Integral points on threefolds and other varieties
Let \(k\) be an algebraic number field and \(S\) denote a finite set of places of \(k\), including the Archimedean ones. Let \(\tilde{\mathcal X}\) be a projective nonsingular irreducible variety of dimension \(q\) defined over \(k\), \(D_1, \dots, D_r\) be ample divisors such that at most \(q\) of them contain a given point, and put \(\mathcal X = \tilde {\mathcal X} \setminus \bigcup_{i=1}^r D_i\). For \(r\) large enough, one wants to show that no set of \(S\)-integral points of \(\mathcal X\) is Zariski-dense, and that there exists a proper Zariski-closed subset \(\mathcal Y \subset \mathcal X\), independent of \(k\) and \(S\), such that \(\mathcal X \setminus \mathcal Y\) contains only finitely many \(S\)-integral points of \(\mathcal X\). For \(q=2\) this was proved by the \textit{authors} [Ann. Math. (2) 160, No. 2, 705--726 (2004; Zbl 1146.11035)] and [Ann. Math. (2) 170, No. 2, 609--655 (2009; Zbl 1250.11067)], and for \(q \geq 2\) and \(r \geq q^2\) by \textit{P. Autissier} [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 2, 221--239 (2009; Zbl 1173.14016)]. In the paper under review, the latter result is now generalized for \(r > q^2-q\) assuming that \(q > 2\) and \(\sum_{i=1}^r D_i\) is a reduced normal crossing divisor. For the proof, the authors develop some new idea to overcome the problem, when an \(S\)-integral point lies on 3 or more divisors \(D_i\), and then apply Schmidt's subspace theorem.
scientific article; zbMATH DE number 5687945
scientific article