stacky wiki - User contributions [en]

Fermat problems via stacks

2011-11-22T05:05:19Z

76.105.127.113: /* Log General type thoughts */

==Goals==
# Pick coefficients A,B,C, use BM obstructions to prove that there are no integral points.
# Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly.
# Is our variety of log general type? If so, we can apply Lang's conjecture.

==Summary of the below==
# etale descent
# questions related to BM obstructions

==Brauer-Manin thoughts==
# Can we calculate the Brauer group of $\X$?
# Via fiber product?
# Via Grothendieck's residue sequence
# Here is a nice list of references
## Skorobogotav: http://www2.imperial.ac.uk/~anskor/IUM-IC.PDF
## Starr: http://www.math.sunysb.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf
## Bright: http://www.warwick.ac.uk/~maseap/arith/notes/brauermanin.pdf
## Frank G: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCEQFjAA&url=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Fgounelas%2Fprojects%2Fbmo.pdf&ei=x-DKTtmtLMaSgwesrqTFDQ&usg=AFQjCNGZgoc9GNHMFquBrvuUYb2VGuepwQ&sig2=euuKVMAkcQJf0IN7x2NZaQ
## Tony V.: http://math.rice.edu/~av15/Files/LeidenLectures.pdf

==Log General type thoughts==
# Asher Auel suggested the following: let V' be the proper, but not smooth, threefold. The $\mathbb{G}_m^2$ action gives a map from V' to $\mathbb{P}^1$ which does not extend to the singular locus. This is kind of a $\mathbb{G}_m$ bundle, but with problems. We should be able to "compactify", i.e. blow up the singular locus until the map is defined. The end result will hopefully be a Del-pezzo bundle or something, and either some standard conjecture will tell us that there is a Brauer-Manin obstruction, OR we will get a counterexample to that conjecture.
# We should check asap whether the three-fold is of log-general type.
# We should also just check what integral points look like over number fields
# We should try to write down the log-albanese map. $\PP^1$ minus 3 points might be a good warm up.
# Some references
## Abromavich: http://arxiv.org/abs/alg-geom/9505038

==Strategy -- etale descent (resp. descent by torsors)==
From email:

So, thinking a little be more about this...

It seems like it will be hard to write down a finite etale cover of XX, at least by a stack with coarse space a non-separated P^1. (Of course, it'd be better if we could write cover with coarse space a higher genus curve).

My reasoning is that the map will probably have to be ramified of degree r and s at the stacky point, but then its ramified too much. I.e., it seems like the problem with the example last night will always happen. And if we divide the stack into separated pieces, the map won't be finite.

Two questions:

# What about smooth covers by surfaces? This wouldn't be optimal, since then we'd need it to be a torsor to have any nice "fiber uniformity" properties.
# [This one is more fun to think about] -- we can produce finite etale covers of $\X$ via finite etale coves of $\Y^3 := [\AA^2/\GG_m]^3$. (Of course, not all have to come from that.) So, what are the finite etale covers of $\Y$ and $\Y^3$? This seems doable.

So, first question: is there an etale cover of A^2/G_m by a curve? What about by a stacky curve with only BGm's as residual gerbes? Also, for the P^1 with a mu_r point, we were able to understand covers via the root stack construction. What about for XX? Does the root construction help us?

-D

What kind of surfaces cover $\X$?

# Tom; what did you talk about? [see [[DZB]] [[User:Anton|Anton]] 13:28, 21 November 2011 (PST)]
# Lang's conjecture
# Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?
# Why is the fiber product the thing that it is? Why $B\GG_m$?
# Can we write down any interesting etale covers of $\X$ by Artin stacks?
# Brauer-Manin obstructions on stacks. I will ask around about this. How does the brauer group change under smooth maps? What about under fiber products?

Let $X$ be a stack, and let $\X$ be $[\AA^1/\GG_m]$

# Is H^1(X,Gm) still Pic? [yes; $H^1(\GG_m)$ parameterizes $\GG_m$-torsors in any topos, which parameterize line bundles on any stack [[User:Anton|Anton]] 13:14, 21 November 2011 (PST)]
# It looks like H^1(XX,G_m) is non-trivial, since A^1 \to XX is a Gm torsor. Is this right? Is it an element of infinite order? [yes; $H^1([\AA^1/\GG_m],\GG_m)$ is $\ZZ$. See this by considering $\GG_m$ actions on $k[x]$ which respect the grading ... given by $t\cdot x^n = t^kt^nx^n$ for $k\in \ZZ$. [[User:Anton|Anton]] 13:14, 21 November 2011 (PST)]
# Is H^2(X,Gm) still gerbes? What is H^2(XX,Gm)?
# Are there any torsion elements in H^2(XX,G_m) or H^1(XX,Gm)?

