×

Density of integral points on algebraic varieties. (English) Zbl 1018.14008

Peyre, Emmanuel (ed.) et al., Rational points on algebraic varieties. Basel: Birkhäuser. Prog. Math. 199, 169-197 (2001).
Denote by \(K\) an algebraic number field, by \(S\) a finite set of valuations of \(K\), including the Archimedean valuations, and by \(\mathfrak O_S\) the ring of \(S\)-integers. Let \(X\) be an algebraic variety defined over \(K\) and \(D\) a divisor on \(X\); \(\mathfrak X\) and \(\mathfrak D\) denote models over \(\text{Spec}({\mathfrak O}_S)\).
The integral points on \((X,D)\) (the paper includes definitions of the concepts involved) are said to be potentially dense if they are Zariski d dense on some model \((\mathfrak{X,D})\), after a finite extension of the ground field and after enlarging \(S\). The authors are concerned with the central problem in arithmetic geometry of obtaining conditions that ensure potential density (or non-density) of integer points. That particular question is a motivation for and is at the root of many problems in classical number theory, for example the determination of integral points by geometric methods analogous to those used for determining rational points, for problems in transcendence theory and other problems in algebraic geometry, examples of which are afforded throughout this paper.
The authors begin with a review of the background and, in particular, with Vojta’s conjecture concerning points \((X,D)\) of ‘log general type’ [see P. Vojta, “Diophantine approximations and value distribution theory”, Lect. Notes Math. 1239 (Berlin 1987; Zbl 0609.14011)]. – There follows a review of the geometric background illuminated by many interesting examples drawn from an impressive bibliography, which certainly afforded the reviewer an absorbing introduction to the subject. – The paper concludes with a review of the problem of potential density for \(\log K3\) surfaces, and in particular with the theorem that if \(X\) is a smooth Del Pezzo surface and \(D\) a smooth anticanoncial divisor, then integral points for \((X,D)\) are potentially dense. There are many other examples drawn, for example, from conic bundles and cubic surfaces.
For the entire collection see [Zbl 0974.00037].

MSC:

14G05 Rational points
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0609.14011
PDFBibTeX XMLCite
Full Text: arXiv