Let $n\geq 0$ and write $\mathbb{P}^{n}=\mathbb{P}^{n}_{\mathbb{Q}}$ for short. Given $a_{r}\geq\dots\geq a_{0}\geq 1$, consider the bundle $\mathbb{P}(E)\rightarrow\mathbb{P}^{n}$ of hyperplanes of the vector bundle

where ${\mathcal{O}}(a_{j})$ denotes the $a_{j}$-th power of the universal line bundle of $\mathbb{P}^{n}$. Equivalently, $\mathbb{P}(E)$ can be defined as the bundle of lines of the dual vector bundle $E^{\vee}$. The fibre of the map $\pi\colon\mathbb{P}(E)\to\mathbb{P}^{n}$ over each point $p\in\mathbb{P}^{n}({\overline{\mathbb{Q}}})$ is a projective space of dimension $r$. This bundle is a smooth toric variety over $\mathbb{Q}$ of dimension $n+r$, see [Oda88, pp. 58-59], [Ful93, p. 42]. The particular case $n=r=1$ corresponds to Hirzebruch surfaces: for $b\geq 0$, we have $\mathbb{F}_{b}=\mathbb{P}({\mathcal{O}}(0)\oplus{\mathcal{O}}(b))\simeq\mathbb{P}({\mathcal{O}}(a_{0})\oplus{\mathcal{O}}(a_{0}+b))$ for any $a_{0}\geq 1$.

The tautological line bundle of $\mathbb{P}(E)$, denoted ${\mathcal{O}}_{\mathbb{P}(E)}(-1)$, is defined as a subbundle of $\pi^{*}E^{\vee}$. Its fibre over a point of $\mathbb{P}(E)$ is the inverse image under $\pi$ of the line in $E^{\vee}$ which is dual to the hyperplane of $E$ defining the given point. The universal line bundle ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ of $\mathbb{P}(E)$ is defined as the dual of the tautological one. Since ${\mathcal{O}}(a_{j})$, $j=0,\dots,r$, is ample, the universal line bundle is also ample [Har66]. This is the line bundle corresponding to the Cartier divisor $a_{0}D_{0}+D_{1}$, where $D_{0}$ denotes the inverse image in $\mathbb{P}(E)$ of the hyperplane at infinity of $\mathbb{P}^{n}$ and $D_{1}=\mathbb{P}(0\oplus{\mathcal{O}}(a_{1})\oplus\dots\oplus{\mathcal{O}}(a_{r}))$. Observe that, although $\mathbb{P}(E)$ is isomorphic to the bundle associated to the family of integers $a_{i}+c$ for any $c\in\mathbb{N}$, this is not the case for the associated universal line bundle, that depends on the choice of $c$.

Following Example 4.3, we regard $\mathbb{P}^{n}$ as a toric variety over $\mathbb{Q}$ equipped with the action of the split torus $\mathbb{G}_{m}^{n}$. Let $s$ be the toric section of ${\mathcal{O}}(1)$ which corresponds to the hyperplane at infinity $H_{0}$ and let $s_{j}=s^{\otimes-a_{j}}$, which is a section of ${\mathcal{O}}(-a_{j})$. Let $U=\mathbb{P}^{n}\setminus H_{0}$. The restriction of $\mathbb{P}(E)$ to $U$ is isomorphic to $U\times\mathbb{P}^{r}$ through the map $\varphi$ defined, for $p\in U$ and $q\in\mathbb{P}^{r}$, as

The torus $\mathbb{T}:=\mathbb{G}_{m}^{n+r}$ can then be included as an open subvariety of $\mathbb{P}(E)$ through the map $\varphi$ composed with the standard inclusion of $\mathbb{G}_{m}^{n+r}$ into $U\times\mathbb{P}^{r}$. The action of $\mathbb{T}$ on itself by translation extends to an action of the torus on the whole of $\mathbb{P}(E)$. Hence $\mathbb{P}(E)$ is a toric variety over $\mathbb{Q}$. With this action the divisor $a_{0}D_{0}+D_{1}$ is a $\mathbb{T}$-Cartier divisor.

