8.1. Height of toric projective curves [02Y0]

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8.1. Height of toric projective curves

In this section, we study the Arakelov invariants of curves which are the image of an equivariant map into a projective space. In the Archimedean case we equip the projective space with the Fubini-Study metric, while in the non-Archimedean case we equip it with the canonical metric. For each of these curves, the metric, measure and toric local height can be computed in terms of the roots of a univariate polynomial associated to the relevant equivariant map.

Let $K$ be either $\mathbb{R},\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. On $\mathbb{P}^{r}$ , we consider the universal line bundle ${\mathcal{O}}(1)$ equipped with the Fubini-Study metric in the Archimedean case, and with the canonical metric in the non-Archimedean case. We write ${\overline{{\mathcal{O}}(1)}}$ for the resulting metrized line bundle. We also consider the toric section $s_{\infty}$ of ${\mathcal{O}}(1)$ whose Weil divisor is the hyperplane at infinity. Next result gives the induced function $\psi$ for a subvariety of $\mathbb{P}^{r}$ which is the image of an equivariant map.

Proposition 8.1.

Let $H\colon N\to\mathbb{Z}^{r}$ be an injective map such that $H(N)$ is a saturated sublattice of $\mathbb{Z}^{r}$ , $p\in\mathbb{P}^{r}_{0}(K)$ . Consider the map $\varphi_{H,p}:\mathbb{T}\to\mathbb{P}^{r}$ , and set ${\overline{L}}=\varphi_{H,p}^{*}{\overline{{\mathcal{O}}(1)}}$ and $s=\varphi_{H,p}^{*}s_{\infty}$ . Let $\psi_{{\overline{L}},s}\colon N_{\mathbb{R}}\to\mathbb{R}$ be the associated concave function, $m_{i}=e_{i}^{\vee}\circ H\in M$ , $i=1,\dots,r$ , and $p=(1:p_{1}:\dots:p_{r})$ with $p_{i}\in K^{\times}$ . Then, for $u\in N_{\mathbb{R}}$ ,

\psi_{{\overline{L}},s}(u)=\begin{cases}-\frac{1}{2}\log(1+\sum_{i=1}^{r}|p_{i}|^{2}\operatorname{e}^{-2\langle m_{i},u\rangle}),&\text{in the Archimedean case},\\ \min_{1\leq i\leq r}\{0,\langle m_{i},u\rangle+{\operatorname{val}}_{K}(p_{i})\}&\text{in the non-Archimedean case}.\end{cases}

Proof.

In the Archimedean case, the expression for the concave function $\psi$ follows from that for $\mathbb{P}^{r}_{K}$ (Example 5.18(2)) and Proposition 5.24. The non-Archimedean case follows from Example 5.26. ∎

Let $Y\subset\mathbb{P}^{r}$ be the closure of the image of the map $\varphi_{H,p}$ . In this situation, the roof function seems difficult to calculate. Hence it is difficult to use it directly to compute the toric local height (see Example 3.57). A more promising approach is to apply the formula of Corollary 6.17. Writing $\psi=\psi_{{\overline{L}},s}$ this formula reads

(8.2)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\lambda_{K}(n+1)!\int_{N_{\mathbb{R}}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{M}(\psi).

To make this formula more explicit in the Archimedean case, we choose a basis of $N$ , hence coordinate systems in $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$ and we write

g=(g_{1},\dots,g_{n}):=\nabla\psi\colon N_{\mathbb{R}}\longrightarrow\Delta,

where $\Delta=\operatorname{stab}(\psi)$ is the associated polytope. Then, from Proposition 3.94 and Example 3.106(1), we derive

	$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)$	$\displaystyle=(n+1)!\int_{N_{\mathbb{R}}}(\langle\nabla\psi(u),u\rangle-\psi(u))\,(-1)^{n}\det(\operatorname{Hess}(\psi))\,\,\text{\rm d}\operatorname{vol}_{N}$
(8.3)			$\displaystyle=(n+1)!\int_{N_{\mathbb{R}}}(\left<g(u),u\right>-\psi(u))\,(-1)^{n}\,\text{\rm d}g_{1}\land\dots\land\,\text{\rm d}g_{n}.$

When $K$ is not Archimedean, we have ${\mathcal{M}}_{M}(\psi)=\sum_{v\in\Pi^{0}(\psi)}\delta_{v}$ and, for $v\in\Pi(\psi)^{0}$ ,

\psi^{\vee}\circ\partial\psi(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}\langle x,v\rangle\,\text{\rm d}\operatorname{vol}_{M}-\psi(v),

see Proposition 3.95 and Example 3.106(2). Thus, if now we denote by $g\colon N_{\mathbb{R}}\to M_{\mathbb{R}}$ the function that sends a point $u$ to the barycentre of $\partial\psi(u)$ , then

(8.4)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\lambda_{K}(n+1)!\sum_{v\in\Pi^{0}(\psi)}(\left<g(v),v\right>-\psi(v)).

