is that mismatch z res to z is an isomorphism over the log smooth locus, so if I'm given z res to z and here I have z log smooth, over sub skin where this guy is log smooth, when actually I'll get isomorphism above this thing, this immediately follows from funtoriality because first of all this morphism should induce resolution of z log smooth and z log smooth is log smooth over point, resolution of point can be only a point, so just I have no choices, contrary to what it implies but I'm not touching with whole locus where this question is already resolved, okay good now maybe I'll just say one word about motivation, motivation is to the result morphisms z to b modifying this we would like to get to find modifications of the base entropy source so with this map is log smooth or lodging out for appropriate, for lodging out log structure on b prime and appropriate log structure on z, now it's known that just what's usually called semi-stable is impossible if the base is of dimension 2 and more and the relative dimension is 2 and more but semi-stability is an initial thing to expect with this possible and in fact such a curem is known in non-colonical folk way, it was first proved by Romesh Karo for Varietis and in a work on Travodegaber with Lukluzy we improved it to classic schemes of characteristic zero or it is a class of classic schemes but it's not factorial and because of that it modifies with whole generic fiber, it can modify generic fiber even if generic fiber is smooth so this weak version of resolution of morphisms does not imply semi-stable reduction and over-versionary and one of our motivations was to improve semi-stable reduction of general work with space and the only curem which I see is to prove phantorial resolution of morphisms it must be phantorial for log smooth, it must have stronger funtoriality than classical and actually the algorithm which we discovered now should be the generic fiber what's going on in hypothetical relative algorithm over generic fiber of zero. So what is the non-phantorial thing you are thinking about just not alteration but modification? Modification yes, I okay let me I just give maybe I'll say one word but we'll never see it again yeah it was just for to motivate the point is that we resolved indeed by modifications of both below any modification I don't care much to be a large enough modification of it and after large enough modification it should be just depend on pullback of Z in a log smooth phantorial way okay I'll yeah proceed good so far is introduction now let's go to classical methods so now I'll tell about hirnakas approach and developed by all the other people I mentioned so two classical methods so let's start with principleization hirnaka reduces in question of resolution to be following to be following question so I'll just say okay maybe I'll just say the words the idea is as follows first of all we want to prove something phantorial so it's enough to construct our algorithm locally because of cononicity it will glue automatically locally with Z can be embedded into a smooth ambient space and up to it our morphisms we is embedding only depends on the dimension of the ambient manifold so in principle there is not much choices here so it's more or less safe to assume that we are given economical embedding into something smooth ambient manifold and we would like to work on this smooth manifold such case we can work with Z or we can work with so embed this is a manifold that is just by this I mean smooth something smooth and work with the ideal which defines the Z inside the restructure so and we'll want to achieve something which is called principleization of this I now let me give a few definitions so first definition boundary is a strict normal crossing divisor e inside x so even if we start with empty boundary it will appear after blowing up when we'll start to blow up x this is very crucial for runacosmetic second we say that v inside x a closed sub scheme has simple normal crossing with e if locally there exists coordinates t1 up to tn at the point such that v is given by vanishing of t1 up to some number t r so in particular it is smooth and e is also given by coordinates but maybe some different coordinates maybe few of them from here and few of them not from here so e is given by vanishing of ti1 times ti2 ti s okay so namely I guess it's very simple we can just pick up local coordinates so with all our data is given as coordinate as linear planes with respect to these coordinates another definition i admissible is the following datum x prime e prime this is just formal notation I don't want I don't have time to define category of triples so this for us it will be just formal notation where x prime is blow up along some v v has simple normal crossings with the boundary we started with so we are only allowed yeah admissibility means that they are only allowed to blow up something with normal crossing and in addition and in addition v of is contained in the vanishing local supply and also we only allowed to modify the locals where I is not trivial e prime equals to let me denote this map to x by f e prime equals to preimage of the old boundary we call this old component union with preimage of v which is new component and i prime is just pullback of i so the ideal is just pullback and now exercise x prime