A research talk at the conference “A panorama of moduli spaces” on Feb. 26 – Mar. 1, 2024 at Goethe University Frankfurt. See
https://sites.google.com/view/panoramaofmodulispaces/home
for details.
This conference is funded by the SFB 326 GAUS. See
for further details.
And the plan for the talk is I start by stating two appearance which won’t appear to you at all uh um then I I’ll talk about Mod Space the main point here and these are the main tools to actually prove these result then um uh and then I’ll talk about the
Group so then some of the things in the the fs to make more sense and then uh I talk about the applications to these spaces and there two things as you will see in the uh in the two the two the one is called splitting brow classes
And the other one is about the index okay so let Mees um and problem so let’s start with a pro field um and the first ter says um the existent over the function field fun so exist right here the fun field such that for every brow
Class every element in the brow group of X exist Al that split split Al that means if I the BR class on X on x class over the fun field so the was def the function and also the function so I can pull back from there and this
The to the tri element in this in this grp and the way to think of thisy terms so this a overx just think of this as a family of of bright X uh but in the end it’s all about about generic so this is the first the and and the second theem is
Um second is they exist they exist number only depends on how to write the X such that all Al in the power the per of Al the index of alha of alha divides the period of alha to the power okay um so this the second the this conjecture conjecture of the
Young prob May year or two years ago and and they proved the theum assuming one of the standard Le conjectures Pro there isure that actually deter the the M I’ll mention this maybe May so I think you would agree if you hav seen at all about these
Two won’t speak to you at all okay uh I do now I’ll go into Mo Stace of this to motivate why it’s interesting to look at this power books then I actually find the power book for you and then these will become okay okay so this brings me to a second point
This so there are various ways of thinking about main For Le about and they have different advant advant okay so the first thing ah and maybe uh there’s a theory that works forties but my main interest is actually in so that simplifies couple of things so may assume or to get and the first part of you starts
With so let a albra what is Algebra locally looks like make maybe let me write down here in red an example a for be that would be one example of course you that would even theis locally be of of this form in generally you assume locally if you work over complex numbers then you can also
Work okay and then look at Cent a modules modes so for inst in this example of Locally sheet and the is given by taking by taking and E and then you see this becomes left a module and it turns out this so in this example so usually is not of this form but when it is this is just okay is by Simpson one of his very first Pap early
90s late 80s maybe and then in POS characteristic is that says there exist a Mod Space of stable a mod one space table a so uh I assume you all know how stability of sheet say Vector bundles um then so you compare the slope of sub
Sheets so you say the slope something is St the slope of any sub is smaller than the slope of the sheep and here you do the same thing but you only look at a sub modules okay well that’s that’s and then if you want to fix you want to
Get Space you have to fix some inance soari a so this will be a stable a modules R of degree of that would be would be be and one finds that the r is always divisible by always as z a a is a see locally the rank of a is always
The squares it’s in squares I can always take a not and any any uh um the r is posi any mod R so the special case the special case would be the case where this Z One a c and that will be m so that’s minimal value for for the r
And that’s why as fored the smallest one is the first is that will be Twisted Twisted um again in the case in the example the example when a is she of course the TR is to some component of the um and then there’s that’s maybe not so
Interesting okay so in this case don’t any new varieties it’s just the good old varieties maybe maybe index maybe I can add one thing whenever I whenever I have a module a module interested by line model STS an ordinary line model it stays St and makes these things zero
Z okay so that’s one way to think about about about TR achieves um so these are just modeled over so the second way of thinking about is which comes close to the name uh is we give give ourselves a COR cycle locally and we look at the Twisted chees yeah
Um G consists of coher sheets on each of subance and then and then we have jeels on on the on the intersections um and those Lings satisfy a certain condition Nam so the US Post like the condition would be the composition the identity and here put said this is
Again of of of pisted cheese and again in this language you can you can Define mod spaces and this comes some qu language us but this come very close to to work of early work of Nish showing again that there is a modeling space of stable sheet oface
She if you fix you get mod spaces um but fixing is not not a be careful how and again case would be case would be a pi the alpha that would be the case of of rank rank rank one sheet um so there way go between one and two that goes as
Follows L one and two May if I take loc she um she and I can find algebra that looks as before but moris algebra of twist that’s and then this category here Becomes of a modules over c b modules and what’s I and To E and you see so since e is a Twist Chief and G is Twist Chief G Twisted with of Al this becomes an untwisted CH at the same time it’s ag ag so these are just different languages of describing the same C the Mod Space correspond to each other but then comparing the
Invariance is a little tricky because it depends on the choice of this here change the choice of the G byange little Technic okay so that was this this and and the last one the last way of thinking going back can you recover alpha or only it’s class I cannot no I
Cannot so I cannot uh uh uh find the cyle itself and that also causes a problem later I talk about commod glasses when I look at those elements inow group the categories will depend on the on the brow class but non canonically which is a okay and the last way last Think so locally are these things look like a uh that’s the projected bundle PN over over over you um and and the example the tri example is I start this before with a vector bundle three and I get get but in general are notation and and once again you can write down the
