@dhy BTW, crazy thing. There exists a curve C over F_q-bar such that for any other curve D, there is a finite etale cover C' of C and a non-constant map C'-->D. So sort of half of my question from last time.

hmmmm

how do you construct C?

i'm curious if you can use something like these sorts of things to get at CH_0 over F_q-bar, all this seems to me vaguely like the conjecture that a K3 surface contains a rational curve through every Qbar point

is it plausible that CH_0(X) is always filtered into degree and Albanese if X is over F_q-bar?

here's the construction: math.nyu.edu/~tschinke/papers/yuri/02genram/genram.pdf

it's kind of "by hand," I can sketch the idea if you want

re: K3 stuff, there is a result on this, let me find it

math.nyu.edu/~tschinke/papers/yuri/05k3/k39.pdf

they prove this for kummer k3s

(that there is a rational curve through every F_q-bar point

(the idea is that Abelian surfaces are isogenous to Jacobians of genus 2 curves, then use the genus two curve to get a rat'l curve after quotienting by the involution on the Abelian surface)

(finiteness of the field is useful b/c it makes every point torsion)

Re: is it plausible that CH_0(X) is always filtered into degree and Albanese if X is over F_q-bar? good question...

There's a theorem of Roitman (at least in char zero) saying that CH_0(X)_{tors}=Alb(X)_tors

if this is true in char p it suffices to show that degree zero 0-cycles are torsion

Hmm but choose a curve C and a map f: C-->X so that your favorite degree zero 0-cycle is f_*x for some zero-cycle x on C. Now x is torsion (b/c we're over a finite field) so f_*x is torsion.

So we just have to check if Roitman's theorem is OK over finite fields.

@dhy Here's the paper: jstor.org/stable/… seems to be an issue w/ p-torsion in char p

hmmm

doesn't burt totaro have a paper ont his? p-torsion in char p

nvm, i think i'm misremembering

ah this paper of milne does it: jmilne.org/math/articles/1982e.pdf

so it seems to be true, nice

@dhy OK this seems useful in spreading out arguments, i wonder if one can get any juice out of it in char zero

i would be really happy if there were any complex analogue of the char p/over Q finiteness statements

but i don't know what kind of statement would be sane to hope for...

@dhy finiteness statements?

e.g. i would like to have some sort of complex analogue of say, bass's conjecture

i have no idea

what the complex analogue

would possibly look like

but it'd be really nice

Well already over Q-bar you're in trouble right?

hmmm

e.g. K_0(E) for E an elliptic curve over Q-bar

is huge

but you still conjecturally have like

a 2-step filtration of all the chow groups over Q-bar

or is that only over Q

all i'm saying is E(Q-bar) is infinitely generated

i.e. CH_0 does have a 2-step filtration, but one of the pieces is really big

right but there's still some amount of niceness to the chow group

OK sure

i don't really know what exactly i would want over C

IIRC the rough line of reasoning i was hoping could work was something like this:

finiteness statements => existence of algebraic cycle statements

as in if you have a bunch of points there have to be some collection of curves which will make them equal in CH_0...

but there are no finiteness statements over C and so i think this is completely hopeless

OK but you can always work over some f.g. Z-algebra

that's true

In any case if you want a prototype finiteness statement over (pretty) arbitrary fields

check out the Lang-Neron theorem

I don't see how to make a version of it for Chow though

good reference: math.stanford.edu/~conrad/papers/Kktrace.pdf

(once you absorb the definition of the K/k-trace, the meat starts at the bottom of p.24)

(starting w/ the line "Let C be the proper smooth connected curve..."

hmmm

so if i have a map X->P^1 with generic fiber an abelian variety

this gives me control over sections of this map

can i get control also over multi-sections of this map?

OK let me make a conjecture along these lines. Let k be an algebraically closed field and let K be the function field of a k-variety. Let X/K be a smooth projective variety. Then there exists a k-variety Y and a correspondence Z on X x Y such that CH(X)/Z_*CH(Y_K) is f.g.

