# Points of bounded height on certain subvarieties of toric varieties

Marta Pieropan  and  Damaris Schindler  Utrecht University, Mathematical
Institute, Budapestlaan 6, 3584 CD Utrecht, the Netherlands<EMAIL_ADDRESS>Mathematisches Institut, Universitaet Goettingen, Bunsenstrasse 3-5, 37073
Goettingen, Germany<EMAIL_ADDRESS>
###### Abstract.

We combine the split torsor method and the hyperbola method for toric
varieties to count rational points and Campana points of bounded height on
certain subvarieties of toric varieties.

###### Key words and phrases:

Hyperbola method, $m$-full numbers, Campana points, toric varieties.

###### 2010 Mathematics Subject Classification:

11P21 (11A25, 11G50, 14G05, 14M25)

###### Contents

  1. 1 Introduction
    1. 1.1 Results
  2. 2 Toric varieties setting
    1. 2.1 Torus invariant divisors
    2. 2.2 Heights
    3. 2.3 Coprimality conditions
    4. 2.4 Möbius function
  3. 3 Subvarieties
  4. 4 Rational points on linear complete intersections
  5. 5 Bihomogeneous hypersurfaces
  6. 6 Campana points on certain diagonal complete intersections

##  1\. Introduction

We combine the split torsor method and the hyperbola method for toric
varieties to count rational points and Campana points of bounded height on
certain subvarieties of smooth split proper toric varieties. This line of
research has been initiated by Blomer and Brüdern [BB18] in the setting of
diagonal hypersurfaces in products of projective spaces. Other results in this
direction include hypersurfaces and complete intersections in products of
projective spaces [Sch16], improvements for bihomogeneous hypersurfaces for
degree $(2,2)$ and $(1,2)$ by Browning and Hu [BH19] and Hu [Hu20], as well as
generalisations to hypersurfaces in certain toric varieties by Mignot [Mig15,
Mig16, Mig18].

The versions of hyperbola method used in all of these articles are rather
close to the original one [BB18] for products of projective spaces. In our
recent work [PS24] we established a very general form of the hyperbola method
for split toric varieties, in which the height condition can also globally be
given by the maximum of several monomials. The goal of this article is to show
applications of our new hyperbola method. We develop a refined framework for
the split torsor method on split smooth proper toric varieties and show that
counting results for subvarieties of projective spaces can be carried over to
toric varieties by a direct application of the hyperbola method [PS24]. With
this we can prove new cases of Manin’s conjecture [FMT89, BM90] on the number
of rational points of bounded height on Fano varieties for certain
subvarieties in toric varieties.

The split torsor method provides a parametrisation of rational points on Fano
varieties via Cox rings [Sal98, DP20]. The Cox ring of a smooth proper toric
variety $X$ is a polynomial ring endowed with a grading by the Picard group of
the toric variety [Cox95]. Subvarieties of toric varieties are intersections
of hypersurfaces, which are defined by $\operatorname{Pic}(X)$-homogeneous
polynomials in the Cox ring of $X$. The subvarieties considered in this paper
are defined by homogeneous elements in the Cox ring of the toric variety such
that each polynomial involves only variables of the same degree. With the
split torsor method parametrisation the height is given by the maximum of a
set of monomials and with this in the correct shape to apply our generalised
version of the hyperbola method [PS24]. The hyperbola method reduces the
counting problem to counting functions over boxes of different shapes. An
advantage of our method is that it is already adapted to the shape of height
functions appearing. Also compared to earlier versions of the hyperbola
method, we do not need estimates for lower dimensional boxes and with this our
proofs are relatively short.

We now illustrate our approach on a number of examples. In a similar fashion,
it is possible to apply counting results such as [Bir62, HB96, RM18, RM19,
BHB17] and many others, to subvarieties of toric varieties defined by elements
of the Cox ring each involving only variables of the same degree.

###  1.1. Results

Let $X$ be a smooth split complete toric $\mathbb{Q}$-variety with open torus
$T$. Let ${\mathbf{D}}_{1},\dots,{\mathbf{D}}_{s}\in\operatorname{Pic}(X)$ be
the pairwise distinct classes of the torus invariant prime divisors on $X$.
For $i\in\\{1,\dots,s\\}$, let
$n_{i}=\dim_{\mathbb{Q}}H^{0}(X,{\mathbf{D}}_{i})$. Let $H_{L}$ be the height
associated to a semiample torus invariant divisor $L$ on $X$ as in Section
2.2.