Hey, let XX be A^1 mod Gm, and let XX^2 \to XX be the (r,r+1) map.

# What is Pic(XX)? Pic(XX)^2?
# Are there any curves that cover (preferably smoothly) XX^2?

[[Category:DZB]]

DZB

2011-11-22T04:59:37Z

76.105.127.113: /* Chat with DZB Anton 21:10, 20 November 2011 (PST) */

== chat with Tom - [[User:Anton|Anton]] 11:58, 16 November 2011 (PST) ==

Have toric $W=[\{x^ry^{r+1}=t\}/\GG_m]$ and $\X=[V/\GG_m^2]=\overline W\times_{\PP^1} \overline W\times_{\PP^1}\overline W$. So try to study integer points of $\overline W$. These are relatively prime pairs $[a:b]$ up to sign, together with a factorization of $a$ as an $r$-th power times an $(r+1)$-th power. Integer points of this thing essentially never miss the $B\GG_m$, so they don't factor through the "layers" of the non-separated line. If they did, then you'd have a generically degree 8 cover of so that integer points all lift. For each of the 8 curves, you could apply the machinery that says you have an finite etale cover by a finite number of higher genus curves, so you have finitely many integer points.

Instead, try to find nice covers of $\overline W$ by surfaces, with the idea of covering $\X$ by a finite number of surfaces so that all integral points lift to one of the guys in the cover. Lang's conjecture for surfaces is wide open, but we may at least be able to reduce to it. {{question|Are there smooth covers by toric surfaces or toric DM stacks?}}

Understand the case of $x^a+y^b+z^c=0$. How do you construct those finite etale curves ... Riemann existence gives you one, but the fact that there are a finite number which capture all the integral points has something to do with universal torsors and Cox rings ... see Bjorn's notes? In particular, something funny is going on between rational points and integer points. The stack has lots of rational points (since $\PP^1$ minus three points has lots of rational points), but not many integer points. The claim is that all the ''integer'' points lift to integer points of the higher genus curves. This is clearly false for rational points since the higher genus curves have finitely many rational points and the stack has infinitely many.

We already know that there are a finite number of rational points on $x^a+y^b+z^c=0$ when $a,b,c$ big. Showing that there are a finite number of rational points on $\X$ shows that there is a ''uniform'' upper bound on the number of solutions when $a,b,c$ are big.

= Chat with DZB [[User:Anton|Anton]] 21:10, 20 November 2011 (PST)=

Suppose $Z\to X$ a $G$-torsor for some algebraic group $G$ ($G$ must be abelian?) and $X$ a variety over $\QQ$. There is some set of primes $S$ in $\ZZ$ away from which $X$, $G$, $Z$ are defined; let $R=\ZZ[1/S]$.

Claim: There are a finite number of twists of $Z$ so that any $R$-point of $X$ lifts to one of the twists.

Proof: The key point is that $H^1(Spec(R),G)$ is always finite. This is explained in 8.2 of Poonen's rational points notes, and is in something by Skorobogatov. So for any $R$-point of $X$, when you pull back $Z$, you can only get a finite number of possible $G$-torsors. By twisting $Z$ by the pullbacks (along the structure morphism $X\to Spec(R)$) of this finite list of torsors, we see that one of the twists is trivial when pulled back to $Spec(R)$, so the $R$-point lifts to $Z$.

If $Z\to X$ is finite etale, but not a torsor, we can choose $R$ similarly. The pullback of $Z$ to an $R$-point is a finite etale extension. There are finitely many of these (follows from Hermite's theorem that there are finitely many number fields with bounded descriminant). The compositum of all those gives us some big finite etale extension $E$ of $R$ so that any $R$-point of $X$ lifts to an $E$-point of $Z$.

[[Category:Note]]
[[Category:DZB]]

Fermat problems via stacks

2011-11-21T16:00:04Z

76.105.127.113:

Goals:
1) Pick coefficients A,B,C, use BM obstructions to prove that there are no integral points.
2) Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly.
3) Is our variety of log general type? If so, we can apply Lang's conjecture.

