Remarks on a conjecture of Batyrev and Manin
type: Item
instance of: scholarly article
title: Remarks on a conjecture of Batyrev and Manin
zbMATH Open document ID: 0927.14014
author: Y. Morita
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1999-11-18T00:00:00Z
review text: 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 \(V\).
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 1140481
zbMATH Keywords: ample line bundle
zbMATH Keywords: Néron-Severi group
zbMATH Keywords: Dirichlet series
Wikidata QID: Q123141677
reviewed by: B. Z. Moroz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225115
OpenAlex ID: W2037133666
cites work: On the number of rational points of bounded height on algebraic varieties
cites work: Q5749404
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Rational points of bounded height on Fano varieties
cites work: Q3328651
cites work: Q4143433
cites work: Distribution of rational points on hyperelliptic surfaces
cites work: Quasi-functions and heights on abelian varieties
cites work: Q3992577
cites work: Q3710637
cites work: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack
DOI: 10.2748/TMJ/1178225115
description: scientific article; zbMATH DE number 1140481