 Good morning, so the first remark is please do not forget to visit this wonderful park nearby, it's really worth it. I could have said that yesterday but I was afraid that you would not come on time to the lecture because no it's a non-trivial matter to go from here to the institute only through the park. You need a good guide like Watanabe for example then you will be successful. Kevin is also good now I'm getting better so next week you can also ask me so what are we dealing with let's remind us of our situation we have a smooth projective curve over a field K or if necessary we go to the algebraic closure of the field and we have a locally free sheath, bond C and we have a homology class there's only the first homology and this gives rise to a torsor, G C is the corresponding torsor which is a fine linear bundle so the fibers are a fine spaces but the transition maps are a fine linear not linear as in the case of a vector bundle and the question we are dealing with is whether this is when is this an a fine scheme more and there are many criteria for being a fine and sometimes one criteria is good sometimes another and we have seen that this can be answered by looking at the oh this question is is much more easier in case S is strongly semi-stable in characteristic zero if it's semi-stable because then it's basically a degree condition and more generally if we look at the strong order Norseman filtration we get a criterion for this property along this this filtration so the picture you should have in mind is look somehow like that no addition no no in general no section no I mean this class is trivial even only if the torsor has a section then it is isomorphic to the locally free sheath itself no and today so now in that situation basically we have answered it now this is answered and no I have explained how this is related to tight closure directly of a normal two-dimensional standard graded ring and we have also shown that in the case over a finite field being an affine scheme is equivalent of not containing any projective curve and today deformations of this problem so we have now let's say some base base scheme and over and now we have a relative curve over the base scheme no so and in fact it will be a one-dimensional base scheme so we have say a generic point we have here close points and usually I would draw my curve here my relative curve but I also want to have space for the torso so I draw my curves in the direction of the guest house so that's the curve no so c over b is now a relative curve no smooth relative projective curve no so the main point is whenever you choose a point here let's say p then no then we have back kappa p and the fiber is no the fiber we will denote by c kappa p no that's then the version of the curve over this field no no for it has to give a no we will have concrete examples later maybe I should draw it even a bit more like that no and now we have instead of just having a locally free sheath on on one curve over a field we now have a locally free sheath on the whole thing no on the on this guy we have a locally free sheath and again we have a comology class and therefore we have everywhere a torso no and basically it's the same picture but now well it's a family you'll just make a family of this picture but it looks somehow like that no and here use some color no here over this point we have this curve and we have this torso and over no that is so in the one dimensional case you have close point and you have a generic point and maybe let's draw this generic curve a little bit thick and the torso also and now we are interested in the question how does no so now you can decorate everything with the field no so we have the curve the version of the curve over this field we have the locally free sheath over this curve no so we will have say s kappa p something and we also have to be complete no the comology class induces well that's a bit too much no the but the comology classes in induces by pull back to the inclusion comology class in the curve for the point p is my point in the locally free sheath to that point no and this gives us a torso over c kappa p no and by because this this construction of the torso is universal this is the same as the torso of c restricted to the to the fiber no so everything goes suits well together no and the question is how does the torso in the point p so how does yeah how does the property property of the torso in the point being a fine or not a fine being a fine very in the family in the deformation well we have a family curves and a family of torsos and we want to know how does this property behave when we vary the family so what is possible what is not possible no and both these we will see two instances of this question which are directly related to tight closure questions okay now if you like to work more in the in the setting of commutative algebra you could also I mean that you can this question you can ask in general for no you have any base scheme you have any not even I don't have to draw it you have a base scheme you have w you can ask how does the property of wb the fiber no no b is what was b is here no no the fiber over point b how does the fineness of wb vary in the family no that you can ask much more general no but may most phenomenon which can occur at all you can already see in the torso case now if you come from more from commutative algebra that you can for example look at the following situation you have a ring and a d algebra you have an ideal in a no and then you have the same situation you have spec a above spec d no this ideal determines an open subset no and if you have again here no maybe I I cannot decide whether I write the p or this thing maybe here I write it like that no to this to a prime ideal in the base you again have the fiber is then given by no by the spectrum of this ring and you have the ideal in this ring no the version of the ideal in this ring and there you can also ask how does the property of this open subset being a fine vary no and if you do not like any geometry at all you forget that and then you ask yourself how does say the local how do I have to say this the how does the