Our first result concerns subvarieties of toric varieties defined by linear
forms.

######  Theorem 1.1.

Let $V\subseteq X$ be a complete intersection of hypersurfaces $H_{i,l}$ with
$1\leq i\leq s$, $1\leq l\leq t_{i}$ such that $[H_{i,l}]={\mathbf{D}}_{i}$ in
$\operatorname{Pic}(X)$ for $i\in\\{1,\dots,s\\}$ with $t_{i}\neq 0$. Assume
that $V\cap T\neq\emptyset$ and $t_{i}\leq n_{i}-2$ for all
$i\in\\{1,\dots,s\\}$. Assume that
$L=-(K_{X}+\sum_{i=1}^{s}\sum_{l=1}^{t_{i}}[H_{i,l}])$ is ample. For $B>0$,
let $N_{V}(B)$ be the number of $\mathbb{Q}$-rational points on $V\cap T$ of
height $H_{L}$ at most $B$. Then

$N_{V}(B)=cB(\log B)^{b-1}+O(B(\log B)^{b-2}(\log\log B)^{s}),$

where $b=\operatorname{rk}\operatorname{Pic}(V)$, and $c$ is a positive
constant, which is defined by (3.7) with $k=b-1$, $C_{M,\mathbf{d}}$ given by
(4.1), and $\varpi_{i}=n_{i}-t_{i}$ for $i\in\\{1,\dots,s\\}$.

We use this result as a toy example to show how to combine the hyperbola
method with the universal torsor method in the context of rather general
smooth split toric varieties. We now move on to results which require deeper
understanding of the underlying Diophantine problems via methods from Fourier
analysis.

We start with a result that concerns subvarieties of toric varieties defined
by bihomogeneous polynomials. It is obtained combining the framework developed
in this paper with the hyperbola method [PS24] and preliminary counting
results in boxes of different side lengths from [Sch16].

######  Theorem 1.2.

Let $V\subseteq X$ be a smooth complete intersection of hypersurfaces
$H_{1},\dots,H_{t}$ of the same degree
$e_{1}{\mathbf{D}}_{1}+e_{2}{\mathbf{D}}_{2}$ in $\operatorname{Pic}(X)$.
Assume that $V\cap T\neq\emptyset$, that $n_{i}-te_{i}\geq 2$ for
$i\in\\{1,2\\}$, and that $n_{1}+n_{2}>\dim V_{1}^{*}+\dim V_{2}^{*}+3\cdot
2^{e_{1}+e_{2}}e_{1}e_{2}t^{3}$, where
$V_{1}^{*},V_{2}^{*}\subseteq\mathbb{A}^{n_{1}+n_{2}}$ are affine varieties
defined in §5. Assume that $L=-([K_{X}]+[H_{1}+\dots+H_{t}])$ is ample. Then
there is an open subset $W\subseteq X$ such that the number $N_{V,W}(B)$ of
$\mathbb{Q}$-rational points on $V\cap W\cap T$ of height $H_{L}$ at most $B$
satisfies

$N_{V,W}(B)=cB(\log B)^{b-1}+O(B(\log B)^{b-2}(\log\log B)^{s})$

for $B>0$, where $b=\operatorname{rk}\operatorname{Pic}(V)$, and $c$ is
defined in (3.7) with $k=b-1$, $C_{M,\mathbf{d}}$ given by (5.1),
$\varpi_{i}=n_{i}-te_{i}$ for $i\in\\{1,2\\}$, and $\varpi_{i}=n_{i}$ for
$i\in\\{3,\dots,s\\}$. The constant $c$ is positive if
$V(\mathbb{Q}_{v})\neq\emptyset$ for all places $v$ of $\mathbb{Q}$.

Theorems 1.1 and 1.2 are compatible with Manin’s conjecture [FMT89], as
$L|_{V}=-K_{V}$ by adjunction. The proofs in Sections 4 and 5 yield an
asymptotic formula even if we drop the ampleness assumption on $L$.

