Remarks on a conjecture of Batyrev and Manin
Let \(L\) be an ample line bundle on a non-singular projective algebraic variety \(V\) defined over a number field \(k\), let \(\text{NS}(V)\) denote the Néron-Severi group of \(\overline V:=V\times_k \overline \mathbb{Q}\), and let \(N^1_{eff}(V)\) be the closed cone in \(\text{NS}(V) \oplus_\mathbb{Z} \mathbb{R}\) generated by the effective divisors on \(\overline V\). Let \[ \alpha(L): =\inf\{ \gamma \mid\gamma \in\mathbb{R},\;\gamma \cdot [L]+ K_V\in N^1_{\text{eff}}(V)\}, \] where \(K_V\) is the canonical class on \textit{V. Batyrev} and \textit{Yu. I. Manin} [Math. Ann. 286, No. 1-3, 27-43 (1990; Zbl 0679.14008)], conjectured that, for any \(\varepsilon >0\), there is an open (dense) subset \(U(L,\varepsilon)\) of \(V\) satisfying the condition \(\beta_U(L)\leq\alpha(L)+\varepsilon\), where \(\beta_U(L)\) denotes the abscissa of convergence of the Dirichlet series \(\sum_{x\in U(k)} H_L(x)^{-s}\), and \(H_L\) is the exponential height associated to \(L\). The author investigates the validity of this conjecture for \(V\) under the assumption that it holds true for some unramified covering of \(V\), and relates the sign of the geometric invariant \(\alpha(L)\) to the Kodaira dimension of \(V\).
scientific article; zbMATH DE number 1140481