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The singular support and the characteristic cycle of étale sheaves

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The singular support and the characteristic cycle of étale sheaves
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I will discuss some recent results of Takeshi Saito and of myself that extend the theory of Kashiwara and Schapira to algebraic varieties over a field of arbitrary characteristic.
complex Actions functionality varieties component sheaf division Spitze <Mathematik> open subsets part sites dimension subset komplexen Geometrie Lecture/Conference feasibility box law Alexander Sum association conditions Graph cycle closed division several means Computer animation PIE singularities normal cycle bundles localization Results since
complex Actions differential equal decision time sheaf ones loss open subsets part sun dimension sites sign plane different box law field algebraic representation theory relation Processing closed moment Computing indicators two-dimensional space several category singularities Right cycle point functionality factor varieties component high resolution characteristics maximal events theories period Lecture/Conference Formula terms operations form integers conditions Multiple response standards shift Graph commutes Content structure analytics lines numbers equivalence formalism gute physicists Universal homomorphic Games bundles objects family localization
constantly Rifle differential decision directions loss part energy dimension sites orders of magnitude subset estimates mathematics plane strategy different Rolling algebraic fibers classes proper moment category means Quantum Right spaces point statistics study Transformers characteristics 3rd events powerful period Lecture/Conference Formula shift matchings Prisoner's Dilemma Content total applications numbers inverted embedding bundles
Actions states corresponds time water open subsets part dimension sites subset derivators plane notation extent systems nuclear decimal category Projective Spaces Right spaces point unramified extension Errors Beats functionality study link civil component characteristics elements period goodness Lecture/Conference Formula form response standards Graph matchings surface forces bracket projects Content applications approximate embedding family localization gradient descent
classical constantly choice point Actions functionality statistics study Transformers time component characteristics 3rd events dimension production quadratic sorting Lecture/Conference position conditions predictive area digital focus Multiple forces projects moment Content sampling attitudes several category formalism inverted component analysis embedding normal iterations bundles cycle family localization source unite spaces
category triangulation Lecture/Conference moment manifolds singularities form open subsets part
Actions states directions time sheaf loss part energy dimension junctions report plane sorting Meeting/Interview field algebraic systems closed moment indicators meetings connections category singularities Right cycle progress Results Bas geometry point classical momentum civil characteristics inequalities theories elements period goodness Lecture/Conference naturally Formula level lies association renewal process noise response matchings Graph forces projects Content structure logic topology Universal bundles Games family spectral localization
point Latin component time characteristics sheaf open subsets regular events dimension period sorting system Lecture/Conference Meeting/Interview naturally field stable systems shift Graph model moment Content infinity meetings rate equivalence proof category Universal Quotients normal objects localization
Meeting/Interview Lecture/Conference maximal
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think you so much for different waiting with real juries to come here for such a wonderful such a wonderful occasion no assistance since I'm the last 1 I probably should think about organizers for the wonderful joke and and it was absolutely going to be as well and we have as a hit man and I will talk about the means look at the politics of the situation but made it a mind-set and the fear is that the result was developed by the charade and support our holiday 30 years ago and the book I should many faults so here is a situation where exhibition complex manifold of demands and then in the then to the support that they have a constructive sheaf of Warren Shifrin there will be immune complexes feasible well was something food and then there it diffusion should ever the singular support both ahead it is a close subset and the contention and at the lyrics are and it is defined as well as the does of the smaller subset which is assigned the following conditions it would have been a player you and whether you will it is an open subset in X and there is a whole lot more thick function on the sites that the effect consider the differential 4 both F and considered in the draft then it doesn't intersect Michael Franks said the vehicles so different from such a player it's a decisive this condition of there is a lot earlier sidekick no difficulty local athletic men that if if world that all assume that the definition of what is known about their does Minnesota and deferral computer-industry side of cycles there will be no more than that of the deserves divisions in the middle between and in Our it's mother's less associates us that while it's easy to this year that going to call blacks some variety to stop the Alex I began to lose the tomorrow at our it's easier to check that out of following through on completely deserted capital that are that the singular supporters onto the same profit growth this year and singular support people's they as equals X and Y and I can't say it's a subset of coupons and bundle this means that the Justice intersection then this means that you have a look a constant nuns until now but there at the end of April 4 years at the following that it is that well it's complex of marital can look at it it is usable components and the Klan infidelity visible components have a dimension In the case of the it's evident in this example is better known for its stringent health and they show that there this India support his blood Benjamin well it's severity the 10th of single men suspect and or on its open part of it is Lagrangian and since it's also clinical subsets this means that there are actually it does every year usable component of singular supported is own become normal but some the ghosts of variety and acts like a normal Mundelein incidents small box to take the usual cannot Wonderland and you take you take the coach well known so that there is this the 1st the the main victims of the investigation and here is another concept let