Theorem 1.2 as well as work of Mignot [Mig16, Mig18] include the case of
certain hypersurfaces in products of projective spaces. However, in comparison
to Mignot’s work we do not require the condition that the effective cone of
the toric variety is simplicial. An example of a split toric variety with non-
simplicial effective cone, where our theorem applies, is the blow-up at a
torus-invariant point of $\mathbb{P}^{n_{1}}\times\mathbb{P}^{n_{2}}\times Y$
where $n_{1},n_{2}$ are sufficiently large and $Y$ is a split del Pezzo
surface of degree 6.

Our last result concerns sets of Campana points in the sense of [PSTVA21] for
subvarieties defined by diagonal equations. We introduce the following
integral models. Let $\mathscr{X}$ be the $\mathbb{Z}$-toric scheme defined by
the fan of $X$. For $i\in\\{1,\dots,s\\}$, let
$\mathscr{D}_{i,1},\dots,\mathscr{D}_{i,n_{i}}$ be the torus invariant prime
divisors on $\mathscr{X}$ of class ${\mathbf{D}}_{i}$.

######  Theorem 1.3.

Let $V\subseteq X$ be an intersection of hypersurfaces $H_{1},\dots,H_{t}$
such that $H_{i}$ is defined by a homogeneous diagonal polynomial in the Cox
ring of $X$ of degree $e_{i}{\mathbf{D}}_{i}$ in $\operatorname{Pic}(X)$ and
with none of the coefficients equal to zero. Let $\mathscr{V}$ be the closure
of $V$ in $\mathscr{X}$. For $i\in\\{1,\dots,s\\}$, fix integers $2\leq
m_{i,1}\leq\dots\leq m_{i,n_{i}}$. Let
$\mathscr{D}_{\mathbf{m}}=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}(1-\frac{1}{m_{i,j}})\mathscr{D}_{i,j}$.
Assume that $V\cap T\neq\emptyset$, that $n_{1},\dots,n_{t}\geq 2$, and for
$i\in\\{1,\dots,s\\}$, that $\sum_{j=1}^{n_{i}}\frac{1}{m_{i,j}}>3$, that
$\sum_{j=1}^{n_{i}-1}\frac{1}{e_{i}m_{i,j}(e_{i}m_{i,j}+1)}\geq 1$ if
$e_{i}=1$, and $\sum_{j=1}^{n_{i}}\frac{1}{2s_{0}(e_{i}m_{i,j})}>1$ if
$e_{i}\geq 2$, where $s_{0}(e_{i}m_{i,j})$ is defined in Lemma 6.1. Let
$L=-(K_{X}+\mathscr{D}_{\mathbf{m}}|_{X}+H_{1}+\dots+H_{t})$ be ample. For
$B>0$, let $N_{V}(B)$ be the number of $\mathbb{Z}$-Campana points on
$(\mathscr{V},\mathscr{D}_{\mathbf{m}}|_{\mathscr{V}})$ that lie in $T$ and
have height $H_{L}$ at most $B$. Then

$N_{V}(B)=cB(\log B)^{b-1}+O(B(\log B)^{b-2}(\log\log B)^{s}),$

where $b=\operatorname{rk}\operatorname{Pic}(V)$, and $c$ is defined in (3.7)
with $k=b-1$, $C_{M,\mathbf{d}}$ given by (6.11), and
$\varpi_{1},\dots,\varpi_{s}$ given by (6.10).

The order of growth in Theorem 1.3 is compatible with the Manin-type
conjecture for Campana points [PSTVA21], as $L|_{V}$ is the log anticanonical
divisor of the pair $(V,\mathscr{D}_{\mathbf{m}}|_{V})$ by adjunction.

