 need relevant or refinements of schemes of points over at the okay so is everything working like for the zoom people okay so right so first of all I would like to thank the organizer for letting me talk here I am gonna talk about a note that I'm writing together with Johannes this is supposed to be a very informal talk and also I'm not like we are not in this note we are not trying to prove like a new technical result or something but we are trying to like point out some relations and draw attention to some phenomena phenomena that happen in this with this animative invariance right so before I start defining stuff and talking about the problem itself I want to give like a quick overview of what I want to talk about so I will say some words that I will define later in the talk but it's gonna be okay so this story starts with this motivic dating variance these are refinements of the dating variance and these are defined in a ring which for now which is called like motivic ring of weight or something like this and this ring is actually a localization of the growth in the ring of varieties which I'm gonna define very soon but elements in this ring are isomorphism classes of varieties and one natural morphism you have here is the Euler characteristic with compact support so when you of course like if you have a variety you can compute its Euler characteristic and then you have like a morphism 2z and of course like by via this morphism you should recover like the classical dating variance from these motivic ones and another thing that you have here is the class of the affine line which we denote by L during this talk and one thing that we realize that happens very often is that when you consider this kind of invariance even though you are working overseas the same formulas you get for this motivic invariance can be used to get results over R and so for example for the main example in this talk which is the Hilbert team of points or the or A3 right because if you want to compute the team variance of A3 the space the modular space you want to like the degree zero invariance the modular space you want to consider is the Hilbert team of points and A3 and in this case so for for A3 what we get is the following formula that I'm going to write here so this should be this so you'll get this formula and this this formula was computed by Behren, Brian, Vendroy but they use the techniques that only work overseas but anyway you get this formula and one thing that we realize is that I mean this could be defined not only here but also in this the same ring but considering varieties over R or even over K right and and then if you compute the same Euler characteristic here but over R you get real invariance which were computed by Vulture, Corruption and Pascacchi so this was one thing that that's me it's not it's kind of surprising because like these these numbers were computed using techniques overseas and of course you could even consider over K and now I can refer to Sabrina's talk because over K you can consider the A1 Euler characteristic with compact support and get some results here and now the thing is that so so in this talk I want to talk about this stuff like I will explain how these are defined modules and technicalities and how these numbers or these formulas were computed but we don't have a good answer for why these accounts work over R but we have some conjectures let's say or ideas and basically you want to kind of pose some questions and see if anyone can answer and help us in this time so right so this is more or less an overview of what I'm gonna talk about today so let's start talking about DT invariance so some people already like talked a little about DT invariance I'm also not gonna spend too much time on this yeah yeah T no no okay so L is my refinement parameter and T is just because this is a generating series yeah sorry I actually didn't explain this properly so of course you get you get this this invariant but you can consider like this for any you are looking at degree zero invariance but you could take the invariance with any for any n right for n number of points and T is the generating series parameter so so this is a generating series of DT invariance for A3 yeah yeah yeah exactly so perfect so if you take L equals one so you're gonna get exactly the MacMahon function right and if you take L equals sorry if you take L equals one you get the MacMahon function if you take L equals minus one then you get the symmetric MacMahon function which are the real yeah sorry sorry I didn't explain thank you for the thank you for the question and okay so DT invariance so I think many people already talked about this but the thing is that basically we are considering for some variety x and some homology class beta we are looking at ideal sheaves and here I'm gonna write y to be the variety the sub-scheme induced by this ideal and we want y to have class beta and we want the Euler characteristic of the sheep to be in I mean this is the classical definition of this this is a modular space and the DT invariance of n beta are defined to be the integral of a virtual class defined on this modular plate and modular space right so of course and now that's the important thing if this space is smooth for example this is simply the Euler characteristic of your space now the thing is that in or up to a sign so there is also a sign here which is minus one to the dimension of that yeah so the thing is that if we consider beta equal zero then we are going to get simply the Hilbert scheme of n points in our x so if beta equals zero this i n is the Hilbert scheme of n points over x of x and and then or maybe I'll write n here and then we can and that's why I mean the Hilbert scheme of points will play a role in our story uh because we we want to to consider I mean the main example we are gonna work is for degree zero invariance so the Hilbert scheme of points will be important and now I'm gonna so okay these are the invariance and now I'm gonna explain how to get this motivic refinement refinement of of this invariant and okay this is where I think start to get little more interesting so the idea is that we have this growth and dig ring of varieties and this ring is simply given by is denoted by this so okay bar k and this is the free abelian group generated by isomorphism classes of varieties over k and and there are some relations actually there is only one relation which is the following so for any variety x if you take a closed sub variety of x then you can write this relation x is y plus the this is the complementary variety so so