@dhy if you fix some invariants maybe, e.g. degree? in general CH_0 of an abelian variety will be infinitely generated so I don't know what kind of control you're hoping for

is there an analogue of mordell-weil in this situation?

I would say Lang-Neron is the analogue of mordell-weil

i guess this is roughly like asking for a mordell-weil theorem dependent only on the degree of the field extension

or like some "variation of mordell-weil" theorem

Hm there's some version for torsion (due to Merel) but I don't know about the torsion-free part

OK so baby case, let E be an elliptic curve over Q

is it's rank over quadratic # fields bounded?

I would expect no...

hmmmm

i'm trying to figure out what sort of statement

might correspond to (algebraic) green-griffiths-lang for X, assuming X general type

because i think if you have strong enough statements like this

then you can maybe hope to combine them into something that will tell you some sort of boundedness of rational curves in X or something...

Hmm OK let me suggest a way of making a counterexample to the elliptic curve question

E is some elliptic curve, say y^2=x^3-x. It admits a 2-to-1 map to P^1; taking preimages of rat'l pts of P^1 gives quadratic points of E.

we just need to find lots of rat'l pts of P^1 whose preimages generate the same field

hmmm

i.e. x, x' such that sqrt(x^3-x), sqrt(x'^3-x') generate the same field

this seems like it shouldn't be insane to rig up...

so you want something like

hmm but so this seems like it should become something like having an elliptic curve dy^2=x^3-x with very large rank

i think that's what it becomes if you try just to take preimages of rational points in P^1

so you need preimages of quadratic points as well

there are some conjectures about quadratic twists of ell. curves having arbitrarily large rank

but I don't see the exact relation

e.g. math.uci.edu/~asilverb/bibliography/rkdist.pdf

anyway i gotta run

interesting conversation as always

seeya!

i'll try to figure out what i can get from neron-lang

*lang-neron

various authors seem to define noncommutative schemes as being some particular dg categories. this smells morally wrong to me. i think the right definition is some particular dg categories together with a choice of object (the "structure sheaf")

this fits into a general pattern where modules over an E_n-algebra naturally form an E_{n-1}-category: the n = 1 case is that modules over an algebra A naturally form an E_0-category, or a category with a distinguished object, namely A itself

a choice of distinguished object, among other things, now gives you the possibility of defining an adjoint to the inclusion of "affine" noncommutative schemes (given by taking an algebra A to the pair (Mod(A), A)), namely endomorphisms of the distinguished object

as well as giving you a "global sections" functor

this point of view resolves a confusion I used to have about what the geometric meaning of Rep(G) is for G a group. since this is modules over the group ring k[G] I used to think this meant "sheaves on the noncommutative scheme Spec k[G]," but it can also mean "sheaves on the commutative stack BG" and I didn't know how to reconcile these two points of view

but these two interpretations just correspond to different choices of distinguished objects! for Spec k[G] the distinguished object is k[G] but for BG it's k

these two things have totally different affinizations, namely k[G] itself vs. cochains C*(BG, k) on BG

and so they should be regarded as totally different noncommutative schemes despite having the same underlying category

i think an issue is that dg category w/ distinguished object is not a nice notion with respect to semi orthogonal decomposition

which i think is e.g. orlov's main motivation?

actually i guess there is still a notion of semiorthogonal decomposition

it's just that sometimes you need the zero object to be your distinguished object

maybe a stupid question: the first symbol that appears in title of section 6 in arxiv.org/pdf/0908.4588v1.pdf What is it? It is not a Greek letter?

I downloaded the source, apparently its a kappa...

@QiaochuYuan to me non-commutative stuff done with triangulated cats seems more like motives than geometry (which would be in line with @dhy 's comment on sods). For example, there is plenty of non-trivial derived equivalences which take the structure to the structure sheaf. So, even in your pointed-sense, a lot of varieties get identified when you go non-commutative.