me and it made on the 2nd board it is no support that might decision well this a field that while the guy then at the claim I used that as well you have a collection of some varieties of middle dimensions of the bunch and bound on the plane was that will it at that .period good-naturedly assigned to every it's irreducible component number amid the chair sold which will make which will make it a cycle which is called characteristic cycle is the noted in that in this manner so that we can have the following properties with will hold so that those brothers did is that support that we have a plan you would have thought so we have an open subset underfunded ,comma FIC function and this and that but at before recounted the situation when differential of did not they in the singular support but no support suppose that that it and the single support single point well well then they know that by Our by the definition of singular support it's true that our function f outside of banks is that local a cyclic so there is no Wensheng cycles and this is so this means that if I will compute diminishing cycles for Khaled Manitoba skyscraper at Alex and so I can computers dimension while dimensions the it's complex that will be a characteristic to my attention well and the claim is but the that of a full-length formula for this dimension calls I should go to the foothills of sang minus her and then here will be the local intersection index of because India which is a section so half of the year for into some right in the gut punch involvement and here still consider this site the new multiplicity 1 can here the characteristic cycle and once overheard doses to cycles which decided by 1 point so there is the notion all local intersection in the car and that point and the claim that the exact a dimensional languishing cycle the use of and the motive is to go to court because it goes over there is a heightened claiming I will I will claim anything but in a short while retaining its equality it's quantity well and so the 2nd assertion is global so I assume that have just come and then the latest computer let's concede that the localities to all that this confusion and so this means that it will compute the become Wallach of fax and computer characteristics and the formal assessed the phone that is just intersection in box office you know the characteristics if so here the separation the and bundled itself in its opinion on company but since you have said 0 section and Eric's itself is compact and suddenly you can invest in this cycle of complimented dimensions of this world the fines In but the integer and the claim that you have the differences across the world now what are the last the robot you can't use their advances to authors are questions and let me put it here but even efforts but lost the U.S. is but because she's the however characteristic cycle is In the boy universities this was a wrong call for about a year and a half years what would be in the same called for for by the companies and their resolution use the services of all relatives of the U.S. in the sense of mentioned good is that this usually only that there happening she's object and I think that is but you don't play that is that said that I single support if the just substance knock aviation that characteristic cycle has and you can't have a single support future but characteristic cycle had been missing for example taking it shifted buy 1 and they batted itself single-A support will not change but rendition cycle will disappear sorry characteristic cycle will visit the it wasn't his son who is the way that yes we have more than a year uniquely determined by an active so that the will stay in the moment came so that this
village condition actually completed Dortmund's characteristics I well it's clear that if you have if you have a new components Americans choose you can find to function which was differential and the sex of the finger to small spot and then and then this form produce your immunity produces among at what not only effective but the people yes that they did that year and moreover that at its peak the positive so the supportable of R C C equals singular and it's so on the 1 1 is that suddenly characteristics cycle depends what gets depends on the she's in of manner so it's an event it's a homomorphism from vacate group K 0 group all of our contemptible shoe store to the group of cycles From the content and bundled that fall was immediately from this the formula for lower the world 3rd but the 2nd thing is that let's consider the example that my shift the school stunned then Our if they told you update Our singular supported a 0 section and by the last property you see that well it also facts and made that nite was a characteristic cycle equals minus 1 poet and then specs well and then are earlier this see so this feels formula that because exactly at Mueller's formula so what spend the variety it is exactly the time must number 1 the singularity of a half so this X means that it's critical .period so if it's in the 6 0 in just understanding the numerous numbers and Houston two-dimensional rendition Sakhalin line festivals and because the animal is the positive for it but it was just the usual ones that you can find it and a similar is the 2nd thing is that this the formality of Taylor of characteristics well here it doses itself and perceptions 0 section of the compounds and bundle and the game because of this signs saying things physicists equivalent of a standard for months until the characteristics of from performing useful to soften perception number about so what Houston this aggression everything goes out perfectly the fighting and still won a last mark about the show lavish apparent is that is that the process of Americans India for example they pictured in the book is that there