in the end like when you are summing varieties here plus is in some sense the union right so you are making the union of two varieties and you can also define a product which is simply the Cartesian product and okay for now this this ring as I as a okay so this ring as I kind of made clear here this can be defined over any k but it's important to say that we are gonna work mainly with characteristic zero because there are some results that only work in characteristic zero here so let's so think of k as a field of characteristic zero but yeah so okay yeah right okay so uh so okay so we have some properties of this ring which are gonna be important so the first thing is that two classes are the same if you have a bijective morphism so even if you don't have an isomorphism but only a bijective morphism you already have equality and also if you have a fiber bundle so e over b with fiber f then you have that the class of e is the same as e and x so of course this should be locally trivial in the Zarisky topology okay and and two important facts is that and another important fact is that this is so this is generated by all all isomorphism plastic of varieties but there is a theorem uh that you can that was shown by by Bittner that this this ring uh can be generated by projective variety with the following relation relations and these relations are that x minus y minus e so this this simply means that okay so it is a relation so this is the blow up of x along y and this is the exception of devices so the theorem just says that you can consider the abelian group generated by only projective or isomorphism plastic of projective varieties with this relation and this is going to be isomorphic to that ring of covariate and for this you need characteristic zero so you can do this without correct it uh yeah so now of course if you look at these rings you can define so now if you take k equals to c you can define of course this Euler characteristic of compact support to z like simply by taking the Euler characteristic if k equals r you also can do that the same thing uh over also to z and well as i'm gonna say in a moment you can as you have this you could also consider this like in the growth and victory ring but this i'm gonna talk about in a moment and now i can also define this mk which is simply gonna be this ring of varieties over k with l so remember that l is the class of the affine line and this is gonna be localized and the square root of this class is gonna be localized so i adjoint this extra class here and then this is gonna be mk and notice that now as we have this out to the minus a half if you want of course you can extend this or maybe mc to z simply by sending this to minus one because well of course this is gonna be one this so the real one let me write like this the real one is gonna be minus one the Euler characteristic with compact support and so here you can just extend by sending l l to the half to minus one but here if you consider the real version you have to consider an extension of the integers because of course this is gonna be square root of minus one and then you have to consider this extension of the integers so but i mean this is not really a problem in this context okay and maybe well maybe i can write some examples of simple things that we can do so the class of the projective space is gonna be the class of the many affine spaces plus one right because you can always consider like the affine chart and then you have like an affine line in the infinity and then you have like another affine line then you sum of course you could also you can also do a similar thing for the graph menu and then you're gonna get like a another combination of else and if you look at this formula of course you're gonna well you can you're gonna see that if you compute it over d you get like n if you compute it over r you get one plus minus one plus one plus minus one and then you get one or minus one depending on the period so i mean yeah so this is a simple example of how you can do this stuff but you can compute more like a little more complicated varieties also in this in this fashion like just looking at unions and products and this kind of stuff here no no it's just square root of minus one uh because yeah it's l yeah i'm adding because yeah exactly because well because this should be a morphine right so you know that this is minus one so of course this should be minus one so no no it's just like a formal formal thing yeah right okay yeah so but yeah it's just like a formal formal way of i mean to extend the morphism because otherwise you have some problems here so i think i'm doing this so before i proceed to define this motivic invariance which are gonna be in this ring i want to tell you about another extension of this ring so how much time do i have so so instead of considering only varieties over k we could consider the following thing so so let this be the group of roots of unity so over k bar so the group of roots of unity and consider this should be the projective limit of all the groups of all the groups of roots of unity and then if you have x you say that a good action of this this group uh is simply an action for which uh every orbit is contained in an affine subvaried um of x of course and if you want and a good action of me a hat is going to be simply an action which factors through a good action of some me and right so this is a good action and then you can consider instead of considering just varieties you can consider varieties equipped with a me action a good me hat action okay so that means that this action will factor through some through one of the me ends and this action is going to be good in the sense that the orbit is contained in an affine subvaried if x is projective this is going to be satisfied and now you can consider the three a billion group generated by these things the same way and you can consider uh the same relations so these relations like this the same idea and again you're gonna get a group there's another relation which is important so if you have like a product x times v where v is an affine space uh then this is going to be the same thing as x times a n where me and x trivially trivially on this a n so any action any product like this with an action so we say that the action can be trivial and this is going to be called k zero of me star bar okay so basically what we are doing is considering instead of looking at only varieties we are looking at varieties with an action of this these roots of unity and this is going to be important for defining this motivic invariance somehow