In the case of curves, the integral of equation (8.2), can be transformed into another integral that will prove useful for explicit computations. We introduce a notation for derivatives of concave functions of one variable. Let $f\colon\mathbb{R}\to\mathbb{R}$ be a concave function. We write

(8.5)

f^{\prime}(u)=\frac{1}{2}(D_{+}f(u)+D_{-}f(u)),

where $D_{+}f$ and $D_{-}f$ denote the right and left derivatives of $f$ respectively, that exist always. Then $f^{\prime}$ is monotone and is continuous almost everywhere (with respect to the Lebesgue measure). The associated distribution agrees with the derivative of $f$ in the sense of distributions. This implies that, if $\{f_{n}\}_{n}$ is a sequence of concave functions converging uniformly to $f$ on compacts, then $\{f^{\prime}_{n}\}$ converges to $f^{\prime}$ almost everywhere.

Lemma 8.6.

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be a concave function whose stability set is an interval $[a,b]$ . Then

2\int_{\mathbb{R}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{\mathbb{Z}}(\psi)=(b-a)(\psi^{\vee}(a)+\psi^{\vee}(b))+\int_{\mathbb{R}}(\psi^{\prime}(u)-a)(b-\psi^{\prime}(u))\,\text{\rm d}u.

Proof.

By the properties of the Monge-Ampère measure (Proposition 3.93) and of the Legendre-Fenchel dual (Proposition 3.18) the left-hand side is continuous with respect to uniform convergence of functions. Again by Proposition 3.18 and the discussion before the lemma, the right-hand side is also continuous with respect to uniform convergence of functions. Therefore it is enough to treat the case when $\psi$ is smooth and strictly concave. Then

2\int_{\mathbb{R}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{\mathbb{Z}}(\psi)=2\int_{\mathbb{R}}(\psi(u)-u\psi^{\prime}(u))\psi^{\prime\prime}(u)\,\text{\rm d}u.

Consider the function

	$\displaystyle\gamma(u)$	$\displaystyle=(\psi^{\prime}(u)-\frac{a+b}{2})\psi(u)-u\frac{(\psi^{\prime})^{2}}{2}+u\frac{ab}{2}$
		$\displaystyle=-(\psi^{\prime}(u)-\frac{a+b}{2})\psi^{\vee}(\psi^{\prime}(u))-\frac{u}{2}(\psi^{\prime}(u)-a)(b-\psi^{\prime}(u)).$

Then

\lim_{u\to\infty}\gamma(u)=\frac{b-a}{2}\psi^{\vee}(a),\qquad\lim_{u\to-\infty}\gamma(u)=\frac{a-b}{2}\psi^{\vee}(b),

and

\,\text{\rm d}\gamma=(\psi-u\psi^{\prime})\psi^{\prime\prime}\,\text{\rm d}u-\frac{1}{2}(\psi^{\prime}-a)(b-\psi^{\prime})\,\text{\rm d}u,

from which the result follows. ∎

With the notation in Proposition 8.1, assume that $N=\mathbb{Z}$ . The elements $m_{j}\in N^{\vee}$ can be identified with integer numbers and the hypothesis that the image of $H$ is a saturated sublattice is equivalent to $\gcd(m_{1},\dots,m_{r})=1$ . Moreover, by reordering the variables of $\mathbb{P}^{r}$ and multiplying the expression of $\varphi_{H,p}$ by a monomial (which does not change the equivariant map), we may assume that $0\leq m_{1}\leq\dots\leq m_{r}$ . We make the further hypothesis that $0<m_{1}<\dots<m_{r}$ . With these conditions, we next obtain explicit expressions for the concave function $\psi$ and the associated measure and toric local height in terms of the roots of a univariate polynomial. We consider the absolute value $|\cdot|$ of the algebraic closure ${\overline{K}}$ extending the absolute value of $K$ . For $\xi\in{\overline{K}}^{\times}$ , we set ${\operatorname{val}}_{{\overline{K}}}(\xi)=-\frac{\log|\xi|}{\lambda_{K}}$ .

