Abelian fibrations and rational points on symmetric products (Q2706846)

The authors write: ``Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which the rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the \(N\)-th symmetric product of a \(K3\) surface, where \(N\) is determined by the geometry of the surface. The basic construction is that for some \(N\), the \(N\)-th symmetric power of a \(K3\) surface is birational to an abelian fibration over \(\mathbb{P}^N\). It is an interesting geometric problem to find the smallest \(N\) with this property. Potential density holds for elliptic \(K3\) surfaces and for all but finitely many families of \(K3\) surfaces with Picard group of rank \(\geq 3\), and consequently for their symmetric products [see \textit{F. A. Bogomolov} and \textit{Yu. Tschinkel}, Asian J. Math. 4, 351-368 (2000; Zbl 0983.14003)]. However, a general \(K3\) surface has Picard group of rank 1. In the following sections we will prove density results for symmetric products of general \(K3\) surfaces.''NEWLINENEWLINENEWLINETo be precise, let \(S\) be a \(K3\) surface, and let \(\varphi: S^{[n]}\to S^{(n)}\) be a crepant resolution of \(S^{(n)}\), the \(n\)-th symmetric power of \(S\), where \(S^{[n]}\) stands for the Hilbert scheme of zero-dimensional subschemes of \(S\) of length \(n\) [cf. \textit{A. Beauville}, J. Differ Geom. 18, 755-782 (1983; Zbl 0537.53056)]. The authors prove thatNEWLINENEWLINENEWLINE(i) if there is a big line bundle \(g\) of degree \(2(n-1)\) on \(S\) such that \(|g|\) contains the class of an irreducible elliptic curve, then the rational points of \(S^{[n]}\) are potentially dense, andNEWLINENEWLINENEWLINE(ii) if \(S\) admits a polarisation of degree \(2(N-1)\), then the rational points of \(S^{[n]}\) are potentially dense for some \(n\leq N\). The potential density of \(S^{[2]}\) is discussed in some detail.