local homology so this will be again then denoted by kappa p how does the local homology of this object depend on the point no what can you say well when is it zero when it is not zero this is equivalent to being a fine no so that would be the no this is isomorphic to h one of the omega are a kappa p structure sheath no no and and uh because this is a quality of fine you can a fineness is that this guy is zero and therefore no if you prefer local homology you work with that side no but it's more natural to work in the geometric setting and so why did I write here D well it could be for domain but in here it's for Dedekin domain so here I think of a one-dimensional base no as I have drawn it here so and for D we basically have two interesting situations namely whether this D has mixed characteristic or has equal characteristic and so D will be either so of course everything can be made more general but it's not really necessary so D can be either the integers you could also take any algebraic number ring instead and then this is then basically the question how the fineness of the torso varies if now now now you can say this p is now a prime number how does this vary with the prime number and so that's the first case so that I I talk about the arithmetic situation or arithmetic family or arithmetic deformation no and the second case is that you fix a field so I will do it in prime characteristic and we just look at the polynomial ring in one variable no and that is the geometric deformation now here in the generic point will be the situation over the rationals the special points no so the closed points no so here is a generic point so this is a closed point special yeah special point no it's a specialization of what happens in the generic situation special point no so in this case rationals is the generic case over a finite field with z mod p is the is the special case and here the generic case not the generic case is here the field of rational functions go to the quotient field no the generic point is just the quotient field and no this is the quotient field of this thing and no the maximal ideals here they are finite extensions of sp so you have the same characteristic no so if you have a maximal ideal in the spectrum of d then the residue field is a finite extension of sp and will have a certain degree no so all the algebraic closed points they correspond to finite extension of this field no and the generic the generic point has the other generic field has a transcendental element in it that's the big difference here no and now quite generally you know that if the if the generic fiber is a fine then almost all fibers are a fine no this is basically not if you I mean what does a fine mean it means that there exists global functions on this variety with which you can embed it as a closed embedding into an affine space no and there are only finitely many of those so if you have found here finitely many functions which embed this this torso on the generic fiber this generic torso into an affine space of course you have used some of the data here but only finitely many and so the fraction you use here they will be defined almost everywhere and that gives them the embedding for almost all fibers only finitely many exceptions not but that's the only general statement you you have here so interesting case is the generic fiber is not a fine but maybe all other fibers are a fine or maybe infinitely other fibers are a fine no so we are interested whether such a thing can happen you can also say whether the local geomological dimension no wherever it varies how it how it behaves okay so let's first deal with the arithmetic deformation no and we did not yet really have the definition of of tight closure in characteristic zero no but that's right set for simplicity no if we have a ring or finite type no then we have this infinitely many prime reductions of the situation and if we have f and i in a tensor q these data that gives you for each a p which is a tensor that mod p gives the corresponding data here in in this ring no and the tight closure in characteristic zero is defined by definition if fp belongs to the tight closure of ip in this ring for almost all p not and this is a nice theory with which you can everything you have produced in results you have produced in positive characteristic you can transfer to characteristic zero equal characteristic zero that's the main idea no and the natural question is well if you have say if this holds for if you only know this for infinitely many prime reductions yeah what then do you know then that uh it's uh it's true for already almost all and the example i will give will show that this is not the case so really i don't want to see say everything can happen but a lot can happen no and the example was given by so it's the example of multi cutsman and myself so no our base is z now we need a curve so we need a a relative two dimensional ring but no it's no it's it's then because of the base no the base also has a dimension we have three dimensional ring one equation no and the equation is so we first worked with the fifth power but that didn't well we were not able to prove it and so it's just a fermat cubic i think we eroded with a negative sign but that does not make a big difference now that's the ring that's a and uh no now we need an ideal no and remember a fineness in the fiber has a lot to do with strong same instability no and uh interesting behavior so from this interpretation with strong same instability you know where where to look at for for interesting uh example for which degrees no in which degree basically you can have interesting behavior no so you can also say the other way around if you have a situation where you have a semi stable a semi stable bundle in characteristic thorough and which is almost say for almost all points strongly semi stable then you will have a uniform behavior of tight closure everywhere with it will be then inside or not inside with finitely many exceptions so for example you cannot have uh if you are over an elliptic curve you cannot have a weird behavior which you can have in degree seven so i mean the the guess is