Theorem 8.7.

Let $0<m_{1}<\dots<m_{r}$ be integer numbers with $\gcd(m_{1},\dots,m_{r})=1$ , and $p_{1},\dots,p_{r}\in K^{\times}$ . Let $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ be the map given by $\varphi(t)=(1:p_{1}t^{m_{1}}:\dots:p_{r}t^{m_{r}})$ and let $Y$ be the closure of the image of $\varphi$ . Consider the polynomial $q\in K[z]$ defined as

\displaystyle q=\begin{cases}1+\sum_{j=1}^{r}|p_{j}|^{2}z^{m_{j}},&\text{ in the Archimedean case},\\ 1+\sum_{j=1}^{r}p_{j}z^{m_{j}},&\text{ in the non-Archimedean case}.\end{cases}

Let $\{\xi_{i}\}_{i}\subset{\overline{K}}^{\times}$ be the set of roots of $q$ and, for each $i$ , let $\ell_{i}\in\mathbb{N}$ be the multiplicity of $\xi_{i}$ . Let ${\overline{L}}$ and $s$ be as in Proposition 8.1. Then, in the Archimedean case,

(1)

$\displaystyle\psi_{{\overline{L}},s}(u)=-\log|p_{r}|-\frac{1}{2}\sum_{i}\ell_{i}\log|\operatorname{e}^{-2u}-\xi_{i}|$ for $u\in\mathbb{R}$ ,
(2)

$\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=-2\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}\,\,\text{\rm d}u$ ,
(3)

$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{\xi_{i}+\xi_{j}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j}))$ , where $\log$ is the principal determination of the logarithm.

While in the non-Archimedean case,

(4)

$\displaystyle\psi_{{\overline{L}},s}(u)={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{u,{\operatorname{val}}_{{\overline{K}}}(\xi_{i})\}$ for $u\in\mathbb{R}$ ,
(5)

$\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=\sum_{i}\ell_{i}\delta_{{\operatorname{val}}_{{\overline{K}}}(\xi_{i})}$ ,
(6)

$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\sum_{i<j}\ell_{i}\ell_{j}\log(\max\{1,|\xi_{i}|/|\xi_{j}|\})$ .

Remark 8.8.

The real roots of the polynomial $q$ are all negative, this allows the use of the principal determination of the logarithm in (3). Introducing the argument $\theta_{i}\in]-\pi,\pi[$ of $-\xi_{i}$ , the last sum in (3) can be rewritten

\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{(|\xi_{i}|^{2}-|\xi_{j}|^{2})\log|\xi_{i}/\xi_{j}|+2|\xi_{i}||\xi_{j}|(\theta_{i}-\theta_{j})\sin(\theta_{i}-\theta_{j})}{|\xi_{i}|^{2}+|\xi_{j}|^{2}-2|\xi_{i}||\xi_{j}|\cos(\theta_{i}-\theta_{j})}

showing that it is real.

Proof.

Write $\psi=\psi_{{\overline{L}},s}$ for short. First we consider the Archimedean case. We have that $q=|p_{r}|^{2}\prod_{i}(z-\xi_{i})^{\ell_{i}}$ . By Proposition 8.1,

\psi(u)=-\frac{1}{2}\log(q(\operatorname{e}^{-2u}))=-\log|p_{r}|-\frac{1}{2}\sum_{i}\ell_{i}\log|\operatorname{e}^{-2u}-\xi_{i}|,

which proves (1). Hence,

\psi^{\prime}(u)=\sum_{i}\ell_{i}\frac{1}{1-\xi_{i}\operatorname{e}^{2u}}\quad\text{and}\quad\psi^{\prime\prime}(u)=\sum_{i}2\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}.

The Monge-Ampère measure of $\psi$ is given by $-\psi^{\prime\prime}\,\text{\rm d}u$ , and so the above proves (2). To prove (3) we apply Lemma 8.6. We have that $\operatorname{stab}(\psi)=[0,m_{r}]$ , $\psi^{\vee}(0)=0$ , and $\psi^{\vee}(m_{r})=\log|p_{r}|$ . Thus,

(8.9)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u.