is a developed it that it works on the eliminated manifolds arrogant arrogant if you consider that eliminated mindful then you can have defined in a new analytics justification are just composition by simply Park and an insight sufficient constructed Bush's have easy and so you can you can work but that's extremely transcendental operation and you can adjust to just push people about supervision over the phone 2 raises the heart of the the loss of the didn't well you can In the years of all itself could 1 yes you come what is to 1 of his or her I want to know if you just use a limited period in 1 of the human use With the yes or I actually voices that although these wounds he he suggests we have here yes I should I should suddenly I should certainly have have told them that their story is as was completed intimated that came not from the bullet-riddled and not from 210 either but from the 3rd of tomatoes and repair of the mothers of the notion of seeing their support of the demand that's 1 of the very 1st motion and basic technical motion how you just develop the theory of demand from the very beginning at the end he got the notion of what it is have various standards the basic factors at Single-A something you can define firing into modern capital said dimensions that alleges on them and for those the modest returns for which indivisible has mansion and this is the basic geometric objects aside from them and demanded and the formula that multiplicity game also defined just from the letter from the very definition like multiplicity of commuters of algebra and the formula One this global for global at 4 3 6 4 million of them in the incidence of what affordable and it as it is I think it's it it was Williams could not summon the Tialata and effeminate mistaken but should should stress that in this definition it's it's it's it's proved is extremely simple interest but does it it as 1 and 1 light Faria can't compare compared to Bill quite complicated but all of that and that is a candle for constructed Bush's will definition actually prove but but it's it's again it's one-way improved but it's extremely 996 it's just the structure that you deal with demolished an opportunity the because whole this would you over the qualifications of the 0 at the close of all those who really want to go through the roof and yesterday approved the response demanded characteristics so it shouldn't have happened that affirmative the history but anyway .period terms and I will let them go and it will couples other parts so well and not so much a stance
that I will not have the basic question that as the a distorted everybody think about is that the budget if you can if you can just just if similar decision called in parts should and that if you if you come if you look at this definition everything possible that would work August algebra was motivated riots of some field of than you can in it's just that I would think in principle makes sense of recognition where Mike Stanton and you can you can you can also Statements of Affairs to make make perfect sense and doing our are in fact answer so what I will be talking about this just that of the 2 it's OK but with minor modifications fitted with the minor modifications 1st and then the the government discuss the associated the minor modifications so let's consider for the case of the taste of quantum shift and then suddenly and then the numerous formula as well it's known that it's true but as usual in case of it's true in cases when the base has 0 characteristic but if it has funded characteristic it slightly modified and this name date here instead of demands of their stood which shows the global demands and that the what totalled means that well that's the event the space in which the Galabru of Of the disk down below for a bunch is down below and case characteristics you can't Hawaii Commissioner Western mitigation and total means that you have to dimension and the dilemma this 1 conductor you can't at the football in because funding formula for constant shoes and it was approved by being in the 2nd of 1 of his Jason there's this stock which is exactly called numerous formula news so meanwhile so that's 1 modifications and that they should be made at the end however another modification should layers of on the Westwood bought into business but if so the secessionist falls into the Internet after sessions yesterday and will too and so
what is left is is banned on the left and blood loss for a while that of and mine automatic rifles that throws listed and put us on that like bullet pierced in In a proponent of the fictitious site which is probably said careful to fill the preference nearest future How do know the now but maybe before and affordable the continues the talk McCormick about this situation light of the thinkers not like and to it is it the fact that you really just the Lagrangian property might disappear into the statistic that he and I think it was but it is his 1st show that had from Edeline a longer but but in Moscow at about 2 some and at that moment you think that there is no food and no and stop thinking about how well anyway so that they can make a living gear produced some examples well maybe made with the last is the thing it notations inside I will it's convenient and they will also use the army afterward social was that you have in mind between and between algebraic where the small vulture prior Querrey differences which is and supposedly has a calling it summarized the inside of the dungeon bundled the what and then he deals hot major information was involved in the debate has done that way a clinical somewhere in the cabins a bundle of cats regional Denault investment and by definition it consists of all points in the Cup from bond also with Fairfax and correct the new and it works such that the and this is why calm but the properties that its believes in the fiber and sites in India .period was applied to new realizing that it's better standard the way the post at the push forward with the clinical the clinical subset Fortin spent by the pro-government against New and a small remark Our that it follows a directive from the definition of the if I have achieved G online but that regional Bellagio why have the get-together of the bomber arose in the cool you go all the way back study Center it's out thank you