Summary of the below:
1) etale descent
2) questions related to BM obstructions

'''Brauer-Manin thoughts''':
1) Can we calculate the Brauer group of XX?
2) Via fiber product?
3) Via Grothendieck's residue sequence

'''Log General type thoughts:'''
1) We should check asap whether the three-fold is of log-general type.
2) We should also just check what integral points look like over number fields
3) We should try to write down the log-albanese map. P^1 minus 3 points might be a good warm up.

'''Strategy -- etale descent (resp. descent by torsors)''', From email:

So, thinking a little be more about this...

It seems like it will be hard to write down a finite etale cover of XX, at least by a stack with coarse space a non-separated P^1. (Of course, it'd be better if we could write cover with coarse space a higher genus curve).

My reasoning is that the map will probably have to be ramified of degree r and s at the stacky point, but then its ramified too much. I.e., it seems like the problem with the example last night will always happen. And if we divide the stack into separated pieces, the map won't be finite.

Two questions:

1) What about smooth covers by surfaces? This wouldn't be optimal, since then we'd need it to be a torsor to have any nice "fiber uniformity" properties.

2) [This one is more fun to think about] -- we can produce finite etale covers of XX via finite etale coves of YY^3 := [A^2/G_m]^3. (Of course, not all have to come from that.) So, what are the finite etale covers of YY and YY^3? This seems doable.

So, first question: is there an etale cover of A^2/G_m by a curve? What about by a stacky curve with only BGm's as residual gerbes? Also, for the P^1 with a mu_r point, we were able to understand covers via the root stack construction. What about for XX? Does the root construction help us?

-D

What kind of surfaces cover X?

1) Tom; what did you talk about?
2) Lang's conjecture
3) Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?
4) Why is the fiber product the thing that it is? Why BG_m?

**4.5) Can we write down any interesting etale covers of XX by Artin stacks?

5) Brauer-Manin obstructions on stacks. I will ask around about this. How does the brauer group change under smooth maps? What about under fiber products?

Let X be a stack, and let XX be A1 mod Gm

1) Is H^1(X,Gm) still Pic?
2) It looks like H^1(XX,G_m) is non-trivial, since A^1 \to XX is a Gm torsor. Is this right? Is it an element of infinite order?
3) Is H^2(X,Gm) still gerbes? What is H^2(XX,Gm)?
4) Are there any torsion elements in H^2(XX,G_m) or H^1(XX,Gm)?

Hey, let XX be A^1 mod Gm, and let XX^2 \to XX be the (r,r+1) map.

1) What is Pic(XX)? Pic(XX)^2?
2) Are there any curves that cover (preferably smoothly) XX^2?

Fermat problems via stacks

2011-11-21T15:51:31Z

76.105.127.113:

Goals:
1) Pick coefficients A,B,C, use BM obstructions to prove that there are no integral points.
2) Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly.
3) Is our variety of log general type? If so, we can apply Lang's conjecture.

Summary of the below:
1) etale descent
2) questions related to BM obstructions

'''Other thoughts:'''
1) We should check asap whether the three-fold is of log-general type.
2) We should also just check what integral points look like over number fields
3) We should try to write down the log-albanese map. P^1 minus 3 points might be a good warm up.

'''Strategy -- etale descent (resp. descent by torsors)''', From email:

So, thinking a little be more about this...

It seems like it will be hard to write down a finite etale cover of XX, at least by a stack with coarse space a non-separated P^1. (Of course, it'd be better if we could write cover with coarse space a higher genus curve).

My reasoning is that the map will probably have to be ramified of degree r and s at the stacky point, but then its ramified too much. I.e., it seems like the problem with the example last night will always happen. And if we divide the stack into separated pieces, the map won't be finite.

Two questions:

1) What about smooth covers by surfaces? This wouldn't be optimal, since then we'd need it to be a torsor to have any nice "fiber uniformity" properties.

2) [This one is more fun to think about] -- we can produce finite etale covers of XX via finite etale coves of YY^3 := [A^2/G_m]^3. (Of course, not all have to come from that.) So, what are the finite etale covers of YY and YY^3? This seems doable.

So, first question: is there an etale cover of A^2/G_m by a curve? What about by a stacky curve with only BGm's as residual gerbes? Also, for the P^1 with a mu_r point, we were able to understand covers via the root stack construction. What about for XX? Does the root construction help us?