Due to the range of application of the circle method, Theorems 1.2 and 1.3
require the Cox ring of the toric variety to have a large number of variables
of the same degree. This holds for toric varieties with several torus
invariant prime divisors of the same degree and for products of such toric
varieties. Here are some examples: the Cox ring of the projective space
$\mathbb{P}^{n}$ has $n+1$ variables of the same degree, the Cox ring of the
blow-up of the projective space $\mathbb{P}^{n}$ at $l<n+1$ torus invariant
points has $n+1-l$ variables of the same degree. Another example is given by
blow-ups of products of toric varieties each with several torus invariant
prime divisors of the same degree. Indeed, if $X$ and $Y$ are smooth split
toric varieties and the Cox ring of $X$ has $n_{X}$ variables of the same
degree $d_{X}$, and the Cox ring of $Y$ has $n_{Y}$ variables of the same
degree $d_{Y}$, and $P\in X\times Y$ is a point where $m_{X}\leq n_{X}$
variables of degree $d_{X}$ vanish and $m_{Y}\leq n_{Y}$ variables of degree
$d_{Y}$ vanish, then the Cox ring of the blow-up of $X\times Y$ at $P$ has
$m_{X}$ variables of the same degree $d_{X}-e$ and $m_{Y}$ variables of the
same degree $d_{Y}-e$, where $e$ is the class of the exceptional divisor.

The structure of this article is as follows. In Section 2 we reformulate the
height function and the multiplicative function $\mu$ for Möbius inversion
according to the principle of grouping variables of the same degree. In
Section 3 we combine the new framework with the hyperbola method developed in
[PS24] to obtain a general counting tool for points of bounded height on
subvarieties of toric varieties. Theorems 1.1, 1.2, and 1.3 are proven in
Sections 4, 5, and 6, respectively.

### Acknowledgements

We thank for their hospitality the organizers of the workshop “Rational Points
2023” at Schney, where we made significant progress on this project. We are
grateful to the Lorentz center in Leiden for their hospitality during the
workshop “Enumerative geometry and arithmetic”. The first named author is
supported by the NWO grants VI.Vidi.213.019 and OCENW.XL21.XL21.011. For the
purpose of open access, a CC BY public copyright license is applied to any
Author Accepted Manuscript version arising from this submission.

##  2\. Toric varieties setting

Let $\Sigma$ be the fan of a complete smooth split toric variety $X$ over a
number field $\mathbb{K}$. We denote by
$\\{{\mathbf{D}}_{1},\dots,{\mathbf{D}}_{s}\\}\subseteq\operatorname{Pic}(X)$
the set of degrees of prime torus invariant divisors of $X$. For each
$i\in\\{1,\dots,s\\}$ we denote by $D_{i,1},\dots,D_{i,n_{i}}$ the torus
invariant divisors of degree ${\mathbf{D}}_{i}$, and by
$\rho_{i,1},\dots,\rho_{i,n_{i}}$ the corresponding rays of $\Sigma$. Let
${\mathcal{I}}:=\\{(i,j)\in\mathbb{N}^{2}:1\leq i\leq s,1\leq j\leq n_{i}\\}$.
Let $\Sigma_{\max}$ be the set of maximal cones of $\Sigma$. For each maximal
cone $\sigma$ of $\Sigma$, let
$\mathcal{J}_{\sigma}:=\\{(i,j)\in{\mathcal{I}}:\rho_{i,j}\subseteq\sigma\\}$,
let $\mathcal{I}_{\sigma}={\mathcal{I}}\smallsetminus\mathcal{J}_{\sigma}$,
and let $I_{\sigma}$ be the set of indices $i\in\\{1,\dots,s\\}$ such that
$\\{(i,1),\dots,(i,n_{i})\\}\cap\mathcal{I}_{\sigma}\neq\emptyset$.

Let $\mathscr{X}$ be the toric scheme defined by $\Sigma$ over
$\mathcal{O}_{\mathbb{K}}$, and for each $(i,j)\in{\mathcal{I}}$, let
$\mathscr{D}_{i,j}$ be the closure of $D_{i,j}$ in $\mathscr{X}$.

Let $R$ be the polynomial ring over $\mathcal{O}_{\mathbb{K}}$ with variables
$x_{i,j}$ for $(i,j)\in{\mathcal{I}}$ and endowed with the
$\operatorname{Pic}(X)$-grading induced by assigning degree ${\mathbf{D}}_{i}$
to the variable $x_{i,j}$ for all $(i,j)\in{\mathcal{I}}$. For every torus
invariant divisor $D=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}a_{i,j}D_{i,j}$ on $X$
and every vector
$\mathbf{x}=(x_{i,j})_{(i,j)\in{\mathcal{I}}}\in\mathbb{C}^{{\mathcal{I}}}$,
we write