We have $\displaystyle m_{r}-\psi^{\prime}(u)=\sum_{i}\ell_{i}\bigg(1-\frac{1}{1-\xi_{i}\operatorname{e}^{2u}}\bigg)=-\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{1-\xi_{i}\operatorname{e}^{2u}}$ . Hence,

(m_{r}-\psi^{\prime}(u))\psi^{\prime}(u)=-\bigg(\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{1-\xi_{i}\operatorname{e}^{2u}}\bigg)\bigg(\sum_{j}\ell_{j}\frac{1}{1-\xi_{j}\operatorname{e}^{2u}}\bigg)\\ =-\sum_{i}\ell_{i}^{2}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}-\sum_{i\neq j}\ell_{i}\ell_{j}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})(1-\xi_{j}\operatorname{e}^{2u})}.

Moreover $\displaystyle\int_{-\infty}^{\infty}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}\,\text{\rm d}u=\bigg[\frac{1}{2(1-\xi_{i}\operatorname{e}^{2u})}\bigg]^{\infty}_{-\infty}=-\frac{1}{2}$ and

\int_{-\infty}^{\infty}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})(1-\xi_{j}\operatorname{e}^{2u})}\,\text{\rm d}u=\bigg[\frac{\xi_{i}}{2(\xi_{i}-\xi_{j})}(\log(1-\xi_{j}\operatorname{e}^{2u}))-\log(1-\xi_{i}\operatorname{e}^{2u})\bigg]^{\infty}_{-\infty}\\ =\frac{\xi_{i}}{2(\xi_{i}-\xi_{j})}(\log(-\xi_{i})-\log(-\xi_{j})),

for the principal determination of $\log$ . These calculations together with equation (8.9) imply that

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i\neq j}\ell_{i}\ell_{j}\frac{\xi_{i}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j}))\\ =m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{\xi_{i}+\xi_{j}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j})),

which proves (3).

Next we consider the non-Archimedean case. Let $U\subset{\overline{K}}^{\times}$ be a sufficiently small open subset and $\zeta\in U$ . For short, write $v_{i}={\operatorname{val}}_{{\overline{K}}}(\xi_{i})$ . By Proposition 8.1, the genericity of $\zeta$ , and the condition $m_{i}\not=m_{j}$ for $i\not=j$ , imply

\psi({\operatorname{val}}_{{\overline{K}}}(\zeta))=\min_{i}\{0,m_{i}{\operatorname{val}}_{{\overline{K}}}(\zeta)+{\operatorname{val}}_{K}(p_{i})\}={\operatorname{val}}_{{\overline{K}}}(q(\zeta)).

By the factorization of $q$ ,

{\operatorname{val}}_{{\overline{K}}}(q(\zeta))={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}{\operatorname{val}}_{{\overline{K}}}(\zeta-\xi_{i})={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{{\operatorname{val}}_{{\overline{K}}}(\zeta),v_{i}\}.

The image of ${\operatorname{val}}_{{\overline{K}}}\colon{\overline{K}}^{\times}\to\mathbb{R}$ is a dense subset. We deduce that, $u\in\mathbb{R}$ ,

\psi(u)={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{u,{\operatorname{val}}_{K}(\xi_{i})\},

which proves (4). The gradient of this function is, for $u\in\mathbb{R}$ ,

\partial\psi(u)=\begin{cases}\Big[\sum_{j:v_{j}>v_{i}}\ell_{j},\sum_{j:v_{j}\geq v_{i}}\ell_{j}\Big]&\text{ if }u=v_{i}\text{ for some }i,\\ \sum_{j:v_{j}>x}\ell_{j}&\text{ otherwise.}\end{cases}

Hence, the associated Monge-Ampère measure is $\sum_{i}\ell_{i}\delta_{v_{i}},$ which proves (5). The derivative of $\psi$ in the sense of (8.5) is, for $u\in\mathbb{R}$ ,

\psi^{\prime}(u)=\begin{cases}\sum_{j:v_{j}>v_{i}}\ell_{j}+\frac{1}{2}\sum_{j:v_{j}=v_{i}}\ell_{j}&\text{ if }u=v_{i}\text{ for some }i,\\ \sum_{j:v_{j}>x}\ell_{j}&\text{ otherwise.}\end{cases}

Moreover, $\operatorname{stab}(\psi)=[0,m_{r}]$ , $\psi^{\vee}(0)=0$ and $\psi^{\vee}(m_{r})=-{\operatorname{val}}_{K}(p_{r})$ . By Lemma 8.6

(8.10)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=-m_{r}\lambda_{K}{\operatorname{val}}_{K}(p_{r})+\lambda_{K}\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u.