[also, using abelian categories seems less popular these days. maybe using tensor products is the way to go.]

@bananastack: the problem isn't that varieties are getting identified, it's that dg categories (or whatever) don't behave the right way as a higher category. for example, the initial dg category is the empty category, whereas the initial object should actually be a "point" (since the relationship to noncommutative schemes is contravariant)

similarly i think if you only take dg categories then there isn't an adjunction between affine noncommutative schemes and arbitrarily noncommutative schemes

etc.

of course this issue disappears once you introduce a monoidal structure because then the unit is your distinguished object

i think you want something coarser than choosing an object though

because i think you want morita equivalent to give the same noncommutative scheme

or else you associate difference noncommutative schemes to a point and to G/G, where G acts by addition

*different

and this compares well with the definition of a noncommutative scheme as a scheme with an equivalence class of azumaya algebras

whoa this room is great! i didn't realize it had gotten going again.

i don't think it's accurate to think of dg-algebras as the affine noncommutative schemes. the point is that you want to work with dg-algebras up to Morita equivalence. it's true that the functor dg-alg -> dg-cat factors through a fully faithful functor dg-alg -> dg-cat_* (pointed dg-categories), but you really want to work in dg-cat (= oo-category of dg-categories up to Morita equivalence) for the purposes of noncommutative geometry

Perhaps someone could give an example of a "real-life application" where one point of view is privileged over another? Otherwise I don't really see this discussion going anywhere...

take two smooth projective varieties which are derived equivalent. their derived categories are generated by compact objects, whose respective dg-algebras of endomorphisms are equivalent in dg-cat but not necessarily as pointed dg-categories

Hi guys, I have a stupid question which has been bothering me for a while

Consider the functor $M_{1,1}$ that sends a scheme $S$ to families of elliptic curves over $S$.

This functor is not a separated presheaf in the fpqc topology.

However, why is the associated fibered category $(M_{1,1})_{cart}$ a stack in the fpqc topology?

@JonBeardsley Bow down to the almighty creator of the room :D

uh, I don't really know algebraic geometry

But you know about stacks

I'm confused because it's a theorem in vistoli that

@BenLim I do not understand what you mean by "it's not a separated presheaf". You mean isomorphism classes or are you thinking about presheaves of groupoids?

(I don't know algebraic geometry either but about stacks I can fake quite convincingly)

@DenisNardin I mean, we consider two elliptic curves $E \to B$ and $E' \to B$ the same if there is a $B$-isomorphism $E \to E'$

@BenLim would that be you?

@JonBeardsley Yea :D

Ok, then you're doing something mightily wrong from the modern perspective

Haha, awesome!

Why should you expect that thing to give anything meaningful?

@DenisNardin Hold on a sec.

Good work.

I am trying to get the theorem from vistoli

(Right now I have the image of you trying to torture the poor Angelo to give you the statement of the theorem :))

That is where I'm confused

Ok I totally agree with this theorem. But the point is that the stack of elliptic curves is not the category of points of this functor

@DenisNardin hahaha, vitaly and i used to joke about kidnapping mathematicians to get them to explain their theorems to us

Ok.

It may be easier if you think of stacks as functors valued in groupoids satisfying some sort of descent condition

Ok, so here is the definition of $M_{1,1}$ that I have. It's the fibered category over Sch

Ok, hold on I'm not so familiar with the pseudofunctor stuff

let me tell you how I learned it from vistoli first.

so objects of $\mathcal{M}_{1,1}$ are elliptic curves $E \to B$. Morphisms between $E \to B$ and $E' \to B'$ are cartesian arrows.

So isn't that just the associated fibered category to the functor $M_{1,1}$ that I originally defined above?

@DenisNardin after doing a master's with Vistoli I expect you can do more than fake... :)

Not at all

Ok so what's wrong with what I said? @DenisNardin

Let me put it this way: the category of points of your functor is what you obtain from the stack if you collapse all vertical edges to identities

Uh.... I'm confused by what you mean by "the category of points of your functor"

The associated Grothendieck fibration

I don't know what that means.