$\mathbf{x}^{D}:=\prod_{i=1}^{s}\prod_{j=1}^{n_{i}}x_{i,j}^{a_{i,j}}.$

By [Sal98, §8], $\mathscr{X}$ has a unique universal torsor
$\pi:\mathscr{Y}\to\mathscr{X}$, and
$\mathscr{Y}\subseteq\mathbb{A}^{\sharp{\mathcal{I}}}_{\mathcal{O}_{\mathbb{K}}}$
is the open subset whose complement is defined by $\mathbf{x}^{D_{\sigma}}=0$
for all maximal cones $\sigma$ of $\Sigma$, where
$D_{\sigma}:=\sum_{(i,j)\in\mathcal{I}_{\sigma}}D_{i,j}$ for all
$\sigma\in\Sigma_{\max}$.

Let $r$ be the rank of $\operatorname{Pic}(X)$. Let $\mathcal{C}$ be a set of
ideals of $\mathcal{O}_{\mathbb{K}}$ that form a system of representatives for
the class group of $\mathbb{K}$. As in [PS24, §6.1], we fix a basis of
$\operatorname{Pic}(X)$, and for every divisor $D$ on $X$ we write
$\mathfrak{c}^{[D]}:=\prod_{i=1}^{r}\mathfrak{c}_{i}^{b_{i}}$ where
$[D]=(b_{1},\dots,b_{r})$ with respect to the fixed basis of
$\operatorname{Pic}(X)$. Then, as in [Pie16, §2],

$X(\mathbb{K})=\mathscr{X}(\mathcal{O}_{\mathbb{K}})=\bigsqcup_{\mathfrak{c}\in\mathcal{C}^{r}}\pi^{\mathfrak{c}}\left(\mathscr{Y}^{\mathfrak{c}}(\mathcal{O}_{\mathbb{K}})\right),$

where $\pi^{\mathfrak{c}}:\mathscr{Y}^{\mathfrak{c}}\to\mathscr{X}$ is the
twist of $\pi$ defined in [FP16, Theorem 2.7]. The fibers of
$\pi|_{\mathscr{Y}^{\mathfrak{c}}(\mathcal{O}_{\mathbb{K}})}$ are all
isomorphic to $(\mathcal{O}_{\mathbb{K}}^{\times})^{r}$, and
$\mathscr{Y}^{\mathfrak{c}}(\mathcal{O}_{\mathbb{K}})\subseteq\mathcal{O}_{\mathbb{K}}^{{\mathcal{I}}}$
is the subset of points
$\mathbf{x}\in\bigoplus_{(i,j)\in{\mathcal{I}}}\mathfrak{c}^{[D_{i,j}]}$ that
satisfy

$\sum_{\sigma\in\Sigma_{\max}}\mathbf{x}^{D_{\sigma}}\mathfrak{c}^{-[D_{\sigma}]}=\mathcal{O}_{\mathbb{K}}.$
(2.1)

Let $N$ be the lattice of cocharacters of $X$. Then $\Sigma\subseteq
N\otimes_{\mathbb{Z}}\mathbb{R}$. For every $(i,j)\in{\mathcal{I}}$, let
$\nu_{i,j}$ be the unique generator of $\rho_{i,j}\cap N$. For every torus
invariant $\mathbb{Q}$-divisor
$D=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}a_{i,j}D_{i,j}$ of $X$ and for every
$\sigma\in\Sigma_{\max}$, let
$u_{\sigma,D}\in\operatorname{Hom}_{\mathbb{Z}}(N,\mathbb{Q})$ be the
character determined by $u_{\sigma,D}(\nu_{i,j})=a_{i,j}$ for all
$(i,j)\in\mathcal{J}_{\sigma}$, and define
$D(\sigma):=D-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}u_{\sigma,D}(\nu_{i,j})D_{i,j}$.
Then $D$ and $D(\sigma)$ are linearly equivalent.

###  2.1. Torus invariant divisors

Here we collect some properties of toric varieties and their torus invariant
divisors.

######  Lemma 2.1.

  1. (i)

Let $\sigma\in\Sigma_{\max}$.