Category p x and on the on the upstairs last condition that that thetion um theun is an is so you single out on addition condition and what is the Oma this comes from the relative sequence which you definely know foration of Lo she but which also exist
For for like this and there a unique extension distinguished extension of this form and and sometimes one Pap it makes already uh and that’s the that’s the condition and again there’s a Mod Space theory for it this now toosh space so simps conr was the first and and and then some
Independently reconstru these spes you s and uh and there’s a way again linking with the other two so for instance if I want to one by three then so start c x and then iction and then I get by taking the direct image of of this of this bundle and I
Can also link two and three by by by saying if I have Twisted by Twisted Chief then I can I can proze uh recognize and since in the definition of a locally of twist Chief I I do have theide condition up to the scaling but the scaling this multiplication with the
AL projected F they don’t matter and so those actually glue to a ground okay so first is that’s a Bund that depends what you mean by projective [Laughter] bundle so this only so if you have that looks like P something three what are your objects are those
Brow so the this is the of all satisfying this condition that’s all that’s sub you start with and right now first of all I just take a look I proded as before when I one and two I start yes okay so the first first remark is if see the curve CL here close
Then every is is that means we don’t gain anything know okay if uh if C is defined over then my w are new that come after spes that me what forms of the after bange we get uh we get uh get back what over curves on
Calles or those mod spes of St rank all of ofd okay so now the question is why why are we doing this espe interested in complex driv I not just take see here and then I don’t G anything anything use so question is why into this um and the point is if consum
C so specifically X the numbers then do exist are not this form not then the a which are not which are not every but for and then we can look at mod spaces before I only deal with a Cas of of Cur but now I’m doing the same for for surface X
Surface for instance over C then I get mod spaces a so that be Mod Space of stable a modules and with aor you don’t know what just fix the turn fix turn and typically if these Vector is chosen nicely so typically um these things are SM project is proed is De formation equivalent
To so like the case a my surf and Vector is primitive generically then then I get just of okay so now in a special case where Vector such that all the sheets are concentrated on yes h let’s look at a special case um R is zero the thir you want
H and then second turn and then point SP stable a modes Vector corresponds to to the forward of G is so I can restrict so and I say could as well think of Twisted cheese or that me so to c x will be C looks like Matrix algebra
And then I look at modules on C scale modul and the E is actually the push forward of a sheet on this um so in the uned case could do the following is I can look at the complete linear system and the complete linear system H C over H
Um um and then looking at the relative tan relative of this family um and that would be an open subset of my modeling space of of sa sheet and the D and C2 they they correspond to each other so that will be an open open uh open subset so my mod of stch
Would be natural compactification of the rela of Universal and the F here of course the F here of course in the trace that Cas I start again with my C of of Earth but now I don’t look at line Buel of C but I look at line Buel a modes C which will
Be space okay and again the fibers now these twied introduced restrict Aur rest a next and now I start like C then take point system then this is curve C so these two things actually is so the fers of these two families are the same but um so same fibus same fibus but
The put together in different ways but this course and the best way to see this is you look at the generic point so if I take the generic Point here generic point so the generic F uhy UHD line bu on the generic fiber and this is the unsted case the twist case it
Is the TST and now the C now is is uh it’s a curve of a nonar go here and these two varieties there’s no reason why should be so you can see that these families are glob because simply the generic are just not the same yes
And so there a major input by Mar and then we use our language to repace this as follows you of information sches plus to give yourself cation which automatically so every of is actually of this form then X to uh to uh some Twisted thing for some
Surface so you start with any the only thing information to to Mar shows is always in this case there always some surface around and then re interpreting this result we can say then X is one of those on on the so that explains thetion for one of the standard series of
Okay so now I should come TOS did you see what a exist that’s that’s Me in this Pap I didn’t want to go into this we develop the so we using by special uh and that’s actually the natural object to look at and then some of this class speci be uni by so if you in this theorem you have this birational right yes so how what how
Much control you have over well I be the different mod yes so you could have different bar modules of the hypera all given all coming with a you could have a p one of the f can move flop this PN and then you get still hyper man still have vibration over the same
PN you hope that if you assume that find everywhere the minimum mod need then will be given by this Stace but that’s not not quite true because okay so now I think time tell something about group um so uh let say more generally but that’s then thereal definition of of which is
Um um so for for for there’s a close to the second the way structure and way this is why and there’s different interpretation this in terms of so this is Grp mod equence relation and say this isation and this here the prop so this how and then so okay so here we explained the group in terms of two of different ways and here’s thei but we know we know um we also can think of if you come from here think of
This set and this would be H2 ofar try to use checkology toes okay um then genus Al then as I told you we can Define this brow class the Brows VAR overx that’s just as projectivization of this of this of this thing and maybe one important
