Fermat problems via stacks - Revision history

76.105.127.113: /* Log General type thoughts */

2011-11-22T05:05:19Z

Log General type thoughts

← Older revision		Revision as of 21:05, 21 November 2011
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	# We should also just check what integral points look like over number fields		# 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.		# 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)==		==Strategy -- etale descent (resp. descent by torsors)==

Dave: /* Brauer-Manin thoughts */

2011-11-21T23:38:44Z

Brauer-Manin thoughts

← Older revision		Revision as of 15:38, 21 November 2011
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	# Via fiber product?		# Via fiber product?
	# Via Grothendieck's residue sequence		# 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==		==Log General type thoughts==

Dave: /* Log General type thoughts */

2011-11-21T21:32:17Z

Log General type thoughts

← Older revision		Revision as of 13:32, 21 November 2011
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	==Log General type thoughts==		==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 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 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.		# We should try to write down the log-albanese map. $\PP^1$ minus 3 points might be a good warm up.


	==Strategy -- etale descent (resp. descent by torsors)==		==Strategy -- etale descent (resp. descent by torsors)==

Anton: /* Strategy -- etale descent (resp. descent by torsors) */

2011-11-21T21:28:44Z

Strategy -- etale descent (resp. descent by torsors)

← Older revision		Revision as of 13:28, 21 November 2011
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	What kind of surfaces cover $\X$?		What kind of surfaces cover $\X$?

	# Tom; what did you talk about?		# Tom; what did you talk about? [see [[DZB]] [[User:Anton\|Anton]] 13:28, 21 November 2011 (PST)]
	# Lang's conjecture		# 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?		# Euler char/Hurwitz formula for stacky curves. Is there some analogue (say of the criteria for a map to be etale) for stacks?

Anton at 21:14, 21 November 2011

2011-11-21T21:14:07Z

← Older revision		Revision as of 13:14, 21 November 2011
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	Goals:		==Goals==
	1) Pick coefficients A,B,C, use BM obstructions to prove that there are no integral points.		# 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.		# 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.		# 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

	~~Summary~~ of ~~the below:~~		==Brauer-Manin thoughts==
	~~1) etale descent~~		# Can we calculate the Brauer group of $\X$?
	~~2) questions related to BM obstructions~~		# Via fiber product?
			# Via Grothendieck's residue sequence

	~~'''Brauer-Manin~~ thoughts~~''':~~		==Log General type thoughts==
	~~1) Can we calculate~~ the ~~Brauer group~~ of ~~XX?~~		# We should check asap whether the three-fold is of log-general type.
	~~2) Via fiber product?~~		# We should also just check what integral points look like over number fields
	3~~) Via Grothendieck's residue sequence~~		# We should try to write down the log-albanese map. $\PP^1$ minus 3 points might be a good warm up.

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

	So, thinking a little be more about this...		So, thinking a little be more about this...
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	Two questions:		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.		# 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.
	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?		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		-D


	What kind of surfaces cover X?		What kind of surfaces cover $\X$?

	1) Tom; what did you talk about?		# Tom; what did you talk about?
	2) Lang's conjecture		# 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?		# 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~~?		# 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?

	**4.5) Can we write down any interesting etale covers of XX by Artin stacks?		Let $X$ be a stack, and let $\X$ be $[\AA^1/\GG_m]$

	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~~?		# 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.

	~~Let X be a stack, and let XX be A1 mod Gm~~		# What is Pic(XX)? Pic(XX)^2?
			# Are there any curves that cover (preferably smoothly) XX^2?
	~~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?~~		[[Category:DZB]]
	~~2) Are there any curves that cover (preferably smoothly) XX^2?~~

76.105.127.113 at 16:00, 21 November 2011

2011-11-21T16:00:04Z

← Older revision		Revision as of 08:00, 21 November 2011
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	2) questions related to BM obstructions		2) questions related to BM obstructions

	'''~~Other~~ thoughts:'''		'''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.		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		2) We should also just check what integral points look like over number fields

76.105.127.113 at 15:51, 21 November 2011

2011-11-21T15:51:31Z

← Older revision		Revision as of 07:51, 21 November 2011
Line 1:		Line 1:
	Goals:		Goals:
	1) Pick coefficients A,B,C, use BM obstructions		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.		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?		3) Is our variety of log general type? If so, we can apply Lang's conjecture.


Line 8:		Line 8:
	1) etale descent		1) etale descent
	2) questions related to BM obstructions		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.

76.105.127.113 at 15:45, 21 November 2011

2011-11-21T15:45:57Z

← Older revision		Revision as of 07:45, 21 November 2011
Line 1:		Line 1:
	~~Fermat:~~


	Goals:		Goals:
	1) Pick coefficients A,B,C, use BM obstructions		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.		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?		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...		So, thinking a little be more about this...

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)..."

2011-11-21T15:41:24Z

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)..."

New page

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?