thank you so much for kids that if I ever achieved on the why event are I the if I consider its direct image our or lives will be issued on ethics and if the computer it's singular support them it lies inside of prior to the match in that sense of the Single-A support "quotation mark so this thing gives you an upper estimate for the singular support at 1 of of a small the demand that this happened estimated can be glamorous In the sense that if artist closed embedding than you have a recorded about even I'm about to go on the other hand it can be extremely stupid for example give artists from being then on this but the whole the whole contention bundles and so it has it has just it it just assume that in the case of characteristics here doesn't always tells you something but the statistics all well know all around I have no examples let's consider matter adjusted art it away and just the core the planes and I want and I wonder if it will be of that magnitude the plants 1 in the wasn't that the 2nd government wouldn't change so it will be given by formula all well let's do not accord here as she and why and here X and Y and this will be same energy of follows consider such a transformation and my shift will be just about match Of the constant chief on when they do consider just the she's here but now will and and even of the things that are in such a situation plenty of all possible things that cannot happen and characteristics Europe so let's consider the stupidest examples where G 12 fueled by the growth in the power of the class to right scoring the strategy just just absolutely absolutely Jesus example now not what he sees as the 1st ever by 50 said that if y the economic also 1st looked up at the 1st suddenly artists fight in a 2nd property that is why he is not euro sold outside the of are widely across a wide equals 0 the fingers OK ballots look what happens almost always that excess right across this year I think we have but these characters I don't care I don't think think you mean the 1st foreign foreign all well uh you can you can give to you can get the differential and the differential business you can you can forget about it and here will be West where guns its plus-plus something can you there will be will be D Y and it's it's invertible but did not but no what happens on the axis on the access it is very strange things happen to have the odd
you have the map and this might that that the full-length if I have caught that effect cancer but the vector along of the axis did the grandsons of good times and the key is centered this year the case is the 2nd straight and then do what I do while the while descent the device so it is it doesn't estrangement preferred error on the on access it is again is differential along the axis itself it was 2 Europe but in the normal in the normal that action dissident now if you apply the system it's suddenly efforts South it's not it's not local not local system it's not more than dozen mitigation and so on and so therefore it's singular support to the locus follows so there will be serious action well because local system holds state of the open think and blossoms and the lives of my something because it's not Smalls and something tells from exotic replies assistant and if you apply the estimated them and look at this formula humidity here that so the people's acts such as Eurocentric stands where see is the corner water that excess wide useful political edge generated why did the idea pro-India excellence and so it is that it's it's absolutely not not Lagrangian soldiers so and all that Our Apple for all forces example well now and maybe I should live formulate a little exercise extension of the current which is due to the meeting and that the assumption that we reconsider my new cultural dimension to surfaces man absolutely years the clinical subset of dimension balloons a contention bungalow can be analyzed a single supply of some constructed Bush insisted he did usable component of of single support of some constructive was she felt so it's it's a fair amount of non-degree ability of characteristics in well known I want to discuss and I don't think that this discussed also grows here are not and that's not difficult at all that but I wanted to go the show as part of a part of distillery and that is a part of the study explains how you see what's in their support and suddenly the singer support for example it's an interesting convention it's world you have some going call things but you don't you don't know how to see them basically because as well as they stand by functions it's not much but not a pleasant thing to do but now I will describe how how 1 can actually see single-A support and for example this year that it has cited his right to mention in well not so everything suddenly singular supporters a local agenda so I can assume that and also a psychologist at the feud with some things my closed in Beijing than it transports and evidence so can assume that idea from the prejudiced and I will do still further in order to shore however think looks like I will use tools are the 2 tools the 1 has brings but instead of on transformed and the 2nd is that owners in Beijing so let me just what Coleman with but I don't transform use so that have behalf might be on behalf of the global projective space and we have the standard respondents Q well into our that abundance for what at on in guide Our so maybe before Oregon transforms an adhesive that are this standard standard and that is a canonical Indentification all the the legitimization of the convention bundled the beat and the picture both of them can only nuclear indentified whiskey and that's that's a very classical thinkers and essentially irrelevant because of what they are appointed in a city give the project civilization of the bonds and bond to be well it is at this point in the end the hype playing the interior found some space to and the times space at this point end certainly he puts his belief and prejudice base ahead proclaimed in the pungent space extends uniquely hope of playing in the whole space by SingTel .period and so they get the point prejudice based on the type of plane passing through it and that's an element of Cuba sentences of investigation of the members of this fingers called by the way Elizondo and transform 1 and now you have to have that I don't transform it as a function art from part of the category of sheets and Bobby to those in the check which is given by this correspondence well that it s some wonderful is a standard derivatives but they will not discuss the wealthiest and approved by the government discussed and new that effort but 1 thing that this ready where it at the chapter is that Our around about this Indentification sensors in some sense it's classical approximation divided on runs for which no befalling that effective achieve here man I can compute it's going to to take a single step that's a call on here so I can consider the respondent budget to it's positive is agent which will just to here and on the other hand they can do the same thing for for of them transform the and the claim that the but a simple a simple