this these actions uh we'll we'll encode some well I mean I'm going to explain what they will where they will play a role now so okay finally we can look at uh what we want to do so finally we can talk about this motivic fitting variant after this ring story motivic right so so what is our idea so remember that uh when this when this uh or zero when this was like nice or it moved we could just consider uh so we can take let's call better here we can take the dating variant and when this was nice this was just the Euler characteristic right so now what we want to do is define an invariant here in this ring or k or but let's say c first uh for which uh well when you apply the Euler characteristic here you want this to commute and of course if if this space is as smooth you could just take the class of this space here as a variety and compute the Euler characteristic and everything would be nice but in general this is not smooth so you can just take uh the the class of it here so you have to find a way to define this invariance and this is what motivic invariance are so you are taking your your modular space you are defining a virtual motive that's how people call it which is an element here and when you take the Euler characteristic of this element here to see you get the dating variant so that's what you want to do here right so you want to make this diagram commute but you don't know very well how to uh compute this but you know for many cases and the case in which I will uh want to the case I want to consider today is the case in which you have a map from x to to c for example so now I'm going to talk about c but most of the things can be generalized later but this this was firstly defined first defined like for for c so let's stick to c for now uh sorry I keep like going from one field to the other all the time but I hope it's not so confused so you have a map like this regular and suppose that x is smooth and then you can consider the critical locus of this of this map and this was already considered in Julia's talk like a similar thing so in this situation you want to define a class a virtual class here which correspond to uh this this the invariant so suppose you can write our modular space as a critical locus then you want to define this class so that's what you want to do and this was done by so so you have the work by denif and looser looser and then in in their work they defined a class sf so far when you have like a map like this which is actually in this ring here and how how you can define this class I mean I'm not gonna give you the the full definition but basically you consider a resolution of so you have x and k you have x zero here which is the central fiber and you can consider a resolution for which y zero is just simple normal crossings and then after you do that using the simple normal crossings you can define an element here and the the the role that this groups play is that when you do that you have some multiplicities here and for each a divisor here with a multiplicity you have like an action of roots of unity and yeah so that's the role this thing plays so that's how you define this this sfs and then uh to using this sfs you can finally define a motivic version of this thing variance for this case and I hope I have time to go so when do I have to finish it's 45 okay okay 45 okay so I have 15 minutes right more or less okay yeah so I think it's gonna be okay so right so after you have this sf you can simply define this virtual class should be simply the product l to the minus dimension of x over two times sf minus the class of x zero so basically what we are doing is the following so let me explain kind of the intuition behind it so this is somehow a a motivic incarnation of the munar fiber of this math right and the idea is that you are taking when you do that you take off this x zero so you somehow take off this move part of x zero and then you stay only with this singular locals so that's what you're doing like intuitively and this l to the minus dimension of x over two this will play the role of the sign because if you recall this l to the minus two is gonna be this l to the minus a half gonna be when you apply the other characteristic this is gonna be minus one and then you have minus one the dimension which is uh what you wanted okay and this definition is in the work of baren brian of and and they have another fact that they they proved and for so and they have this theory that if you have uh uh if there is a torus action on x so recall that you are looking at the situation like this so you have x smooth and then you have z inside of x so if you have a torus action on x uh a covariant or f and sorry and f is a covariant with respect to this action or to to a character um then then you then this virtual class can be computed simply by taking uh x one minus x zero so you you have like this generic fiber and then you can instead of having to compute this this this class here you can just compute x one and x zero and but of course you have some conditions here on this torus action it's it can't be any torus action there are some conditions uh which are related to compactness but i uh i don't want to enter in the details of that and what i want to talk about now is how how you can do this for the hubert scheme of points so how can you look at the hubert scheme of points of a three s uh a critical locus like this i think this is somewhat i don't know how classical this is but i am gonna do it anyway just to show you how how this works uh but and then in the last five minutes i want to say some words about uh all these things like how what we think that might be happening and how one could address this uh this problems relating these definitions over these things i didn't really really have had time to talk about the growth and okay so i'm gonna maybe explain this about the hubert scheme and then i'm gonna try to say some words of things i want you to say so okay so uh well so the hubert scheme so let h n be the hubert scheme of n points of a three and this is simply as you know uh ideals of so dimension zero sub schemes which are simply ideals here uh for which the dimension x one and three over i equals n okay and uh the thing is that you can write this as a critical locus of a function and i'm gonna just say what it is and then explain why so you consider this variety here or k three a three times k n uh and we ask that v generates k n via this under this action under uh the action of a b and c we take the quotient by g l n this is a moot variety okay so basically so let me explain where this