Tamagawa numbers of polarized algebraic varieties
| dc.creator | Batyrev, Victor V. | |
| dc.creator | Tschinkel, Yu. | |
| dc.date | 1997-12-01 | |
| dc.date | 1997-12-17 | |
| dc.date.accessioned | 2026-07-07T01:51:22Z | |
| dc.date.available | 2026-07-07T01:51:22Z | |
| dc.description | Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ${\cal L}$-primitive varieties and ${\cal L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number $τ_{\cal L}(V)$ 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 $N(V,{\cal L},B)$ on the choice of $v$-adic metrics on ${\cal L}$. | |
| dc.description | 54 pages, minor corrections | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9712002 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9712002 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/280 | |
| dc.subject | Algebraic Geometry | |
| dc.title | Tamagawa numbers of polarized algebraic varieties | |
| dc.type | text |