## 1. Introduction

Let *X* be an algebraic variety defined over a number field *F*. We will say that rational points are *potentially dense* if there exists a finite extension *K/F* such that the set of *K*-rational points *X*(*K*) is Zariski dense in *X*. The main problem is to relate this property to geometric invariants of *X*. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

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