is smooth a manifold and e prime is a boundary just because our definition immediately allows to go to local coordinates you just compute everything locally and what's it okay now theorem 2 principalization says that for any such a triple e i where exists a sequence of admissible I'll denote it x n e n i n this guy is sort of result what do I mean result that means we call it i n is invertible and monomial the vanishing locals of i n is contained in e n so after this procedure we manage to make the ideal to be just product of few components of e n with some multiplicity okay and now maybe just remark why this theorem is stronger than theorem 1 so if z is vanishing locals of i in this situation and we start with empty boundary then let z n equal to empty set goes to z to z n minus one and so on to z sequence of strict transform where the total strict transform is empty why because the pullback of i is supported on the exceptional boundary so it necessarily means that and we started with empty boundary so it means that some fish strict transform was killed and then take maximum such that the i is not empty easy to see I will go to detail but it's very easy but in such case the eye is just the eye eye center and hence the eye has simple normal crossing with the eye and this implies that first of all the eye is smooth but even more it implies that the restriction of the eye is bounded is is the simple normal crossing device so in a sense this theorem implicitly gives indication that you are not just resolving z you are resolving a logarithmic scheme you solve strong the problem you resolve logarithmic scheme z i and the restriction of the boundary so even in classical situations there is a flavor of logarithmic things and another maybe at least I shall say the question what the boundary is so usually at least when I started to deal with resolution I thought it's just some nice subscreen and it's not good to think about this as a subscreen because because materiality is wrong there's no map from e n to empty to e which is empty so this is not this morphism they're not morphisms of pairs if you want to view this voluntarily we should think of e as a lock structure we should consider the lock structure associated with the simple normal crossing device and when indeed we have such a map from between logarithmic schemes and on the gravity of the ideal is direct okay good yeah I'll just say it without without writing on the board moreover we can a little bit it was not written anywhere but one can prove a little bit stronger result instead of resolving z with empty lock structure you can consider z with a deline falting lock structure lock structure whose monoids are free any such lock scheme can be embedded by strict closed immersion into some pair x e where x is loose and e is simple normal crossing device in such case actually you'll get a strengthening of pure one to lock schemes with deline falting's lock structure so it was very natural question also if the classical algorithm which in fact results deline falting's lock schemes can be extended to the algorithm which works with general lock schemes at cost of working here with general lock smooth lock operators and this is indeed what's done in our algorithm so this is in other motivation sort of absolute orthogonal to the post I started this to construct logarithmic extension of this algorithm good now next topic is order reduction so main variant of the algorithm is order of i at the point okay I gave a lecture a little bit before I stated this a little bit differently but again I allowed myself to cheat a little bit order of i which is which is just a minimum over all f inside the stock of i at x uh excuse me minimum of order f at x where f belongs to the stock of i at x and order of a function on a manifold is just the order of minimal um non-venetian term in Taylor series for example what did you write main invariant f at f what is after of f is order of i at x what is f f is the algorithm let me say okay of the algorithm okay okay I didn't just let's give a definition I would definitely prefer weighted but people coined you know we weren't marked so now it's always marked ideal it's just id where i is ideal and d is a number which is like one zero and it will be just formal combination of but so so I don't pretend to give any deep meaning to this but I will consider a singular locus of such guy as all points in the vanishing locus of i such that the order of i at x is at least so we would like to reduce d so essentially we look at places where the order is at least some number and the formal will work in such way and in addition we say with a blow up let's say an admissible such guy is called id admissible blow up if is contained in id singular that is everywhere where we blow up the order is at least d and f is i admissible in all sense but this we blow up the multiplicity is d and the the center has simple normal crosses with the boundary okay and moreover we define transform for a such we define transform i prime which is smarter when what we have done before before we just took pull back but now I say let's take pull back and let's divide it by this power of a new exceptional divisor what is I take ideal of v pull it back this is the ideal of the new exceptional divisor and take it to minus d so is here there is an