fact but we do not have time to describe the Boston studies that I want to know the 2nd just blank with stock and bond transform coconut think well if you don't know a bunch of no boats and their support before she face and then you don't know what the just on for not helping needed to but because after the Mariners indeed so let's do the polling it's considered the embedding of my projective space let's call it now small projector space and wedding but was some dignity are more than anything and let me know and then I will build but I don't transform wanted a larger budget space so I have be led bed so this will be the winners embrace link and when I consider the larger of space and the abundance for on the sludge space around noon now notation 4 as simple as that I have a cold see the inside of the stock he well and then I can consider its the match by 5 extended the quantity as well and then I would like to take its station and just just notation will visit several denoted by sea in school and I think it's leave here still here and there well the details notation and at noon Oregon 2nd are also would have enough knowledge he's here and what I want to do is to apply this form so I can see that but going transform "quotation mark I started so this is achieved here and let's look at its ramifications of us and the mitigation devices in the following that they just restricted to the genetics genetic .period so the local system and the local system expenses local system will know whether we can expand and then but candidates unification of communication devices for 5 well so consider need for defining the need to know the thinking of the genetic .period my prejudice based go unnoticed it seemed United In this all Noonan agenda just just just plain subsets of its obsessed with dimension 1 with normal to please that is an issue for the people on smoking do you think that because the right of yes I know that I take I'll stop than they the are in fact on and not divide their I'm so there is no idle funds in this little days so after the complex issue well I know of the Salem so informally that it is the way how you can start from from the richest somehow visible in matters of that you can content the singular support before and played adults who cast this simpatico tells us that the president dimension what is often so the 1st version use that but the itself can a candidate ,comma worked for from the same name is just the image Of the in square brackets of 30 citizens so now from now 1 books he equalled the singular support that adds equals the magical of the this is really 3rd love that's so this was also the a final scene he started doing all the the In announcing the start of the year the problem is the use of the site on this that is the heart of the matter is that that as the next Notimex next is projected as a nation is legitimization of the call so I have to see which he
felt somewhat small an extended and expanded way the corn and in particular the virtues of the people on the use absurdity thinking well here this patients were without heat about 20 now so that is the 1st thing things 2nd on the 2nd assertion is at moreover so my GE has a different irreducible compliments and he has different irreducible components and the claim is that in this manner they could respond 1 to another family of for evidence continues to be a component 20 times for you there is a new car is principal component the and folk scene such as the iPhone people so we think that this there another thing and then you bill legend of transformers and prospects component that could here they project or something the same year position and you but when you do the things they will project but absolutely different yes that defective embedding that's exactly method on any old Egremont and 1 evening to take dignity is not alone in the world and well there is the fear is that a lot work this probably along however zeal of the modern and now feel is that actually this condition it's uniquely defined the so but see I'm far inverted C for years the unique call them in peace plan B while the dimensions amendments and again on his property with protesters 1 OK and maybe I should say also well maybe I should sales of properties for sale at the Map this projection from CIA files to India His genetically during his testimony .period this will be were on the value this more than the Nolte known and unknown alloy which I'm sorry so he should be directed and say that I don't know here is serious I think think so they think is genetically DHL in art classical separation of classical certain characteristics recitation and in fact and in the 1st half the members by what certainly is about irrational and also the thing is what it is in the transplant unbundled digitalization of the contention bundle the BTU yes India is the divisive and that is projected additional become normal today but in case of a characteristic speed and in the face of fine characteristics obscenity doesn't need to be it and bitter also since its name under John the singular support well but somehow you can you can become aware from the whole world move was is and will about multiplicity a little bit later I had occasion so I will say well maybe I have a question is that I would have a very much want to know how protocol which we think he has genetically as possible even in case of a projection of sources specified projection of 1 surfaced another better image of constant she's so here we would like is a ticket to by the stupid situation but it would just iteration of those 2 things of GDP if that it built at a lunch at the reservations that you just can't commit coverage from Germany I can but promised and deeper and deeper German this is 1 of Asia's most wanted real solutions this individual in our Our in all our view arrested a University of Virginia was the move Moreau Rojas his users here noon enough it you it and using if you have invented the event map of than hour other than the thinking is just the injustice of major action between between quadratic or so if you have absolutely any quadratic call on it if didn't have you which is used Nunavut devices hidden in and that I do show the image and so new Kansas just thinking of thinking that then go back and by producing the prediction so that another the best something domestic but it now
