On the number of points of bounded height on arithmetic projective spaces
Let \(K\) be a number field and \(\mathcal{O}_K\) its ring of integers. Let \(\overline{E}\) be a Hermitian vector bundle over \(\text{Spec}(\mathcal{O}_K)\) of rank \(N\) and \(\mathbb{P}(\overline{E})\) its associated projective bundle. The first result of the paper uses Arakelov theory to prove an analogue of Schanuel's theorem on the number of points of bounded height on in a projective space over a number field [\textit{S. Schanuel}, Bull. Soc. Math. Fr. 107, 433-449 (1979; Zbl 0428.12009)]. Namely, an explicit upper bound depending on \(N\) and \(K\) for the number of points of \(\mathbb{P}(\overline{E})(K)\) whose normalized Arakelov height is at most \(T\) as \(T\to\infty\). Three applications of this result are given. In the first one, there is an estimate for the number of hyperplanes in \(\mathbb{P}(\overline{E})\) whose normalized Arakelov height is at most \(T\) containing a fixed linear subspace of \(\mathbb{P}(\overline{E})\) as \(T\to\infty\). The second one is an estimate for the number of hypersurfaces in \(\mathbb{P}(\overline{E})\) of degree at most \(T\) containing a fixed linear subspace as \(T\to\infty\). The third application is inspired by previous results [\textit{J. Franke, Y. Manin} and \textit{Y. Tschinkel}, Invent. Math. 95, 421-435 (1989; Zbl 0674.14012) and \textit{V. Batyrev} and \textit{Y. Manin}, Math. Ann. 286, 27-43 (1990; Zbl 0679.14008)] on the asymptotic estimate of points of bounded height on Fano varieties. Given the blow-up \(\widetilde{X}\) of \(\mathbb{P}^N_K\) at a linear subspace \(F\) of codimension two and \(\widetilde{T}\) the anticanonical line bundle of \(\widetilde{X}\), an estimate for the number of points in \(\widetilde{X}(K)\) of normalized Arakelov height with respect to \(\widetilde{T}\) at most \(T\) as \(T\to\infty\) is obtained.
scientific article; zbMATH DE number 1310838