If we write

f_{i}(u)=\begin{cases}0,&\text{ if }x\leq v_{i}\\ \ell_{i},&\text{ if }x>v_{i},\end{cases}

then, we have that, almost everywhere $\psi^{\prime}(u)=\sum_{i}\ell_{i}-f_{i}(u)$ and $m_{r}-\psi^{\prime}(u)=\sum_{i}f_{i}$ . Therefore

(8.11)

\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u=\sum_{i,j}\int_{-\infty}^{\infty}f_{i}(\ell_{j}-f_{j})\,\text{\rm d}u=\sum_{i,j}\ell_{i}\ell_{j}\max\{0,v_{j}-v_{i}\}.

Thus, joining together (8.10), (8.11) and the relation $\log(|\zeta|)=-\lambda_{K}{\operatorname{val}}_{K}(\zeta)$ we deduce

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\sum_{i,j}\ell_{i}\ell_{j}\max\{0,\log(|\xi_{i}|/|\xi_{j}|)\},

finishing the proof of the theorem. ∎

We now treat the global case.

Corollary 8.12.

Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be a global field. Let $0<m_{1}<\dots<m_{r}$ be integer numbers with $\gcd(m_{1},\dots,m_{r})=1$ , and $p_{1},\dots,p_{r}\in\mathbb{K}^{\times}$ . Let $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ be the map given by $\varphi(t)=(1:p_{1}t^{m_{1}}:\dots:p_{r}t^{m_{r}})$ , $Y$ the closure of the image of $\varphi$ , and ${\overline{L}}=\varphi^{*}{\overline{{\mathcal{O}}(1)}}$ , where ${\overline{{\mathcal{O}}(1)}}$ is equipped with the Fubini-Study metric for the Archimedean places and with the canonical metric for the non-Archimedean places. For $v\in\mathfrak{M}_{K}$ , set

\displaystyle q_{v}=\begin{cases}1+\sum_{j=1}^{r}|p_{j}|_{v}^{2}z^{m_{j}},&\text{ if }v\text{ is Archimedean},\\ 1+\sum_{j=1}^{r}p_{j}z^{m_{j}},&\text{ if }v\text{ is not Archimedean}.\end{cases}

Let $\{\xi_{v,i}\}\subset{\overline{\mathbb{K}}}^{\times}$ be the set of roots of $q_{v}$ and, for each $i$ , let $\ell_{v,i}\in\mathbb{N}$ denote the multiplicity of $\xi_{v,i}$ . Then

\operatorname{h}_{{\overline{L}}}(Y)=\sum_{v|\infty}n_{v}\bigg(\frac{1}{2}\sum_{i}\ell_{v,i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{v,i}\ell_{v,j}\frac{\xi_{v,i}+\xi_{v,j}}{\xi_{v,i}-\xi_{v,j}}(\log(-\xi_{v,i})-\log(-\xi_{v,j}))\bigg)\\ +\sum_{v\nmid\infty}n_{v}\bigg(\sum_{i<j}\ell_{v,i}\ell_{v,j}\log(\max\{1,|\xi_{v,i}|_{v}/|\xi_{v,j}|_{v}\})\bigg).

Proof.

This follows readily from Proposition 6.35, Theorem 8.7, and the product formula. ∎

Corollary 8.13.

Let $C_{r}\subset\mathbb{P}^{r}_{\mathbb{Q}}$ be the Veronese curve of degree $r$ and ${\overline{{\mathcal{O}}(1)}}$ the universal line bundle on $\mathbb{P}^{r}_{\mathbb{Q}}$ equipped with the Fubini-Study metric at the Archimedean place and with the canonical metric at the non-Archimedean ones. Then

(8.14)

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=\frac{r}{2}+\pi\sum_{j=1}^{\lfloor r/2\rfloor}\bigg(1-\frac{2\,j}{r+1}\bigg)\,\cot\bigg(\frac{\pi\,j}{r+1}\bigg)\in\frac{r}{2}+\pi\,{\overline{\mathbb{Q}}}.

Proof.