e of boundary or not in the definition of mark day in the definition of mark ideal there is no there is no d there is no d it is okay e is not is not okay but the the invisibility are required to be simple normal crossing with the power yeah when I say that it is i admissible it means that it's simple normal crossing with the power okay so such a thing is defined only because of the condition that my center is inside d multiple locus it's very easy again simple exercise check that this is defined that we can divide by this okay and then if you're in free it's called order reduction of marked ideals and it says this following it says that any marked ideal e i possess a sequence d admissible blowups you know the x prime d prime d but it's a sequence it's not a single one it's a sequence and each time we transform ideal by such a such way this guy is a result it's order it's momenty again as in theorem 1 or theorem 2 this sequence should be factorial and not spelled without but it's natural continuity for arbitrary smooth morphisms between initial that and maybe i'll just mention that theorem 2 is just theorem 3 for d equal 1 so theorem 2 is just part of your case and it was something for inductive reasons it's much much much more convenient to prove okay now in a logarithmic setting if you get to some details it will be the same structure we'll have fewer in one logarithmic, fewer in two logarithmic and fewer in three one so here x is smooth always it's manifold okay it's a manifold is a boundary i is any ideal and d is any number after reduction of theorem 1 to theorem 2 we only work with smooth things and ideals inside well we don't see any non-smooth geometry okay all singularities are encoded in the ideal which is just algebraic work so if the ideal is zero you just blow up everything and get the amplitude yes yes yes and in induction this sometimes this happens so it can't happen with the other if you ask it now when yeah it can happen when the other is infinity for ideal zero and indeed in such case you just should blow up everything and in logarithmic setting we'll have what infinity not only for zero it will be more interesting but okay we'll get good now next so let me explain a little bit how one proves theorem 3 so remind me if d equal yes equal one then i prime is the pre-major of i or but you each time you subtract one copy of exception divisor okay and you won't invent to get to empty guy okay if this one when the resolving it means just with its locus is empty okay so by subtracting each time just one copy of exception divisor you invent manage to completely resolve it and obviously this implies theorem 2 because it means that your public of your initial ideal becomes just union of exception okay so maximal contact and induction on the dimension okay so what i said so far actually is characteristic 3 this is an indication that it's not that deep but maximal contact is the first notion which will be really good okay so my miracle is that the maximal order that is the case when order of i at any point x is less or equal to a reduction to small dimension and it goes as follows the problem of resolving i and d is equivalent to the problem of resolving ideal c of i called coefficient ideal restricted to h called maximal contact hyper surface and beyond this defector so the induction on the mention runs as follows we replace our data by some other data and any sequence which results we knew that there is also real data in vice versa so the problem is sort of equivalent we encode everything which we had about original question into a question which lives in dimension one less not the way we can run induction and this is what you're an hacker did on idealistic this is a there is an exponent it's more about definition of market ideal okay this is okay but anyway idea maximal contact was formalized by zero but ideas going to run out about you know it took a lot of time to refine them and to understand what is real in gene self-contained in gene yeah and we thought out that there is an odd reduction part ideals is the minimal self-contained block which can prove itself by induction though the rest was sort of unnecessary in original but okay okay anyway now main example just to illustrate how this works uh so let's assume that i is just given by t to some power d plus a1 uh td minus one uh sorry a2 td minus two plus a d where t is t1 top coordinate and a i actually depend on t2 up to t on the other coordinates yeah for example locally we can write some Taylor series which gives something like this so let's assume that we are given such a just particular case or for the and then the maximum content is just vanished a lot of softy and coefficient ideal is just the ideal generated by a2 d factorial over two ad de factorial over d so it's just generated by coefficients but in weighted weight we should take each of them with correct weight and it's more or less clear if the weight of this guy should be two and the weight of this guy should be d so we weigh them and uh i'll just uh show one thing i claim with id singular with this we play the points where the multiplicity is d is the same as co5 restricted on two page de factorial singular that is the other obvious guy is d even though only the other obvious guy is at least two the order of next guy is at least pre and the