### 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_{ℒ}(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 language | English (US) |
---|---|

Pages (from-to) | 299-340 |

Number of pages | 42 |

Journal | Asterisque |

Volume | 251 |

State | Published - 1998 |

### Fingerprint

### Keywords

- Height zeta functions
- Rational points
- Tamagawa numbers

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Asterisque*,

*251*, 299-340.

**Tamagawa numbers of polarized algebraic varieties.** / Batyrev, Victor V.; Tschinkel, Yuri.

Research output: Contribution to journal › Article

*Asterisque*, vol. 251, pp. 299-340.

}

TY - JOUR

T1 - Tamagawa numbers of polarized algebraic varieties

AU - Batyrev, Victor V.

AU - Tschinkel, Yuri

PY - 1998

Y1 - 1998

N2 - 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 τℒ(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.

AB - 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 τℒ(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.

KW - Height zeta functions

KW - Rational points

KW - Tamagawa numbers

UR - http://www.scopus.com/inward/record.url?scp=0007276563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0007276563&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0007276563

VL - 251

SP - 299

EP - 340

JO - Asterisque

JF - Asterisque

SN - 0303-1179

ER -