-D

What kind of surfaces cover X?

1) Tom; what did you talk about?
2) Lang's conjecture
3) Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?
4) Why is the fiber product the thing that it is? Why BG_m?

**4.5) Can we write down any interesting etale covers of XX by Artin stacks?

5) Brauer-Manin obstructions on stacks. I will ask around about this. How does the brauer group change under smooth maps? What about under fiber products?

Let X be a stack, and let XX be A1 mod Gm

1) Is H^1(X,Gm) still Pic?
2) It looks like H^1(XX,G_m) is non-trivial, since A^1 \to XX is a Gm torsor. Is this right? Is it an element of infinite order?
3) Is H^2(X,Gm) still gerbes? What is H^2(XX,Gm)?
4) Are there any torsion elements in H^2(XX,G_m) or H^1(XX,Gm)?

Hey, let XX be A^1 mod Gm, and let XX^2 \to XX be the (r,r+1) map.

1) What is Pic(XX)? Pic(XX)^2?
2) Are there any curves that cover (preferably smoothly) XX^2?

Fermat problems via stacks

2011-11-21T15:45:57Z

76.105.127.113:

Goals:
1) Pick coefficients A,B,C, use BM obstructions
2) Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly.
3) Is our variety of log general type?

Summary of the below:
1) etale descent
2) questions related to BM obstructions

'''Strategy -- etale descent (resp. descent by torsors)''', From email:

So, thinking a little be more about this...

It seems like it will be hard to write down a finite etale cover of XX, at least by a stack with coarse space a non-separated P^1. (Of course, it'd be better if we could write cover with coarse space a higher genus curve).

My reasoning is that the map will probably have to be ramified of degree r and s at the stacky point, but then its ramified too much. I.e., it seems like the problem with the example last night will always happen. And if we divide the stack into separated pieces, the map won't be finite.

Two questions:

1) What about smooth covers by surfaces? This wouldn't be optimal, since then we'd need it to be a torsor to have any nice "fiber uniformity" properties.

2) [This one is more fun to think about] -- we can produce finite etale covers of XX via finite etale coves of YY^3 := [A^2/G_m]^3. (Of course, not all have to come from that.) So, what are the finite etale covers of YY and YY^3? This seems doable.

So, first question: is there an etale cover of A^2/G_m by a curve? What about by a stacky curve with only BGm's as residual gerbes? Also, for the P^1 with a mu_r point, we were able to understand covers via the root stack construction. What about for XX? Does the root construction help us?

-D

What kind of surfaces cover X?

1) Tom; what did you talk about?
2) Lang's conjecture
3) Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?
4) Why is the fiber product the thing that it is? Why BG_m?

**4.5) Can we write down any interesting etale covers of XX by Artin stacks?

5) Brauer-Manin obstructions on stacks. I will ask around about this. How does the brauer group change under smooth maps? What about under fiber products?

Let X be a stack, and let XX be A1 mod Gm

1) Is H^1(X,Gm) still Pic?
2) It looks like H^1(XX,G_m) is non-trivial, since A^1 \to XX is a Gm torsor. Is this right? Is it an element of infinite order?
3) Is H^2(X,Gm) still gerbes? What is H^2(XX,Gm)?
4) Are there any torsion elements in H^2(XX,G_m) or H^1(XX,Gm)?

Hey, let XX be A^1 mod Gm, and let XX^2 \to XX be the (r,r+1) map.

1) What is Pic(XX)? Pic(XX)^2?
2) Are there any curves that cover (preferably smoothly) XX^2?

GET for GMS

2011-11-21T15:42:52Z

76.105.127.113: Created page with "GET -- finish it! -- read full faithfulness -- read the rest"

GET -- finish it!
-- read full faithfulness
-- read the rest

BAT-MAN for TAS's

2011-11-21T15:42:37Z

76.105.127.113: Created page with "a) Lets think up a list of interesting toric Artin stacks and think about what points of bounded height look like. various A^2 mod G_m's (like the one that's a P^1 with som..."

a) Lets think up a list of interesting toric Artin stacks and think about what points of bounded height look like.

various A^2 mod G_m's (like the one that's a P^1 with some extra BGm's (so the quotient by the (1,-1) action).

b) Adelic points. What do they look like?