that you can have that already in degree four but it was worked out in degree seven not and uh so the ideal is given by the fourth powers so that's the ideal and not four four four gives 12 12 divided by two we have to look for something of degree uh six not at least so that's the element no and then we proved yeah what did we prove so we proved um that the behavior depends on the no the the modulus of the prime number modulus seven and we proved it in two cases so noation is here so f belongs to the tight closure of this ideal and the three mott seven inside the tight closure for p being two mott seven not and uh so here that is what we proved in in the other uh residue classes we we have evidence how how it behaves and in three here in that that case we even have it inside the the Frobenius closure no you know by the theorem of directly that for for both types there are infinitely many prime numbers no so you have infinitely many prime numbers where it belongs to the tight closure and infinitely prime numbers where it does not belong to the tight closure no and therefore by the definition of tight closure and characteristic zero it does not belong to the tight closure and characteristic zero no and the interpretation here would be no now the base is back set we have not a fine in the over the rationals the torso is not a fine but for infinitely many prime numbers it is a fine and for infinitely many prime numbers it's not a fine no no and so in the in the affine situation you can embed it into an affine space but but these embeddings do not uh yeah you cannot globalize these embeddings they have nothing to do with each other in in for different primes no so that's the that's these examples which settled this question and so now we look at this geometric deformation then now oh no the base is just an affine line and so we have here an extra parameter I call it t where the curves depends on this extra parameter t and uh we also are interested in the same question well before I give the relevant example that there's also strange behavior let me mention how this is related to the localization problem no so the localization problem to remind you so we have a ring in positive characteristic we have a multiplicatively closed and we have an element and an ideal no we compare maybe we don't need the element no we compare the we look at the tight closure and then we localize at this system or we go to the we extend the ideal into this localization it's not really a localization it's not a local ring and then we get the tight closure no and so we always have this inclusion just by persistence of tight closure no and the localization problem is whether this equality holds no so what does that have to do with such families that's very easy um so that's the following uh proposition I formulated for one dimension it's a bit easier so d no characteristic p one dimensional domain and we have r r is is is a d algebra and we have an ideal in r no and suppose that localization holds in general or in in this setting suppose localization property holds of all multiplicity in fact we only need it for one multiplicative system um f b an element and the suppose f belongs to the tight closure in tensor with the quotient field of of d no so this is just the multiplicative system is just d without zero that's the multiplicative system no then the statement is then n f belongs to the tight closure of i in r over no r kappa p for almost all spec e no and that is quite direct and easy to prove no I mean starting from there you you go back to r you have some factor h times f so that tight closure even holds in r itself and then by persistence it's it holds almost everywhere where where h becomes a unit no no so if localization no let's translate it in this situation so we have tight closure inclusion generically that would mean here that uh the generic torso is not a fine then that statement would say then almost everywhere the torso is not a fine no it's the opposite here if it's a fine it will be almost a fine everywhere but now localization would imply generic fiber not a fine almost everywhere not a fine no and we will give an example showing that uh this is not uh true in fact in the example shows not a fine generically but a fine everywhere okay so so now we need a curve and that is a little bit more complicated and built on work of monst's key in the context of Hilbert Kuhn's multiplicity so the equation is and I should say it's in uh so now we work in characteristic two and so our has now become s so that's now we need a f2t algebra so t is inside that's the parameter with which we deform and now we have three variables now it looks exactly the same but now we have again one equation and now g is z4 plus z square x y plus z times x three plus y three so it's it's homogeneous of degree four so t has degree one one one t has degree zero and plus t plus t squared no that is the place where the parameter occurs x squared y squared no and monst's key came to this equation in studying Hilbert Kuhn's multiplicity in Hilbert Kuhn's function now and the result of monst's key was that's I think from 98 no now you can you plug in for t but either either you consider t as a transcendental element and you consider it over the field of rational function or you plug in for t no so either t is the t in the function field or t goes to uh I think we call this alpha to a generator in fq a generator and then the degree of alpha is given by the degree of the field extension fq over f f2 no and in both situations now we are over over uh we have a two-dimensional standard graded normal ring over a field of positive characteristic no and uh monst's key showed that uh the Hilbert Kuhn's multiplicity of depends on the field and it is either well it is free for t transcendental so in this case and in the other situation it depends on the degree of alpha no so here it's uh three plus four three plus one over four d in that case no and and d is the degree of alpha no alpha is algebraic element over f2 so it has a degree and so only in one case we have three in all the other uh cases we have uh a bigger Hilbert Kuhn's multiplicity and that was