    1. (a)

For $i\in I_{\sigma}$, there is a unique index
$j_{i,\sigma}\in\\{1,\dots,n_{i}\\}$ such that
$(i,j_{i,\sigma})\in\mathcal{I}_{\sigma}$. So $\sharp
I_{\sigma}=\sharp\mathcal{I}_{\sigma}=r$.

    2. (b)

For $i\in I_{\sigma}$, $(i,j^{\prime})\in\mathcal{J}_{\sigma}$ for all
$j^{\prime}\in\\{1,\dots,n_{i}\\}\smallsetminus\\{j_{i,\sigma}\\}$.

    3. (c)

For $i\in\\{1,\dots,s\\}\smallsetminus I_{\sigma}$,
$\\{(i,1),\dots,(i,n_{i})\\}\subseteq\mathcal{J}_{\sigma}$.

Let $D$ be a torus invariant $\mathbb{Q}$-divisor on $X$. For
$\sigma\in\Sigma_{\max}$, write

$D(\sigma)=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}D_{i,j}.$

For $i\in\\{1,\dots,s\\}$, let
$\alpha_{i,\sigma}=\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}$.

  1. (ii)

Let $\sigma\in\Sigma_{\max}$. Then $D(\sigma)=\sum_{i\in
I_{\sigma}}\alpha_{i,\sigma}D_{i,j_{i,\sigma}}$.

  2. (iii)

Let $\sigma,\sigma^{\prime}\in\Sigma_{\max}$. If there are $i\in I_{\sigma}$
and $j\in\\{1,\dots,n_{i}\\}$ such that
$\mathcal{J}_{\sigma}\cap\mathcal{J}_{\sigma^{\prime}}=\mathcal{J}_{\sigma}\smallsetminus\\{(i,j)\\}$,
then $I_{\sigma}=I_{\sigma^{\prime}}$ and
$\alpha_{i^{\prime},\sigma}=\alpha_{i^{\prime},\sigma^{\prime}}$ for all
$i^{\prime}\in\\{1,\dots,s\\}$.

  3. (iv)

Let $\sigma\in\Sigma_{\max}$ and for every $i\in I_{\sigma}$, let
$j_{i}\in\\{1,\dots,n_{i}\\}$. Then there exists a unique
$\sigma^{\prime}\in\Sigma_{\max}$ such that $I_{\sigma^{\prime}}=I_{\sigma}$,
$(i,j_{i})\in\mathcal{I}_{\sigma^{\prime}}$ for $i\in I_{\sigma}$, and
$\alpha_{i,\sigma^{\prime}}=\alpha_{i,\sigma}$ for $i\in\\{1,\dots,s\\}$.

  4. (v)

The relation $\sigma\sim\sigma^{\prime}$ if and only if
$I_{\sigma}=I_{\sigma^{\prime}}$ defines an equivalence relation on
$\Sigma_{\max}$, and the equivalence class of $\sigma$ has cardinality
$\prod_{i\in I_{\sigma}}n_{i}$.

  5. (vi)

Let $\mathcal{J}\subseteq\mathcal{I}$ minimal for inclusion and such that
$\mathcal{J}\cap\mathcal{I}_{\sigma}\neq\emptyset$ for all
$\sigma\in\Sigma_{\max}$. Let $i\in\\{1,\dots,s\\}$ such that
$\\{(i,1),\dots,(i,n_{i})\\}\cap\mathcal{J}\neq\emptyset$. Then
$\\{(i,1),\dots,(i,n_{i})\\}\subseteq\mathcal{J}$.

###### Proof.

Part (i) follows from the fact that $[D_{i,j}]={\mathbf{D}}_{i}$ for all
$j\in\\{1,\dots,n_{i}\\}$ and that the set $\\{{\mathbf{D}}_{i}:i\in
I_{\sigma}\\}$ is a basis of $\operatorname{Pic}(X)$ by [CLS11, Theorem 4.2.8]
as $X$ is smooth and proper.

Part (ii) follows from part (i) and the fact that by construction
$\alpha_{i,j,\sigma}=0$ whenever $(i,j)\in\mathcal{J}_{\sigma}$.