ballots past prettiest exactly as well as ever at petitions Monday at this product sample and it uses many it imports the focus of the studies splits are often in the back and moved by that and I can I can discuss adjusted just because of the absence of time but instead of that I will ever try while various senses something a bit the redundant area it's not the story and at 1 think that comes in the Justice Party is the characteristic cycle in what the condo units has not in proficient but there could be a processed beans and and that's what it is that compliments of seeking have tended beauty insupportable forwarded the image and the multiplicity and intersection in numerous formerly will have unavoidable pulse of and in the end he said so that's 1 thing said 1 would like very much to do not what I would like to do the same sort of how well maybe hoped-for for formerly of that would explain the story also I hope it will provide understanding called things like you can consider for global intersection formal Ford is a characteristic you can ask for finest thankful for example comes from compute the government of college and I would like to have to have just at the story simultaneously and Boca definition of characteristic some fine definition of statistics cycle which will answer also is the 2nd question and that would not be involved in that sort of thing that will visit would be in this proves that doesn't depend on the choice of 4 functions in the formal and so on but somehow Miller formerly will be just curled so intelligent tried to put the ball players in the remaining 2 minutes 11 . 2 points to put it on the black people so there is an what of the moment to have the notion of single Paul you have the notion of a neutral locals she well you her usual shaves a year our space and a force and the
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which she forefinger later objected that MySpace it can consider for every opening and parts of the corresponding category and this will be sold From shit and go at it again on but at the moment have emotions of singular support and have Our Arabic and into the fall and think we can consider the concerns and bundled his way the hazardous and the roar with half of my our manifold and VAD is well on its achieved chief also strangulated Pentagon assignments so I signed the Wilpon sentence the get-together of she's the triangulated category of sheets and but this year or well it's in in the in the form of in modern in modern parlance the tangle indicated that it means what about infinity and accessible all but I know I have at the moment I have the notion of single support you can make a localized D forwards the
contention by I well so considered punch bundle but I will consider not this plane that is capable at about 1 only those opens up since Richard Quinn on the clinical well how it how would define a 12 so if you have our if you have an open political steps in to stop work then we can put energy in the who this will be the cautioned off until the end of the monolog the fixed subject getting all those she used whose single herself for lies in the complement of you do you if so then this will sheaf of Tonga later considers that has nature of 2 structure was 1st the whisperer structure here and you can ask about things like good dimension the conjecture but that it evident that don't want to discuss but just consider this thing now he added that no 1 would know what they want and what they want to concede that are what sort of questions I wanted the Bridle Path use the phone Associates was at my desk I will assume that it has come back from and then I went with him consider the founder of drama thrown in the hot to just the London more jails since then the . 3 kb algebraic Lycos and then attend through the respondent member convicted and that's what I wanted all the noise is defined to find this defenders in this from multiple and look at the map of suspect so you know what the escape edits at Cleveland's Case the particularly if I have a sheaf than I have ever so she can have a defines you .period here and so I if I know the think I will know the commodity it's the match it will become what people here and such a commodity .period defines you defines whatever you want to differentiate celebrities to give your boss connect connected components if you look at the progress in eliminating bond category political defined in the diagram of and older so so that's actually what went was not I let me title title put on 1 The blackboard what I want to say that we will all year said Bill Miller In report that I want him so I I won't go basically I want to have a localization of the liquor this month and I want localize twice and went collided with respect and then I want to into Michael localized to the contention by and the climate that is all that is needed from the federal of characteristics cycle so so let's say so because so behalf .period text and the companies and people thaksin and that McDonald's by the bye bye buying and this will be the well I'm here on this 0 level because this story in this matter that I want to I understand what does it mean the life at the local either thing book acts as well as I told you the itself it forms chief of categories finger later stated of on and from then on I can apply to the case and it says that this is a terrible get issue suspect that's quality OK seen the chief of spectacle where ABC's case now what they wanted only defined 1st and my help now it's map of