order of this guy is at least d and this precisely means what is the definition so at least uh equality of initial sin singular and local lots of it's clear so uh uh id admissible block is any block who centralized here it's the same as block which is admissible with respect to new data now it's much more difficult but possible to prove it this miracle persists after any admissible block so after first block again we'll have a quality of singular lots of transformations so and you can also ask what where is a1 so the answer is that because of characteristic zero i don't need a1 i can always get rid of a1 and this is the case but i can't take this if uh not reveal a1 completely now let me say a few words about general case yeah now what i wrote now is sort of very coordinate dependent and i would like to construct something canonical something coordinative so this is achieved to large extent by derivations so one considers d just ideal of derivations on x over k the i is generated by blocker is generated by dt1 okay and uh one defines d of i to be equal let me say d of ideal generated by let me let me say i is generated by f1 fm then d of i is generated by f1 fm dt1 f1 dt1 fm dt2 and so on so derivation of ideal is generated by the ideal itself and all derivations of elements inside and then we can iterate and divide uh iterated derivation of good and uh when we can encode almost all basic uh of the algorithm which i described so far by use of derivations so it goes as follows first of all order of i at the point x is just minimal number d such that with this derivation of ix is trivial so derivation just reduces order by 1 it's a very simple exercise in local coordinate okay two uh maximum compact is any h which is v of t where order of t is one that is this is a really smooth guy t is a coordinate and t is contained in d minus first so we know where d minus first derivation has order one yeah because order reduces by one each time and so it contains elements of order one and these are naturally also this is very intuitive but such an h has maximal contact with our initial problem it's as close to initial problem as possible and free uh coefficient ideal is just weighted sum of derivations power d factorial over d minus i yeah so it's just weighted sum of derivations okay now i managed to define order in coordinate independent way i managed to define coefficient ideal in coordinate independent way i have a choice of t here and this choice is really here they can the only real issue to prove independence of the construction is independence of maximal contact this really this was an issue and it solved in few ways but no secret so in a sense this is maybe one of main technical uh problems in constructing our uh okay good uh i think okay maybe just one what are the complications of this method this as i see them yeah uh because of time restrictions i you'll have to believe me i i didn't give enough details that you you'll feel it but uh so uh first of all complication one is that uh p can be non-transversal in such case uh i cannot restrict my problem to h because restriction of e to h is not a boundary uh to solve this one actually first of all takes care of the of the old boundary new boundary will not pose a problem it will be always transferred to h but old boundary is a is a problem and one actually solves uh first of all resolves i the order of i reduces order of i uh it places where there is a maximal multiplicity of the old boundary so there is a secondary invariant already in here like a paper multiplicity of the old boundary and because of this invariant of the algorithm looks as d one s one d two s two and so on where this is because order we start with the order of i or maybe order of its non non non non part and then the number of exceptional divisors for a point after that the order of d uh the order of uh coefficient ideal restricted to uh maximal contact and uh again number of components of exceptional divisor and so on good and second i'll just say by worse uh in principle it would be much better to work with logarithmic derivations because all formulas even what happens to derivations after uh admissible plot they're easier for logarithmic derivations and not for usual derivations there is one point where uh use of usual derivations is completely critical and it is for the definition of the order our definition of order does not separate uh coordinates corresponding to exceptional divisors and other coordinates so we do not separate different uh logarithmic derivations they are different with respect to usual coordinates at logarithmic class but the order is uh completely not sensitive to this uh in a sense all combinatorial complications of the classical algorithm those which i put on the verac are actually because of uh this non-separating of two types of variables exceptional and non-exception okay good now let me uh start uh the logarithmic option so what is this conflict i didn't understand about this conflict about this complication so if the maximal contact hypersurface is not transversal to the boundary when i just i can i cannot i cannot use this h i can restrict i but i cannot restrict the boundary because i cannot restrict the boundary i cannot guarantee that uh the uh i can resolve i with empty boundary on this h yes but it can be no non