is coming from so this is a vector space of dimension n so that's why you have k n here each of this x one x two and x three is one of this a b and c so it defines an action in this vector state and this v corresponds to a choice of one here so you choose one here so you want one to generate everything under the action of this x one x two three so that's why you choose this and then of course you have to uh take the quotient by the action of g l n which is conjugating in the first three coordinates and then apply the i mean acting on the usually and this is a moot variety and then you can define a map here which is just computing uh the trace of a times the commutator of b and c notice that this is the same thing as computing uh b times a c or c or a b times c these are all the same uh function so in the end what happens is that when you compute the derivative of this function you're going to get that uh the critical locus is exactly where the matrix is a b and c commute and if the matrix is commute this is going to mean that they are they they this action like factors through the symmetric group which is simply this polynomial so uh so that's how you see the hubert scheme of points as a critical locus and now i want to so the hubert scheme of points is going to be a very good example if you want to check these things so now i want to just say how you can kind of try to start generalizing this for any k using the growth and bit ring uh if you have five minutes so the so we have the following so the first thing is that this construction works over any k and also like you can define so if you are on this ring or maybe in this ring first i want to explain that you can define a map which is the a1 Euler characteristic with compact support here how you define this map so you recall that uh this uh this this map this this ring can be generated so you can see it as generated by projective varieties and on projective varieties this Euler characteristic it's simply uh it can be seen as as the the degree map of of the Euler class so you have an Euler class in a if you are on a projective variety you have an Euler class you can define this over uh in this over any field in in this a1 setting using what are called uh show bit groups i i don't really have time to define this but basically you have show groups in in the normative geometry or intersection theory and you have show bit groups in this a1 setting so the same way sabrin explained that you can define a degree here from the topological degree you have in the usual setting we can define this show this twisted show groups and things that are a bit more complicated you have to in order to compute the degrees here you have to make sure that your classes are orientable but for the for the Euler class of the tangent tangent bundle it's always orientable so you can always define this for projective varieties and as this is generated by projective varieties you can define this model okay you could you could also define this by other means in this a1 homotopy theory but i want to stick to to this show idea because it's maybe easier in some sense but okay so the point is that you can define this morphism here and and and you could also define uh that sf so this sf this class could also be defined over any field k by simply because i mean this is the work of thenf and lozair they actually work it over any field k of characteristic zero and so in principle you could try to define like a a virtual class using this sf over any field and using this Euler a1 Euler characteristic to define some a1 dt invariant for example but we we still can't do this in this example because in order to compute that formula for the Hilbert scheme you need to use this result here so you actually need and this result only worked over c this star is action result and uh so we actually are still not very sure of how to how can we like make these definitions over any field but the point is that we can of course take the formula we we for for the the dt invariant of the Hilbert scheme which i wrote in the beginning and we can of course compute uh compute this uh this this this numbers this dt invariant over r and we can also compute some invariants here just by like making l uh equals to minus one or making l equals to uh minus one in the growth index bit so uh the the thing is that so let me write some stuff we can compute uh real and gw valued uh real and gw value counts but uh we don't yet know know uh how to properly explain this and we have some ideas for to explain this so and this is gonna with this idea that i'm gonna end my talk so first thing is that uh classically this dt invariant for Hilbert schemes are computed by localization so i think that this theorem here has something to do with localization because you have a torus action here and of course this torus action will induce a torus action in your uh singular locus so one idea would be like to try to find some relations uh with levinas work on localization so so there is some work made by mark levin on localization in this a one setting so this would be an idea so if you can prove a similar theorem to this one in this a one setting then you would explain everything especially the real counts and another idea would be to relate uh echio classes with the munar fiber this motivic munar fiber s f so we have s f we have this echio classes which are defined also in this grotendic bit ring for for a map and this this is a work by vichyogram and cats in which they they were able to even compute some munar numbers in in this in this ring so we all it is also another uh way to try to to to explain why this real and complex counts are walking together and well yeah that's basically what i had to think thank you very much and we have time for a few short questions hey what are you uh i didn't define it so the names of the third the people i can well uh uh so yeah i mean i i didn't define these classes but basically uh the names isenberg kim shashville and levin the names are isenberg kim shashville and levin and basically the this thing is if you have a so the idea is that if you have a map from an to an uh you want to to show that the degree of this map uh is somehow related to to the jacobian ideal of this f so this is done like in the real setting and in the complex setting like a long time ago but in the work of vichyogram and cats they did it for this context in today in growth in the v of k so this is relating jacobian ideal and degree topological degree yeah so basically anybody else have a question let's thank the speaker again and be resumed at 415