Fermat problems via stacks

2011-11-21T15:41:24Z

76.105.127.113: Created page with "Fermat: Goals: 1) Pick coefficients A,B,C, use BM obstructions 2) Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly. 3)..."

Fermat:

Goals:
1) Pick coefficients A,B,C, use BM obstructions
2) Find a cover by a surface. Either apply Lang's conjecture to the surface, or study the surface directly.
3) Is our variety of log general type?

So, thinking a little be more about this...

It seems like it will be hard to write down a finite etale cover of XX, at least by a stack with coarse space a non-separated P^1. (Of course, it'd be better if we could write cover with coarse space a higher genus curve).

My reasoning is that the map will probably have to be ramified of degree r and s at the stacky point, but then its ramified too much. I.e., it seems like the problem with the example last night will always happen. And if we divide the stack into separated pieces, the map won't be finite.

Two questions:

1) What about smooth covers by surfaces? This wouldn't be optimal, since then we'd need it to be a torsor to have any nice "fiber uniformity" properties.

2) [This one is more fun to think about] -- we can produce finite etale covers of XX via finite etale coves of YY^3 := [A^2/G_m]^3. (Of course, not all have to come from that.) So, what are the finite etale covers of YY and YY^3? This seems doable.

So, first question: is there an etale cover of A^2/G_m by a curve? What about by a stacky curve with only BGm's as residual gerbes? Also, for the P^1 with a mu_r point, we were able to understand covers via the root stack construction. What about for XX? Does the root construction help us?

-D

What kind of surfaces cover X?

1) Tom; what did you talk about?
2) Lang's conjecture
3) Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?
4) Why is the fiber product the thing that it is? Why BG_m?

**4.5) Can we write down any interesting etale covers of XX by Artin stacks?

5) Brauer-Manin obstructions on stacks. I will ask around about this. How does the brauer group change under smooth maps? What about under fiber products?

Let X be a stack, and let XX be A1 mod Gm

1) Is H^1(X,Gm) still Pic?
2) It looks like H^1(XX,G_m) is non-trivial, since A^1 \to XX is a Gm torsor. Is this right? Is it an element of infinite order?
3) Is H^2(X,Gm) still gerbes? What is H^2(XX,Gm)?
4) Are there any torsion elements in H^2(XX,G_m) or H^1(XX,Gm)?

Hey, let XX be A^1 mod Gm, and let XX^2 \to XX be the (r,r+1) map.

1) What is Pic(XX)? Pic(XX)^2?
2) Are there any curves that cover (preferably smoothly) XX^2?

Composition laws via stacks

2011-11-21T15:40:41Z

76.105.127.113:

'''Composition law paper todo''':

a) Go over composition law proofs

b) Think about applications to composition laws

do comp laws hold over arbitrary rings?

try to use the "kernel thing" to generate new composition laws. (Need non-special groups. The usual setup uses pre-homogenous spaces. Maybe find some 'near misses' -- i.e., setups that people don't study because they don't give PHV's

I don't really understand yet how people use composition laws to prove other cool theorems.

Melanie and Dan's paper, references to Melanie's thesis, etc

Composition laws via stacks

2011-11-21T15:40:26Z

76.105.127.113:

'''Composition law paper todo''':

a) Go over composition law proofs

b) Think about applications to composition laws

- do comp laws hold over arbitrary rings?

- try to use the "kernel thing" to generate new composition laws. (Need non-special groups. The usual setup uses pre-homogenous spaces. Maybe find some 'near misses' -- i.e., setups that people don't study because they don't give PHV's

- I don't really understand yet how people use composition laws to prove other cool theorems.

- Melanie and Dan's paper, references to Melanie's thesis, etc

Composition laws via stacks

2011-11-21T15:40:09Z

76.105.127.113: Created page with "'''Composition law paper todo''': a) Go over composition law proofs b) Think about applications to composition laws -- do comp laws hold over arbitrary rings? -- try to u..."

'''Composition law paper todo''':

a) Go over composition law proofs

b) Think about applications to composition laws

-- do comp laws hold over arbitrary rings?

-- try to use the "kernel thing" to generate new composition laws. (Need non-special groups. The usual setup uses pre-homogenous spaces. Maybe find some 'near misses' -- i.e., setups that people don't study because they don't give PHV's

-- I don't really understand yet how people use composition laws to prove other cool theorems.

-- Melanie and Dan's paper, references to Melanie's thesis, etc