sheaf of spectrum for the following things so so this is game as a suspect them all with the ball and and I want to consider its upper should Bolek but acts so I will say in a moment with me well as such things as well he lets consider usual usual spent the part and what is expected of you the impetus to is the values of a yes yes yes yes Spector Foreign is solace and the sense of apology might expected and the sense of it 1 from multiple it 1 from process so that I think is that I think is much expect ship of might've expected over at out and it looks a full assortment if instead if instead of they're flounder this will be easy in for example if lenders if fields you have snapped to man after she back looks as follows so they should be commodity and then we should shift but I took and what to do next year so that will be situations in cases so this will be well now so what I want to have a look to get this basically they're basically should come by junction so what sits here is essential the map from direct image of Texas comeback from direct the image of this this is just a part of that image and you have a map to kill floundered and want to have to the better to get such a map and by by and with the way it is not known "quotation mark usual assessing exists in the in classical usually usually the bulletin but in in algebraic geometry this it's much more interesting women will give up there have been 2 minutes again but it so we have this picture now that so here I will I will put small at ?question mark and here's the Vrabel larger question came as a legit ?question mark is uh this so let's consider this Airoso assume that it exists and now considered it's just back by the close here will get not that well so that this she it was a strip of suspected it has made naturalness I recall that there was a genius and that is have the greatest finding the she's well that is a map from pullback of the dingy Maryland that says that some mental indicators point in case they don't know the name of Jesus death yes now here so here so here relief over the boy really Fuller X and which is the next what made 11th with question mark on the small engines and with light and the logic is that there is an absolute a canonical man he well you know the fact that what they
know is that if you look the Evian classical in this edition of Question in Japan on and work with classical topology instead of government and what that meant such a construction it is he but now that I believe that it is well that it should come if you're the 1 understands how all of this started with your your support and later to study this month when Ishant cycle for multidimensional ways isn't that much would come by itself just just for citizens of an impression ridership recitations essentially blamed with some Wengen cycle of multidimensional as well but some governments some notion of it for use of purple too will now I would let me just says the claim is that when we have such a former than you have and you have all of the things that you wish to have so for example you have so let me just say a wide-ranging cycles ,comma said profit after half a year but and help comes from the speech and attend and credit and at times like this when 56 as so so that have before the chiefs of the going with him .period here so new missile defense section here and there and on and on about because it comes from a section of the store and what does it mean I have to have a section of the thing so if because it have a sheaves of so this thing and as a sentiment invitees the the complement of the singular support of my shoes it's still there is a section just as an element of that portion control and so on it's given such an effort by the system for shares on that they've ever so consider that there is 1 exception here and the producer said the realization of this section when you pull it back to the contention mandolins the and the speaker the compliment doll also for all states in all the Single-A support now this project the story from the theory that he didn't by the American artistic merit and here was this year and and then the pool in here and then you know that sections that sections of such strength of the famous cycles of condiment could dimension and pictured as a child but they reconsider the section supported on some sobriety this will be the child group and cycle from on the somewhat I attended the sobriety our singular support for the mentioned on this month's exact is at the the moment most appreciative of genetic such it out so so investment for the thinking if you replace state a the theory by producers in particular the cycle and that believe me it also produces fuel for us as a global inequalities that formula and so on and so a game but is my whole but at the end of this book you just have to recognize war the you have a whole lot of University of World War I wouldn't close and you can she said that the losses to the company he well if it's if it's well at that that nobody and if you're somewhat small so that it a good normal ,comma you might think that is not small but I'm not with an also depends on the singularity on the single and what level of the year the I know I can I can I cannot and that's why even if if it's not complete and section than this and even you don't even know if it would be a a Milazzo said In if view you listen to all of you in the future if bring in the use of more than half the size of the wasn't it what do you mean on Sunday ,comma from this tangle of Katie and hundreds of phone calls from In the case so if I have if have if I have achieved she phoned me so I have a point here so I will have a sections here and our In can't just entered the instability image its image he has them since it is the same as if I passed here to hear this will be a section of this sheaf of spectral which is stabilized on the complement of the singular support just because the image of the 2nd period allies on the complications on the complement of the singular support because Michael local should just finish and insoluble have a section of this fellow we just could be so you're replaces lazy yes a equipped with the realization on the complement this means that you have come Wallach a colossus deficient in the thing support in the singular support and business absolutely the same as young please