Tamagawa numbers of polarized algebraic varieties

Victor V. Batyrev, Yuri Tschinkel

Research output: Contribution to journalArticle

Abstract

Let ℒ= (L, ∥ · ∥υ) be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety V over a number field F. Denote by N(V, ℒ, B) the number of rational points in V having ℒ-height ≤ B. In this paper we consider the problem of a geometric and arithmetic interpretation of the asymptotic for N(V, ℒ, B) as B → ∞ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ℒ-primitive varieties and ℒ-primitive fibrations. For ℒ-primitive varieties V over F we propose a method to define an adelic Tamagawa number τ(V) which is a generalization of the Tamagawa number τ(V) introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for Q-Fano varieties with at worst canonical singularities. In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of N(V, ℒ, B) on the choice of υ-adic metrics on ℒ.

Original languageEnglish (US)
Pages (from-to)299-340
Number of pages42
JournalAsterisque
Volume251
StatePublished - 1998

Keywords

  • Height zeta functions
  • Rational points
  • Tamagawa numbers

ASJC Scopus subject areas

  • Mathematics(all)

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