By abuse of notation, we also denote $E^{\vee}$ the total space associated to the vector bundle $E^{\vee}$. The map $\mathbb{G}_{m}^{n+r}\to E^{\vee}$ defined as

induces a no-where vanishing section of the tautological line bundle of $\mathbb{P}(E)$ over the open subset $\mathbb{T}$. Its inverse, denoted $s$, is a no-where vanishing section of ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ over $\mathbb{T}$. In particular, this section induces a structure of toric line bundle on ${\mathcal{O}}_{\mathbb{P}(E)}(1)$. The divisor of the section $s$ is precisely the $\mathbb{T}$-Cartier divisor $a_{0}D_{0}+D_{1}$ considered above.

We now introduce an adelic toric metric on ${\mathcal{O}}_{\mathbb{P}(E)}(1)$. For $v=\infty$, we consider the complex vector bundle $E(\mathbb{C})$ that can be naturally metrized by the direct sum of the Fubiny-Study metric on each factor ${\mathcal{O}}(a_{j})(\mathbb{C})$. By duality, this gives a metric on $E^{\vee}(\mathbb{C})$, which induces by restriction a metric on the tautological line bundle. Applying duality once more time, we obtain a smooth metric, denoted $\|\cdot\|_{\infty}$, on $O_{\mathbb{P}(E)(\mathbb{C})}(1)$. For $v\in M_{\mathbb{Q}}\setminus\{\infty\}$, we equip $O_{\mathbb{P}(E)}(1)$ with the canonical metric (Proposition-Definition 5.20). We write $\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}=({\mathcal{O}}_{\mathbb{P}(E)}(1),(\|\cdot\|_{v})_{v\in M_{\mathbb{Q}}})$ for the obtained adelic metrized toric line bundle.

We have made a choice of splitting of $\mathbb{T}$ and therefore a choice of an identification $N=\mathbb{Z}^{n+r}$. Thus we obtain a system of coordinates in the real vector space associated to the toric variety $\mathbb{P}(E)$, $N_{\mathbb{R}}=\mathbb{R}^{n+r}=\mathbb{R}^{n}\times\mathbb{R}^{r}$. Since the metric considered at each non-Archimedean place is the canonical one, the only nontrivial contribution to the global height will come from the Archimedean place. The restriction to the principal open subset $\mathbb{P}(E)_{0}(\mathbb{C})\simeq(\mathbb{C}^{\times})^{n+r}=(\mathbb{C}^{\times})^{n}\times(\mathbb{C}^{\times})^{r}$ of the valuation map is expressed, in these coordinates, as the map ${\operatorname{val}}\colon(\mathbb{C}^{\times})^{n+r}\to N_{\mathbb{R}}$ defined by

Let $\theta_{0}$ be the natural inclusion of real variety $\mathbb{P}(E)_{0}(\mathbb{R}_{\geq 0})\simeq(\mathbb{R}_{>0})^{n+r}$ in $\mathbb{P}(E)_{0}(\mathbb{C})$ and let ${\operatorname{\mathbf{e}}}_{\mathbb{C}}$ be the homeomorphism $N_{\mathbb{R}}\to\mathbb{P}(E)_{0}(\mathbb{R}_{\geq 0})$, both defined in §5.1. In these coordinates, the composition map $\theta_{0}\circ{\operatorname{\mathbf{e}}}_{\mathbb{C}}\colon N_{\mathbb{R}}\to\mathbb{P}(E)_{0}(\mathbb{C})$ is given by $(u,v)\mapsto(\operatorname{e}^{-u_{1}},\dots,\operatorname{e}^{-u_{n}},\operatorname{e}^{-v_{1}},\dots,\operatorname{e}^{-v_{r}})$.

Write $\psi_{\infty}\colon N_{\mathbb{R}}\to\mathbb{R}$ for the function corresponding to the metric $\|\cdot\|_{\infty}$ and the toric section $s$ defined above.

The following result summarizes the toric structure of $\mathbb{P}(E)$ and of ${\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}$.

Proposition 4.37 and Theorem 6.37 imply

where, for short, $\,\text{\rm d}x$ and $\,\text{\rm d}y$ stand for $\,\text{\rm d}x_{1}\dots\,\text{\rm d}x_{n}$ and $\,\text{\rm d}y_{1}\dots\,\text{\rm d}y_{r}$, respectively.

We now compute these volume and integral giving the degree and the height of $\mathbb{P}(E)$. We show, in particular, that the height is a rational number. Recall that $\Delta^{r}$ and $\Delta^{n}$ are the standard simplexes of $\mathbb{R}^{r}$ and $\mathbb{R}^{n}$, respectively.