then the the problem with this year's soldiers that Israel has no no to get the government otherwise it will be element of the child of the channel group of the cycles on and if you consider that come knowledgeable the thinks supported them somewhere it is the same as show group of the subway consensus has could dimension and that this was just another cycle of the generic .period nothing else on has wanting to be on the screen and and 50 years since the if I speak in half and the the good ties with family we will carry on behalf of the lead and you I hope he isn't actually a look some cold I think that I should say that there should be a relative picture for more system so far at this meeting which is very much possible and I would say that we will blast the politics please tell me how to spell it out he's conjecture but again support for this is that it is that you have absolutely canonical canonical pitching the status quo for real you analytical contact which is even nicer in case of the result is that a review of all races and political problems this book that by the singular support was always playing the managers called on it's not complex if she if it is a section of the span of the European forces the U.S. from substituting exactly on the news about the course no the whole thing on the whole thing of some of the views of the losses than the the Europeans and some scientists you wish you can really see you take ,comma cake lost carried exceeds kickoffs metropolitan areas with relatively easily ,comma quarters in the room that has the Yuanmingyuan was removed .period the socially some someone you know or you Pa probably every produce but I believe they that they don't understand that at single board and they are entered as I would like the 3rd it will be at the office he interviewed on the U.S. yes I wanted the it's everywhere since probably knew Benoit have anywhere to have actual picture on the level of cool case they would probably not noted for electorate thinking it only takes a genetic points and about but 1st subtler the request was because of this we don't view on the coffers of the assessment yet there is little doubt that is so was similarly responses saying you you will use the Welsh civil rights group there is so much and their businesses that it eventually result renewal his Ohio "quotation mark that's through the use
of convenience it's an infinite stable infinite circuit again but the beneficiary she feel and then there the case with the Louvre you just listen to the people of the world the same extended and status and being sold to the government university is irreversible and the there is the problems the people the amendment only tickets available for the limited success of the season that you have enough you and all of them some with some kind normal air land and it's it's it's it's sort of a funnier finally she's every section supported the well it's important include dimension them so most of the conjecture you mentioned that there was no a problem the distractions of hundreds of half of the soul as that's just the way so as as a culture that every object here just judge if you start with the with the Chouf than as a section of the new system like Michael locally it supports opponents single-A support here and now what now supports that you're playing Notre strangulated categories but was by machine the fact so let's consider it but promotions now there are the following :colon the so-so think again so it's supported include dimension and now what what you want to know is that the fund that was is stationed outside of "quotation mark dimensions dentists all this get-together will be faithful the PlayStation 2 good I know yes and plus it will be fully faithful Our environment variable 1 along 1 dimension less it's an equivalent of it so that so this means that if I have if have a Michael local panelists she's then on the cool contention bundled on the Sunday only that it is the same as Michael local parish on it's open substance obtained by removing its mania the point of going Creekwood dimension and blustery you and Policy think plus Florida this is over about 2 years now this this and event old conjecture in the context of the model segmented wonders with regular single entity and that was of Fred and 2 maybe 3 years ago by Christian Latin and willing to help you in that profits it's called something quick to mention the conjecture and so but an amazing can throw the people would actually for the statement the modern somewhat but will shift but with efficiency in the field of characteristics you for promotion office is even better efficiency and thinking about how to put interpret the proof of this is the nature of what is the status of the world to unify this much of the world the quotient the in a Rosenbloom who use the region as something that makes sense don't look so what so what happened so simple that you have much yes and a look at its at the generic .period obsessing exists a singular support there you have some Sunday percentages is that it would be nice to describe now he or she is as if yeshivas non-zero lenses is
nonscientific all no no now you want to look at it construct your shift from Sicily what you should do you should get blitzed on Sunday that could dimensional 1 what some sort of reviewing data if the components in the Senate by something called dimension and minus 1 then you should get there some sort of abusing the no and this data sitting blasting uniquely defined defines your ship but in order to reconstruct the shift just from the steering you should have our net completed meeting which will sit in and wanted mention more a genetic point 1 could dimensions and the moment you haven't you don't need the to go any deeper than the that because of the fire is the view of you the BSO the success rate of interest of political you with the complement of viewers that's what the rest of the not at the think that you justified you should devise a stolen so when when you define it using the squatters inevitably sheaf of Bishop of getting is there now you consider that the 2 sheep and for sister ship in principle it could have the sort of revealing data which even deeper and deeper I believe that this is the time of the years I thought that that was lower precision so I can't say console made things
thank you