chat with Tom - 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 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 $Z$.