The category whose objects are pairs $(U,a)$ with $a\in F(U)$

How do you go from a functor to a category fibered in sets?

Ok so $F$ is the functor that sends a scheme to elliptic curves over it?

To the set of isomorphism classes of elliptic curves, yes

Ok, where we consider $E \to B$ and $E' \to B$ to be isomorphic if there is a $B$-isomorphism $E \to E'$

Yep, so you can form the category of points of F, which is a category over $C$

Oh I think my problem is that I'm getting mixed up between categories fibered in sets and categories fibered in groupoids @DenisNardin

The category formed out of $F$ is only fibered in sets

I think so. You should really keep straight in your head when you consider the set of isomorphism classes and when you consider the groupoid

Oh man this is confusing.......

They are different and their difference is exactly what makes stacks work

@BenLim The point is that isotrivial elliptic curves are etale-locally trivial, so that the sheafification of M_{1,1}/isom identifies isotrivial elliptic curves w/ trivial ones

On the other hand fpqc descent is effective for elliptic curves, since they have a natural polarization (given by the Cartier divisor associated to the unit section)

which is why M_{1,1} is a stack

@DanielLitt That's what I originally thought. In my mind, I wanna say that the elliptic curves $y^2 = x^3 + x + 1$ and $3y^2= x^3 + x + 1$ have descent data. I wanna say something like the descent data of these two things under the functor $M_{1,1}(Spec Q) \to M_{1,1}(Spec Q(\sqrt{3}) \to spec Q)$ is different precisely because there is no Q-iso between them @DanielLitt

(maybe you mean \sqrt{3}?) Yes, that's exactly right, those are two different (but geometrically isomorphic) elliptic curves over Q

Sorry yes

@DanielLitt So apparently the definition of the fibered category $M_{1,1}$ that I gave above is wrong?

So the sheafification of M_{1,1}/isom would identify those two elliptic cuves, b/c they are locally the same

@DanielLitt the non-commutative discussion doesn't have to have a point, it's just there to throw around words. Or do you not like derived categories?

@DenisNardin "poor Angelo" he's a big boy, I think he can defend himself quite well (especially when it's scheme theory we are talking about...)

@DenisNardin So really, my mistake is that out of the functor $F$ I really want a category fibered in groupoids and not in sets?

I don't know what you mean when you say "the functor $F$". In my mind when you say "isomorphism classes of" you have already lost all good properties you can hope for

Ok. But remember, that's the n00b definition that ppl had before stacks were invented (IIRC)

You really should work with the pseudofunctor (or the category fibered in groupoids if you prefer). That is the thing that has good properties

Ok. So a pseudofunctor is like a generalized functor?

Yes and then people invented stacks because that definition didn't work :)

It's something where a $B$-point is category and not a set yea?

I call them just "functors", but then I am an homotopy theorist and not an algebraic geometer

More like, you don't have a set of points, you have a groupoid of them

ok.

i don't know why there are people still not using this definition of stacks: math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/…

Or if you prefer, points have automorphisms

I don't like that one because it often gets conflated with the popular belief that a stack is a scheme where points have autos (see MO thread)

what

@AdeelKhan I think it's on some MO thread, hold on

@Adeel I know how to talk of sheaves of ($\infty$)-groupods and I love it, but people seem to be scared of $\infty$-categories for some reason...

@bananastack I certainly do, I just think that a discussion about what definitions should be probably benefits from some connection to motivating applications...

@BenLim It is true that a stack is not just a scheme with points with autos, because weird things happen when you descend

there's some thread about common misconceptions in math @AdeelKhan

the linked notes are all 1-categorical though

They define the homotopy limit, they are at least 2-categorical even if they try to hide it

Not that there's anything wrong with it

Ok can we stay away from the homotopy stuff from the moment @DenisNardin :D

Sure, sorry

@DenisNardin Keep in mind, for applications the functor is not a stupid thing. For example, a perfectly reasonable question is "count elliptic curves up to isomorphism over F_q"

@DenisNardin So what is the pseudofunctor $M_{1,1}$?

