Fermat problems via stacks

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 $\X$?
2. Via fiber product?
3. Via Grothendieck's residue sequence
4. Here is a nice list of references
   1. Skorobogotav: http://www2.imperial.ac.uk/~anskor/IUM-IC.PDF
   2. Starr: http://www.math.sunysb.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf
   3. Bright: http://www.warwick.ac.uk/~maseap/arith/notes/brauermanin.pdf
   4. 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
   5. Tony V.: http://math.rice.edu/~av15/Files/LeidenLectures.pdf

Log General type thoughts

1. 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.
2. We should check asap whether the three-fold is of log-general type.
3. We should also just check what integral points look like over number fields
4. We should try to write down the log-albanese map. $\PP^1$ minus 3 points might be a good warm up.
5. Some references
   1. 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:

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 $\X$ 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 $\X$?

1. Tom; what did you talk about? [see DZB Anton 13:28, 21 November 2011 (PST)]
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 $B\GG_m$?
5. Can we write down any interesting etale covers of $\X$ by Artin stacks?
6. 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 $[\AA^1/\GG_m]$

1. 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 Anton 13:14, 21 November 2011 (PST)]
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? [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 $k\in \ZZ$. Anton 13:14, 21 November 2011 (PST)]
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?