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 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?