I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.


For example, consider $X = \mathbb{A}\_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine $n$-space over $\mathbb{Z}$ with the origin removed. Naively, one would guess that $X(\mathbb{Z})$ is the set of integers $\{ (x\_0,x\_1,\ldots,x\_n) \in \mathbb{Z}^{n+1} \setminus \{0\}\}$.


However, some work that I have been doing on recently with universal torsors has in fact led me to believe that $X(\mathbb{Z})$ should equal $\mathbb{P}^n(\mathbb{Z})$, at least modulo the action of $\mathbb{G}\_m$. That is $X(\mathbb{Z})$ is actually the set of integers $\{ (x\_0,x\_1,\ldots,x\_n) \in \mathbb{Z}^{n+1}\setminus \{0\}\}$ such that $\gcd(x\_0,x\_1,\ldots,x\_n)=1$.


Is there a simple explanation for why this is the case?


Thanks!
Dan



Integral points on varieties
Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}\_p$-point for all $p$, which kills examples with gcd > 1. 


If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}\_{\mathbb{Z}}^1 \backslash 0$: this is the spectrum of $\mathbb{Z}[X, Y] / (XY - 1)$.


