Heights and Tamagawa measures on Fano varieties
type: Item
instance of: scholarly article
title: Heights and Tamagawa measures on Fano varieties
zbMATH Open document ID: 0901.14025
DOI: 10.1215/S0012-7094-95-07904-6
author: Emmanuel Peyre
published in: Duke Mathematical Journal
publication date: 1995-12-03T00:00:00Z
review text: Let \(V\) be a Fano variety over a number field \(K\) such that the anticanonical class \(-K_V\) is very ample. To every basis of the \(K\)-vector space \(H^0(V,{\mathcal O}_V(-K_V))\) one can naturally associate a height \({\mathbf h}\) on \(V\). For every non-empty open subset \(U\) of \(V\) one denotes by \(n_U(H)\) the cardinal of the set \(\{P\in U(K)\mid {\mathbf h}(P)\leq H\}\). Then Manin conjectured that the asymptotic behaviour of \(n_U(H)\) is of the form \(n_U(H)\sim C\cdot H\cdot\log^{t-1}(H)\), where \(t\) is the rang of the Picard group of \(V\), provided \(U\) is sufficiently small. The first aim of the paper is to refine this conjecture of Manin by giving a conjectural expression of the constant \(C\) occurring in this formula. The advantage of this is that the refined conjecture thus obtained becomes stable under products of varieties, and, in particular, the author is able to recover the previous constants found by \textit{S. H. Schanuel} [Bull. Soc. Math. Fr. 107, 433--449 (1979; Zbl 0428.12009)] (in the case of projective spaces), and by \textit{J. Franke}, \textit{Yu. I. Manin} and \textit{Y. Tschinkel} [Invent. Math. 95, 421--435 (1989; Zbl 0674.14012)] for generalized flag manifolds. In the last part of the paper the author proves that the refined Manin conjecture holds true in some significant special cases, namely for the del Pezzo surfaces obtained by blowing up one, two, or three points of the rational projective plane \(P^2_{\mathbb Q}\), as well as for the variety obtained by blowing up \(P^n_{\mathbb Q}\) along some subspaces of \(P^n_{\mathbb Q}\).
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 809309
zbMATH Keywords: heights
zbMATH Keywords: Tamagawa measures
zbMATH Keywords: Fano variety
zbMATH Keywords: rational points
zbMATH Keywords: del Pezzo surfaces
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07904-6
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