once we did it by quite explicit quite complicated uh computations so that does mean so if we now look at the corresponding family you know so now now we take the brooch of s the brooch of s will be the relative family the base is a one over f2 no and then in terms of uh same instability in terms of same instability no we had a formula relating Hilbert Kuhn's multiplicity with the slopes in the the strong slopes of the scissi g bundle and if you translate that back that means that uh of s which is the relative curve then we have on c yeah in a transcendental case uh the scissi g's x y z uh is strongly semi-stable c in an algebraic point it's never strongly semi-stable and in fact the d tells you which Frobenius pullback you have to go back to destabilize no so if you because in particular you have strong semi-stable in uh in the generic fiber you will have semi-stable almost everywhere only finitely many exceptions so the first Frobenius only destroys semi-stability for finitely many points but the second Frobenius destroys stability same stability in more points and in the in the intersection you only get the the generic point remains nothing else no okay so that's the that's the curve together with the Hilbert Kuhn's and the the same instability criteria and these properties you need in order to build now a counter example to the localization problem in that form no so now we we have to say what the ideal is no i mean no i just said that these scissi g's are of the of the the maximal ideal have this semi-stability behavior which we need to have in order to find a counter example but of course x y z you cannot use directly because that's a maximal ideal and that's tightly closed but we we just take the second Frobenius pullback of it which is this guy now we are in characteristic two so second Frobenius pullback is just raising to the fourth power and here we take no so now the equation is not symmetric so we take f is so no this has the same behavior like like the originals you think so the ideal is the fourth powers and our element again we need something of degree three and because it's not symmetric here we need y3 z3 no okay no and now yeah basically we have again uh no i mean the proof is a mixture of of concrete computations and uh yeah a more conceptual approach and mixing this somehow together right but at the end you have to do some hard work to show here anything no i mean think of that you have really a degree extension with a certain alpha and you want to have to say something about it whatever something belongs to an ideal or not um so the lemma the main work lies in this lemma um so fq is f2 generated by alpha the degree of alpha is d and capital q is 2 to the power d minus 1 now then we show that x y times fq does not belong to the uh puse bracket power of the ideal no that's the explicit computation we have have to do and the good thing was monski could remember the computation and the theory he had to develop here to make this computation and after a while he could remember what he did and so he was able to to show this no and now we have to know the the the theorem is that so localization does not hold no and the only thing is uh um no you remember tight closure you have a non-zero element times f to the power q must always belong to i to the power bracket q so but of course now here we have shown this only for one candidate no but there's the theory of test elements and uh so what's remaining is that x y is a test element which was in that case not trivial no test elements in concrete rings concrete characteristic no small characteristic so it's it's quite easy that that x y will be a test element in a if you make the characteristic bigger but that's not allowed here so you really have to work in this characteristic no so this is a test element therefore this guy shows that f in the algebraic case f does not belong to the tight closure in the fq version of our ring no because of the for but then it can also be done directly because of the Hilbert Kunz multiplicity is three we know that the that in the transcendental case we have a strongly semi-stable bundle and therefore by the degree formula i i showed you two days ago we know that f belongs to the tight closure in s f2 in the transcendental case and no so here you have a really strong behavior not a fine generically a fine everywhere in the tight closure generically not in a tight closure nowhere and the corollary is that so tight closure is not plus closure for two-dimensional normal stand-up graded and if you were paying attention yesterday i i proved more or less the same thing but i had the extra condition over a finite field now over a finite field it is the same but if you drop the condition over a finite field so finite i mean algebraic closure of a finite field this is not true anymore not and the example is just you look at the at the generic situation not we know here it's in the tight closure so we have to show somehow that it's not in the plus closure but that it's is easy no if we would have here in the generic fiber a curve then this curve would extend almost everywhere and then it would be in the plus closure almost everywhere and in the tight closure almost everywhere not so that cannot not be so here we have that cannot exist and so do i have something more to say ah maybe one so in other characteristics what are what should you look at in other characteristics so the equation to look at so if p is at least three the equation to look at is something like that or this thing no now we need a parameter x plus ty which is also a curve which has been studied by by monski and in characteristic three but i'm quite sure that so no so here you have the parameter again and that here you will have a similar behavior so my general impression is that that localization sales is rather a generic phenomenon i mean in each case it might be difficult to write it down but if you really have that localization holds you are in a very special situation as soon as the data become sufficiently generic there's no reason to believe in localization at all okay well i think that was it so thank you very much and enjoy the school