Integral points of bounded height on a log Fano threefold

TL;DR

This work determines the asymptotic number of integral points of bounded height on a smooth log Fano threefold $X$ with boundary $D$ using universal torsors. It builds an explicit adelic framework: adelic metrics on Cox-graded line bundles yield a log-anticanonical height $H$, Tamagawa measures with finite and infinite components, and the alpha constants, predicting a main term of order $B\log B$. For two open subvarieties $U_1=X-D_1$ and $U_2=X-D_2$, the paper proves precise asymptotics $N_1(B)=\frac{20}{3\zeta(2)}B\log B+O(B)$ and $N_2(B)=\frac{20}{3}\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) B\log B+O(B(\log\log B)^2)$, respectively, by combining the torsor parametrization with explicit volume calculations and Möbius inversion. The results illustrate the applicability of the universal torsor method to integral points on non-toric log Fano varieties and provide explicit Tamagawa-type constants for these cases.

Abstract

We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of $\mathbb{P}^3$ outside certain planes using universal torsors.

Theorems & Definitions (32)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5