non-admissible it can be non so no simple crossing to h but if i always blow up also in the maximal multiplicity locals of the boundary when i automatically uh have simple normal crossings with the boundary okay offer let me yeah yeah i guess everybody else is often enjoying but i'll explain i'll explain you later but just private okay good and i'll look where it becomes okay uh so main idea is uh favorite one look just replace look uh for example instead of xe just consider any uh log manifold which i'll sometimes maybe write as x and mx just look smooth right or sometimes we can represent it as x e but this time e is just x minus the reality for the log structure in addition replace d by logarithmic derivations by d log and replace other by log order and so on so just put log everywhere you can so let me start with such a procedure log order log manifold is just minimal d so with logarithmic derivation of this top of i becomes trivial so this guy belongs to n and infinity yeah it can be infinite it may happen but we never get but already in classical situation this was the case for zero ideal and only for zero ideal here it can happen more frequently if you take any uh monomial when any log derivation uh just multiplies it by number yeah all monomials they are eigen uh functions of log derivations so log order of any monomial is actually infinite so there are a lot of uh guys of infinite order but this will not be a problem in a sense this gives us this separation of two types of coordinates we have usual coordinates of what about and we have monomial coordinates which have infinite order you should not deal with them at all by use of derivations or by use of the classical immigrants they are completely in combinatorial side of algorithm so we'll get distinguishing between combinatorial parts coming from log structure and geometric part which is maximal contact okay well again just put here walk and you are done coefficient ideal maybe i'll put here logarithmic coefficient again put here log and you are done uh uh four as i said this time x is any log main point to go to another board what about e the boundary boundary is what i wrote where e is just x minus x three so log manifold is a f s uh smooth or what is log manifold yes f s smooth all all yeah yes log smooth log smooth maybe i'll just give uh six local uh let me see local picture completion uh formal completion uh looks as follows it looks as k you are joining uh some monoid and also you are joining t one up to t n so we call these guys regular coordinates we have other one and we go everything here monomial coordinates and this time i don't have any other for monomial coordinates all monomial coordinates are equally good for me and uh for example my algorithm should be compatible with taking extracting roots of monomial coordinates so i have no chance to have any or any reasonable lot of monomial coordinates we adjust combinatorial part of it okay and uh j uh d equal differential focus of j is admissible if j is over four t one up to t i let's say r comma m one comma m s where these guys are just every every terminology so now we are allowed to blow up any guy for this form so uh r of t one t n is a log sub manifold this guy is uh what's is a monoidal uh separate and uh no restrictions on numbers so uh obviously uh even if i start with something smooth after doing such blow up i immediately get something which is only log smooth but not smooth exercise again exercise check the blow up of x if need is log smooth again a very simple exercise but i just only should play suffer blow up and then suffer okay i should go to conclusion so i one t i something not not here j is t one up to t few t's and then some monomials okay the indices are not what you want no indices this is aren't yeah yeah in this okay yeah yeah i want to i am yeah okay and uh now uh in fact uh so uh the proof theorems uh two log and three log with respect to this notion of order and uh obvious uh generalization of uh marked ideals and uh the by the same by the same procedure it even uh the main invariant is just d one d two n where d i leaves inside n and infinity so the algorithm simplifies we don't have to separate from the old boundary uh we only have usual uh passage to maximum contact and that's it but we must explain how about uh plus one more stage which we call monomial stage which happens if d is in let me just illustrate how this may look like this may look like if i is generated by functions like m i j t to some power j by elements uh which for example informal completion look like this and all these m i j are non invertible in such case the other is yeah uh in this case just blow up just blow up the ideal generated by m i j by the monomial coefficients just blow up this guy and may transform with respect just subtract the exceptional device what you get will be ideal of finite order and after that you can run so m i j are where are in the here are monomial are monomial coefficients of uh unusual coordinates in yeah i i i i i am working in in this yeah so you have six in the monomial okay times uh powers of the t's yes yes and you just form ideal monomials this is where this is in fact this is the minimal ideal monomial ideal gene containing i this is invariant definition yeah if you want invariant you just consider minimal monomial