Infinite Sets of D-integral Points on Projective Algebrain Varieties

Abstract

Let <em>X</em>(<em>K</em>) &sub; <strong>P</strong>^<em>n</em> (<em>K</em>) be a projective algebraic variety over <em>K</em>, and let <em>D</em> be a subset of <strong>P</strong>^<em>n</em>_<em>OK</em> such that the codimension of <em>D</em> with respect to <em>X</em> &sub; <strong>P</strong>^<em>n</em>_<em>OK</em> is two. We are interested in points <em>P</em> on <em>X</em>(<em>K</em>) with the property that the intersection of the closure of <em>P</em> and <em>D</em> is empty in <strong>P</strong>^<em>n</em>_<em>OK</em>, we call such points <em>D</em>-integral points on <em>X</em>(<em>K</em>). First we prove that certain algebraic varieties have infinitely many <em>D</em>-integral points. Then we find an explicit description of the complete set of all <em>D</em>-integral points in projective n-space over Q for several types of <em>D</em>.