Link between integral points on varieties and solutions to Diophantine equations

Question

Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.

I am looking for notes, books or surveys detailing the link between the $S$-integers solutions of equations of the type $F(X_1,\dots,X_n)=c$ and quasi-$S$-integral points (following Serre) on the variety $\mathbb P_n(k)\setminus\{F=0\}$.

I am aware that many aspects are trivial (or almost trivial) and I have already worked on the subject by myself. I would like to see if my conclusions are correct, if I am missing something, if there are point of views I ignored, if there are striking examples...

Answer 1

As Felipe mentions, I discuss this a bit in the listed paper. There's also a reasonable discussion, I believe, in Vojta's Springer Lecture Note. But since, as you say, many (maybe even most) of the relations are easy, people don't tend to include the details in papers, and even in a book they'd likely be exercises. There's a brief discussion in my Diophantine Geometry book with Hindry (page 483). In order to make sense of $S$-integral points, you really need to take a model for your affine variety over Spec($R_S$), i.e., a scheme. So for an affine variety $U/k$, it makes sense to ask if the "set" of $S$-integral points in $U(k)$ is finite or infinite, since the answer doesn't depend on the choice of model. For $\mathbb{P}^n(k)\setminus\{F=0\}$, there's a natural model to take (especially if you require that $F$ have $S$-integral coefficients). Anyway, if you have a specific question after looking at the references, feel free to post it.