# Integral points on varieties and solutions to Diophantine equations

I am looking for a book (or article, or notes...) explaining details about the link between integral points on varieties defined as complement of certain divisors and integral solutions to the equation(s) defining the divisor itself.

Namely, if I have a projective space defined over a number field and a divisor defined on it by a polynomial, I can define integral points on the complement of such divisor (quasi-S-integral points, following Serre).

I know I could do (and I have done) the work by myself, but I am pretty sure that an expert review could give me examples, insights and references I am missing.

Thanks!

• I do not understand something as divisors can determine solutions of equations? This is not always the case. It happens also that decisions are determined by other equations. There may be cases when the transition to the equivalent form provides a formula for the solution. Although my own experience I know that math is not very fond of solving Diophantine equations. May 21 '14 at 14:25
• @individ I am looking for a link between "integral points on..." and "integral solutions to...". Divisors define the variety as complement of a projective space (as I am going to edit!) May 21 '14 at 21:49
• Here's a simple equation: $X^2+Y^2=aZ^2$ number $a$ - known and integer. Well with the help of its divisors such an equation in the general ideal decide? A solution I understand recording formula which will appear this number. If we abandon these ideas and begin to use other ideas then we can write the formula for solving complex equations much. Such example: math.stackexchange.com/questions/466077/… Consider and compare which method better. May 22 '14 at 4:15