definition containing i blow it up and you get finite order and uh i'll just say by worst yeah unfortunately yeah we are completely out of time uh the real complication yeah which i wanted yeah ten minutes to talk about is that uh sometimes the algorithm insists that you must blow up something like m to one over d for invincibility reasons and because of that in fact uh one such thing kills you must go to kumer uh italic apology and in fact we work not with ideals on what we want is kumer uh ideals and uh blow up of such things actually provides stacks so this is the reason why stacks appear and the i stop here i think i guess you can start with the linear monomorphs logarithmic stack from the beginning yes absolutely yes uh that's that's correct uh but uh i i i think we formulate in this general in generality of the algorithm but to formulate first of all the the theorem it's easier to start this this this right also maybe just as a remark if you start this variety you can also want to finish this right no this is a step so after that you can there is also a step how to pass back to varieties but it is less factorial it's only factorial with respect to saturated uh locksmith this is a statement so you have it you have a better statement for saturated morphisms or i have a statement which says that if you start with variety you can end with variety with law with locksmith writing in the process will be factorial for saturated logmore and it means not for kumer covers but about other questions so in the right in case you need to introduce the doing of the stack which point you need right at which point i need step okay uh 35 minutes okay okay so i'll i'll try to very very briefly yeah so the point is involved i i skipped one more condition condition i know six seven six admissibility or id admissibility and this actually uh in this case it's not so clear what should be this condition of admissibility because my center is complicated i contain cinnamon but it turns out that the condition actually is very simple we just want i to be contained in this power of j this is the correct generalization of kernakas condition which we have blown up uh demultiple centers if this is satisfied when after the transform i can divide transform of a public of i by this power of public of j so this is the the correct condition now in case i apply this monomial stage for example uh i want uh let's take this big k of m monomial coordinate and i have ideal m2 for inductive reasons it can happen with starting from something innocent i i have to resolve something like this in such case obviously i would like to blow up m but j equal m is not admissible after blowing up such ideal i cannot divide by m square what is admissible is j to one half with this m to one half when i can blow blow it up and divide by by by the square so here it looks completely as a trick but if i if this happens only on h and i started from some x or x this can be highly non trivial just just one example which completely illustrates this is a suppose takes m one coordinate is remember and one is monomial yeah so e is vanishing of m and let's take i equal x square minus m so in classical situation and in classical situation uh it's only one but all the other is two because m is of infinity so uh so we'll consider this guy is order two now resolution says that okay you should go to h which is k of spec k of m and then either uh and you restrict i and you still have you restrict i to h and you have multiplicity so uh on h you must blow up m to one half and on x you must blow up something like x comma x to one half this is the center you must now how can we understand such a thing there are two ways first of all we can try say okay let's try a trick let's try to blow up x square comma m problem is that what we get will not be walk smooth uh we insert the bed singularity and this will be very difficult to control so blow up of this guy is not smooth not walk smooth not walk smooth but instead of this you can say okay my algorithm is factorial for loxmas covers so instead of uh walking is x let's go to y just adjoin a root of here we can easily blow up yeah so let's call is n and when my yield is x square minus n square it's very easy to resolve idea x square minus n square just blow up blow up x comma n we get by prime this is zero to cover so we can now divide this by zero two and that's what we get here if i just divide as a course model space i get the bed blow up i describe here so i cannot divide back with skipping lock structure but i can divide this as a stack this guy's and this is the sort of blow up what we call kumar blow up of x and one half so it is a stack for the log kumar log tantopology no it's stack for usual though it's enough to work as usual for policy we don't have to go stacks are just conventional stacks the ideals ideals for kumar tantopology yeah it's a little bit confusing but again we wanted to keep everything as simple as possible so for stacks it was possible to work on it is uh yeah but the y to x original is not an attire it's a tile kumar attire and not usually an attire for the stack quotient of y but okay okay maybe maybe we should discuss questions from paging yeah no questions from him to question from paris maybe yeah question i can discuss this later we see so you can say the speaker again