The curve $C_{r}$ coincides with the closure of the image of the map $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ given by $\varphi(t)=(1:t:t^{2}:\dots:t^{r})$ . With the notation in Corollary 8.12, this map correspond to $m_{i}=i$ and $p_{i}=1$ , for $i=1,\dots,r$ . Then $q_{v}=\sum_{j=0}^{r}z^{j}$ for all $v\in\mathfrak{M}_{\mathbb{Q}}$ . Consider the primitive $(r+1)$ -th root of unity $\omega=\operatorname{e}^{\frac{2\pi i}{r+1}}$ . The polynomial $q_{v}$ is separable and its set of roots is $\{\omega^{l}\}_{l=1,\dots,r}$ . Since $|\omega^{l}|_{v}=1$ for all $v$ , Corollary 8.12 implies that

(8.15)

\operatorname{h}_{{\overline{L}}}(Y)=\frac{r}{2}+\frac{1}{2}\sum_{l<j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}(\log(-\omega^{l})-\log(-\omega^{j}))\\ =\frac{r}{2}+\frac{1}{2}\sum_{l\neq j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}\log(-\omega^{l}).

We have that

\sum_{j=1}^{r}\frac{\omega^{j}+1}{\omega^{j}-1}=\sum_{j=1}^{r}\frac{\omega^{j}}{\omega^{j}-1}+\sum_{j=1}^{r}\frac{1}{\omega^{j}-1}=\sum_{j=1}^{r}\frac{1}{1-\omega^{-j}}+\sum_{j=1}^{r}\frac{1}{\omega^{j}-1}=0.

This implies that, for $l=1,\dots,r$ ,

\sum_{1\leq j\leq r,j\neq l}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}=-\frac{\omega^{l}+1}{\omega^{l}-1}=i\cot\Big(\frac{\pi l}{r+1}\Big)

Hence,

\frac{1}{2}\sum_{l\neq j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}\log(-\omega^{l})=-\frac{i}{2}\sum_{l=1}^{r}\cot\Big(\frac{\pi l}{r+1}\Big)\log(-\omega^{l})\\ =\pi\sum_{l=1}^{\lfloor r/2\rfloor}\cot\Big(\frac{\pi l}{r+1}\Big)\Big(1-\frac{2l}{r+1}\Big),

since $\cot(\frac{\pi(r+1-l)}{r+1})\log(-\omega^{r+1-l})=\cot(\frac{\pi l}{r+1})\log(-\omega^{l})$ for $l=1,\dots,\lfloor r/2\rfloor$ and $\log(-\omega^{\frac{r+1}{2}})=0$ whenever $r$ is odd. The statement follows from this calculations together with (8.15). ∎

Here follow some special values:

$r$	1	2	3	5	7
$\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})$	$\displaystyle\frac{1}{2}$	$\displaystyle 1+\frac{1}{3\,\sqrt{3}}\,\pi$	$\displaystyle\frac{3}{2}+\frac{1}{2}\,\pi$	$\displaystyle\frac{5}{2}+\frac{7}{3\,\sqrt{3}}\,\pi$	$\displaystyle\frac{7}{2}+(1+\sqrt{2})\,\pi$

Corollary 8.16.

With the notation of Corollary 8.13, $\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=r\log r+O(r)$ for $r\to\infty$ .

Proof.

We have that $\displaystyle\pi\cot(\pi x)=\frac{1}{x}+O(1)$ for $x\to 0$ . Hence,

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=\sum_{j=1}^{\lfloor r/2\rfloor}\bigg(1-\frac{2\,j}{r+1}\bigg)\,\frac{j}{r+1}+O(r)=r\bigg(\sum_{j=1}^{\lfloor r/2\rfloor}\frac{1}{j}\bigg)+O(r)=r\log r+O(r).

∎

By the theorem of algebraic successive minima [Zha95a],

\mu^{\operatorname{ess}}(C_{r})\leq\frac{\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})}{\deg_{{{\mathcal{O}}(1)}}(C_{r})}\leq 2\mu^{\operatorname{ess}}(C_{r})

The essential minimum of $C_{r}$ is $\mu^{\operatorname{ess}}(C_{r})=\frac{1}{2}\log(r+1)$ [Som05]. Hence, the quotient $\frac{\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})}{\deg_{{{\mathcal{O}}(1)}}(C_{r})}$ is asymptotically closer to the upper bound than to the lower bound.