For part (iii) we observe that if $\sigma\neq\sigma^{\prime}$, then
$\mathcal{J}_{\sigma}=(\mathcal{J}_{\sigma}\cap\mathcal{J}_{\sigma^{\prime}})\sqcup\\{(i,j)\\}$
and
$\mathcal{J}_{\sigma^{\prime}}=(\mathcal{J}_{\sigma}\cap\mathcal{J}_{\sigma^{\prime}})\sqcup\\{(i,j_{i,\sigma})\\}$,
where $j_{i,\sigma}$ is the index defined in part (i). Thus $i\in
I_{\sigma}\cap I_{\sigma^{\prime}}$, and for every index
$i^{\prime}\in\\{1,\dots,s\\}$ with $i^{\prime}\neq i$ we have
$\mathcal{J}_{\sigma}\cap\\{(i^{\prime},1),\dots,(i^{\prime},n_{i^{\prime}})\\}=\mathcal{J}_{\sigma^{\prime}}\cap\\{(i^{\prime},1),\dots,(i^{\prime},n_{i^{\prime}})\\}\subseteq\mathcal{J}_{\sigma}\cap\mathcal{J}_{\sigma^{\prime}}$.
Recall that $[D(\sigma)]=\sum_{i\in
I_{\sigma}}\alpha_{i,\sigma}{\mathbf{D}}_{i}$ and
$[D(\sigma^{\prime})]=\sum_{i\in
I_{\sigma^{\prime}}}\alpha_{i,\sigma^{\prime}}{\mathbf{D}}_{i}$. Now the
result follows as $[D(\sigma)]=[D(\sigma^{\prime})]$ in
$\operatorname{Pic}(X)$ and $\\{{\mathbf{D}}_{i}:i\in I_{\sigma}\\}$ is a
basis of $\operatorname{Pic}(X)$.

For part (iv), write $I_{\sigma}=\\{i_{1},\dots,i_{r}\\}$. We construct by
induction $\sigma_{1},\dots,\sigma_{r}$ such that for each
$l\in\\{1,\dots,r\\}$,
$(i_{1},j_{i_{1}}),\dots,(i_{l},j_{i_{l}})\in\mathcal{I}_{\sigma_{l}}$,
$I_{\sigma_{l}}=I_{\sigma}$ and $\alpha_{i,\sigma_{l}}=\alpha_{i,\sigma}$ for
all $i\in\\{1,\dots,s\\}$. If $(i_{1},j_{i_{1}})\in\mathcal{I}_{\sigma}$, let
$\sigma_{1}=\sigma$. Otherwise, $(i_{1},j_{i_{1}})\in\mathcal{J}_{\sigma}$ and
by [Sal98, Lemma 8.9] there is $\sigma_{1}\in\Sigma_{\max}$ such that
$\mathcal{J}_{\sigma_{1}}\cap\mathcal{J}_{\sigma}=\mathcal{J}_{\sigma}\smallsetminus\\{(i_{1},j_{i_{1}})\\}$.
Since $i_{1}\in I_{\sigma}$, by part (iii) we have $I_{\sigma_{1}}=I_{\sigma}$
and $\alpha_{i,\sigma_{1}}=\alpha_{i,\sigma}$ for all $i\in\\{1,\dots,s\\}$.
Assume that we have constructed $\sigma_{l-1}$ for given $l\leq r$. If
$(i_{l},j_{i_{l}})\in\mathcal{I}_{\sigma_{l-1}}$, let
$\sigma_{l}=\sigma_{l-1}$. Otherwise,
$(i_{l},j_{i_{l}})\in\mathcal{J}_{\sigma_{l-1}}$ and by [Sal98, Lemma 8.9]
there is $\sigma_{l}\in\Sigma_{\max}$ such that
$\mathcal{J}_{\sigma_{l}}\cap\mathcal{J}_{\sigma_{l-1}}=\mathcal{J}_{\sigma_{l-1}}\smallsetminus\\{(i_{l},j_{i_{l}})\\}$.
Since $i_{l}\in I_{\sigma_{l-1}}$, by part (iii) we have
$I_{\sigma_{l}}=I_{\sigma_{l-1}}=I_{\sigma}$ and
$\alpha_{i,\sigma_{l}}=\alpha_{i,\sigma_{l-1}}=\alpha_{i,\sigma}$ for all
$i\in\\{1,\dots,s\\}$. Since
$(i_{1},j_{i_{1}}),\dots,(i_{l-1},j_{i_{l-1})}\in\mathcal{I}_{\sigma_{l-1}}$
and
$\mathcal{J}_{\sigma_{l}}=(\mathcal{J}_{\sigma_{l-1}}\cap\mathcal{J}_{\sigma_{l}})\cup\\{(i_{l},j_{i_{l},\sigma_{l-1}})\\}$,
where $j_{i_{l},\sigma_{l-1}}$ is the index defined in part (i), we conclude
that $(i_{1},j_{i_{1}}),\dots,(i_{l},j_{i_{l}})\in\mathcal{I}_{\sigma_{l}}$.
Take $\sigma^{\prime}=\sigma_{r}$. The uniqueness of $\sigma^{\prime}$ follows
from part (i), as $\sigma^{\prime}$ is completely determined by
$\mathcal{I}_{\sigma^{\prime}}$.