You might say, "well we should count them weighted by the sizer of their automorphism groups"

It's the functor from schemes to groupoids

@BenLim No!

It is a functor from schemes^op to groupoids

@BenLim Check out Mumford's paper "Picard groups of moduli problems"

Ok sorry, clearly I meant to say groupoids.

To each scheme $B$, we associated the groupoid of elliptic curves over $B$?

@DanielLitt I do not believe in groupoid cardinality. As an homotopy theorist I read that question as asking to compute the $\pi_0$ of some space, which is a perfectly reasonable thing to ask

@BenLim Yep, that's all that there is to it

@DenisNardin Ok, so an object of this groupoid is an elliptic curve $E \to B$

@DenisNardin Ah excellent. Well, \pi_0\circ M_{1,1} is precisely that old stupid definition...

and a morphism $(E \to B)$ to $(E' \to B)$ is a $B$-isomorphism $E \to E'$?

@Ben Yes, just that

@DanielLitt I was just looking at that today. It's not written in the modern language of stacks though.

@DanielLitt but making definitions is so much easier than proving theorems (and we can all pretend to be Grothendieck doing so. Can I have grants now?

@Ben No we don't

Oh ok. I see now.

We don't make such identifications.

Hmmm I need to learn this pseudofunctor crap

And just to conclude and then I'll return under my rock, we use cartesian fibrations because they're easier to define than functors to groupoids, but they are exactly the same amount of data

Ok.

(because to define functors to groupoids you have a bunch of annoying compatibilities that you have to keep track of, while the cartesian fibration formalism does it for you)

@DanielLitt I was just about the print out the expository stuff you wrote about that paper too :D

@DenisNardin Clearly I'm not so good with the formalism....

@BenLim Nice! It's worth reading a more formal exposition too, e.g. Vistoli's notes on descent and fibered categories in FGA explained or Neron Models on descent...

It might be a worthy exercise to translate everything to stacks

The trick is remembering you're trying to define a category-valued functor, then many definitions that sound weird start to make sense

I was actually quite surprised how much easier it was to read that paper than say his GIT book

@DenisNardin Ok. Like I said in the beginning, I think I was confused between something valued in sets and something valued in groupoids.

@DanielLitt Yes, my point is that usually to compute $\pi_0\circ F$ you often need to consider $\pi_n\circ F$ for every $n$ (well in this case $n=0,1$), so even if you're only interested in $n=0$ it's worthwhile to carry all $n$'s around. And now I will really go away :)

@DanielLitt I saw in your paper that you wrote the moduli stack of line bundles as something like $\pi_0 BG$....

@DenisNardin Well I think we agree :). I just like to defend a more classical point of view; after all, the motivations for these schmancy definitions are pretty concrete questions.

@BenLim I certainly don't. See Example 2.

Ok I'm looking at example 2. You have $G= G_m$ there.

@BenLim The point is that this is not BG_m, it's \pi_0(BG_m)

Ok, hold on I'm not familiar with this BG_m thing. As a pseudofunctor, $BG_m$ sends a scheme $X$ to the groupoid....

of line bundles

on X

Ok, and these line bundles have like a $G_m$-action or something

Anyway I gotta grab some lunch

good luck :)

Ok see you. I'll see if Vistoli has anything on BG.

@DenisNardin i'm off to bed. 3.30am here.

Ok, good night :) I'd better go back to work

So here's a question: Are there any examples of irrational behaviour of nef/effective cones of surfaces over F_p-bar?

All of the examples I know basically involves picking non-torsion points on an elliptic curve, and this wouldn't work here.

@B.Wellington do abelian varieties not work for this?