Hyperbola method on toric varieties. (Méthode de l’hyperbole sur les variétés toriques.) (English. French summary) Zbl 07811890
Pieropan and Schindler obtain here estimates for the number of Campana points of bounded height on toric varieties over \(\mathbb{Q}\). Recall that a Campana point is a notion that “interpolates” between a rational and an integral point on orbifolds. Their method extends the work of V. Blomer and J. Brüdern [J. Reine Angew. Math. 737, 255–300 (2018; Zbl 1408.11099)] who treated the case of products of projective spaces.
More precisely, let \(X\) be a complete smooth split toric variety over \(\mathbb{Q}\). Denote by \(D_1,\cdots,D_s\) the prime torus invariant divisors corresponding to rays of the fan that defines \(X\). Assume that \(L=\sum_{i=1}^s\frac{1}{m_i}D_i\) is ample and let \(H_L:X(\mathbb{Q})\rightarrow\mathbb{R}_+\) be the height associated to \(L\). Let \(N(B)\) be the number of Campana points on the orbifold \(\bigl(X,\sum_{i=1}^s(1-1/m_i)D_i\bigr)\) that have height \(H_L\) at most \(B\) and do not lie in \(\bigcup_{i=1}^sD_i\). The present authors prove that \[ N(B)=cB(\ln B)^{r-1}+O\bigl(B(\ln B)^{r-2}(\ln\ln D)^s\bigr)\ , \] where \(r\) is the rank of the Picard group of \(X\) and \(c\) is a positive constant.
This result matches the prediction of a Manin-type conjecture on Fano orbifolds [M. Pieropan et al., Proc. Lond. Math. Soc. (3) 123, No. 1, 57–101 (2021; Zbl 1479.11116)].
More precisely, let \(X\) be a complete smooth split toric variety over \(\mathbb{Q}\). Denote by \(D_1,\cdots,D_s\) the prime torus invariant divisors corresponding to rays of the fan that defines \(X\). Assume that \(L=\sum_{i=1}^s\frac{1}{m_i}D_i\) is ample and let \(H_L:X(\mathbb{Q})\rightarrow\mathbb{R}_+\) be the height associated to \(L\). Let \(N(B)\) be the number of Campana points on the orbifold \(\bigl(X,\sum_{i=1}^s(1-1/m_i)D_i\bigr)\) that have height \(H_L\) at most \(B\) and do not lie in \(\bigcup_{i=1}^sD_i\). The present authors prove that \[ N(B)=cB(\ln B)^{r-1}+O\bigl(B(\ln B)^{r-2}(\ln\ln D)^s\bigr)\ , \] where \(r\) is the rank of the Picard group of \(X\) and \(c\) is a positive constant.
This result matches the prediction of a Manin-type conjecture on Fano orbifolds [M. Pieropan et al., Proc. Lond. Math. Soc. (3) 123, No. 1, 57–101 (2021; Zbl 1479.11116)].
Reviewer: Pascal Autissier (Bordeaux)
MSC:
| 11P21 | Lattice points in specified regions |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11G50 | Heights |
| 14G05 | Rational points |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
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