On the number of rational points of bounded height on algebraic varieties (Q1124645)

Let \(k\) be a global field, \(V\) a projective variety defined over \(k\), \(h_ L\) an exponential height associated to \(L\). For a subset \(U\subset V(k)\), we denote by \(\beta_ U(L)\) the abscissa of convergence of \(\sum_{x\in U}h_ L(x)^{-s} \). We define also the function \(\alpha(L)=\inf\{\gamma \in {\mathbb R}| \quad \gamma L+K_ V\) is effective modulo Néron- Severi equivalence\}. The paper states some conjectures to the effect that \(\beta_ U(L)\) and \(\alpha(L)\) are comparable (sometimes equal) if one stabilizes the situation taking \(k\) sufficiently large and \(U\) sufficiently small and Zariski-open. These conjectures are proved for homogeneous Fano varieties and some del Pezzo surfaces.