Remark is if you find a sheep of rank one of r one that’s Alpha Twisted then if you project it then you just get X something of rank one of course you get X that means you get AOW and that’s the identity in this gr so if rank of is one then
Class the unit theow and more generally uh the uh the uh order of alha always divides always divides R okay um now there was one page of History just say so there two parts in the history all started in the 100 years ago so exk then of and this is can just as
Alles of simple certain certain uh relation and and thenation period of all and the index of alpha is the minimal is the dimension of D dimension and and also be computed as the minimal degree of of fin extension of split Al so you have a division Al then you can TST by
L Matrix on then L split say l splits splits and and the general fact is that the period always the order of the always divides the index and more these two are the same time so that means the index of Al divides the some power of of the period
Of and uh so this is the this is the uh the situation there a beautiful book by and for the the the goes back to B early early 60s uh there are three B talks and then they are printed in the this and my you also find this important thing is there link
Between the two by the function field so access or enough then then uh restricting thee you can restrict to the generic point that element the of function here and that this one so the problem don’t lose any information by passing from X to the of function here but this group here is
Much much bigger and also think maybe if I pass different mod mod Then I then I take different glasses of the but that’s not true so the r varant of the of the I’m afraid this is uh this is all I want to see about the uh now talk about
Uh so I mentioned the other already there’s this this thing about the uh Computing the index by looking for find an extension that split the class splitting fields of class um and in the geometric the geometric set where the capital K is the function field ofy the
Final extension of the function field of corresponds to finite coloring of the vary and that means drawr that think of sign covering of the surface typically you don’t lose any information for you look at the structure will be injected but that’s not quite true for the group although
It’s detered by so for every class is extens cover that KS that splits the so this is EXT but also but also ask second example is class corresponds to then uh the pack of the class is also Tri element the of upstairs and the of the is generated by this by this
CL so it’s notp thing that find as that splits P all of a sudden of locally f um and then there’s question um which uh actually goes back one of the main Al and the student of was and he looked at this the following question in the case of local Fields but more
Recently it was really placed like this by by one gener genus one curve so you think of genus one curve thei genus one curve tou that t that the of alpha is can Only okay and this is this is okay by results of Then ear my that’s okay if the index order is smaller than six and order order seven I think in general do not expect this to be true in general but it’s interesting question you could even ask for more maybe whether you can like say something about the J
Of so X is the most general setting right okay so I minutes left I better come back to this the original so maybe one do this if if I have a curve it’s function field and i p p the class such that the the of Al
Is then the period of the of alha divides 2 G of the curus 2 you see instead of Cur why not one bundles X or Curves ofus then you can’t expect uni answer because the all the all of will always div divide this thing so so when the order grows
Then the genus has to grow except when the genus is one then in principle it’s possible that by and there a result by leish 21 and a exists of Alles that’s Cur one curve that’s splits that such that rest the point then from the B to
Of but this the the a depends on the class Alpha and more Dimension depends on the IND and in our the one says exist a a is independent of Al and then you only have to choose the to over this over this a to split the the a independent
Splits okay uh I have few minutes to explain to you how the come in yeah so as I said I don’t think gener believe that there always existed soting is best thing thing we can do but very people for other of writing youry but the tell you you have to be
Careful you wish because or something similar in the in the high now the idea is is very simple you take in in a way I think the actually just Overlook the this this construction so I take a family of cures so think of a of a left for surface and then this then
Then then I look at I look at um the let’s Pi out this of cures um okay so this is this the mod St of sheet from the FIB of this family ofs but of Twisted sheet with respect the Mod Space Mod Space of Twi sheet on the
Fess okay uh and as in an ordinary case for for you hope that there’s a coary bundle and then we have to work a little bit so assume but now theary bundle par is Twist the Chiefs on C okay so this is will be twist TST one
Twist on this on this product so the alpha comes because that’s a space of achs the one year is the condition that exist this Mod Space the F mod for this you have to work little bit but now you restrict this whole thing to the
Function field of C which is the same as the function field of of X and because that’s a and what you get is I I do this the Alp C take a generic fi of B of the b a base change the point of c and do this
Over the function f c fun x then I see that this PO bundle is a twisted CH on this thing so Twisted with respect to the P of the glass but a PO bundle is a Bund of r one because these are line bules so what I found as a twisted line
Bu twist alha twist the bundle of rank one and then there’s the remark which is probably hidden behind here it says whenever you have a locally fish of rank one that twisted and this twist this class has to be is I’m using this model space of crystaline models um
To TST to to SP our classes you could try to use other model spaces but some you always have to work with something this is section that’s the thing that’s always there and so this is the idea from theorem one and Theorem two then we Ober you can use the
Same the same argument and produce by taking a section of the to use Alpha modules of certain Dimension and that eventually you found on the index in terms of the Peri thank you