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Manin’s conjecture for toric varieties. (English) Zbl 0946.14009

This paper proves a very precise asymptotic estimate for the number \(N\big(T(K), {\mathcal K}_P^{-1},B)\) of \(K\)-rational points in the torus part \(T\) of a smooth toric variety \(P\) of height (relative to the anticanonical bundle \({\mathcal K}_P^{-1}\)) bounded by \(B\). The result is in agreement with a conjecture made by Yu. Manin [J. Franke, Yu. I. Manin and Yu. Tschinkel, Invent. Math. 95, No. 2, 421-435 (1989; Zbl 0674.14012)], although the authors of the paper under review themselves [V. V. Batyrev and Yu. Tschinkel, C. R. Acad. Sci., Paris, Sér. I 323, No. 1, 41-46 (1996; Zbl 0879.14007)] have found a counterexample to the original form of that conjecture.
The estimate is of the form \[ N\big(T(K), {\mathcal K}_P^{-1},B)\leq {{\Theta}\over{(k-1)!}}B(\log B)^{k-1}\big(1+o(1)\big) \] as \(B\to\infty\), where \(k\) is the rank of \(\operatorname {Pic} P\). The constant \(\Theta=\alpha\beta\tau_{\mathcal K}\), where \(\alpha\) depends on the cone of effective divisors in \(P\), \(\tau_{\mathcal K}\) is the Tamagawa number introduced by E. Peyre and \(\beta\) is the rank of the nontrivial part \(\operatorname {Br}P/\operatorname {Br}K\) of the Brauer group.

MSC:

14G05 Rational points
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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