Part (v) is a direct consequence of part (iv).

For part (vi), let $j\in\\{1,\dots,n_{i}\\}$ such that $(i,j)\in\mathcal{J}$.
By minimality of $\mathcal{J}$, there exists $\sigma\in\Sigma_{\max}$ such
that $\mathcal{J}\cap\mathcal{I}_{\sigma}=\\{(i,j)\\}$. If $n_{i}>1$, let
$j^{\prime}\in\\{1,\dots,n_{i}\\}\smallsetminus\\{j\\}$. By [Sal98, Lemma 8.9]
there is $\sigma^{\prime}\in\Sigma_{\max}$ such that
$\mathcal{J}_{\sigma^{\prime}}\cap\mathcal{J}_{\sigma}=\mathcal{J}_{\sigma}\smallsetminus\\{(i,j^{\prime})\\}$.
Hence,
$\mathcal{I}_{\sigma^{\prime}}=(\mathcal{I}_{\sigma}\smallsetminus\\{(i,j)\\})\cup\\{(i,j^{\prime})\\}$.
Since
$\mathcal{J}\cap(\mathcal{I}_{\sigma}\smallsetminus\\{(i,j)\\})=\emptyset$ and
$\mathcal{J}\cap\mathcal{I}_{\sigma^{\prime}}\neq\emptyset$, we conclude that
$(i,j^{\prime})\in\mathcal{J}$. ∎

###  2.2. Heights

Let $L$ be a semiample torus invariant $\mathbb{Q}$-divisor on $X$. Let
$H_{L}$ be the height on $X$ defined by $L$ as in [PS24, §6.3]. For
$\sigma\in\Sigma_{\max}$, write
$L(\sigma)=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}D_{i,j}$ and
$\alpha_{i,\sigma}=\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}$ for all
$i\in\\{1,\dots,s\\}$. Let $\Omega_{\mathbb{K}}$ be the set of places of
$\mathbb{K}$.

######  Lemma 2.2.

For every $\nu\in\Omega_{\mathbb{K}}$ and every
$\mathbf{x}\in\mathscr{Y}(\mathbb{K})$, we have

$\sup_{\sigma\in\Sigma_{\max}}|x^{L(\sigma)}|_{\nu}=\sup_{\sigma\in\Sigma_{\max}}\prod_{i=1}^{s}\sup_{1\leq
j\leq n_{i}}|x_{i,j}|_{\nu}^{\alpha_{i,\sigma}}.$

###### Proof.

Fix $\nu\in\Omega_{\mathbb{K}}$ and $\mathbf{x}\in\mathscr{Y}(\mathbb{K})$. By
Lemma 2.1(ii), we have

$\displaystyle\sup_{\sigma\in\Sigma_{\max}}|x^{L(\sigma)}|_{\nu}=\sup_{\sigma\in\Sigma_{\max}}\prod_{i\in
I_{\sigma}}|x_{i,{j_{i,\sigma}}}|_{\nu}^{\alpha_{i,\sigma}}.$

For every $i\in\\{1,\dots,s\\}$, let $j_{i}\in\\{1,\dots,n_{i}\\}$ such that
$|x_{i,j_{i}}|_{\nu}=\sup_{1\leq j\leq n_{i}}|x_{i,j}|_{\nu}$. Let
$\sigma\in\Sigma_{\max}$. By Lemma 2.1(iv) there is
$\sigma^{\prime}\in\Sigma_{\max}$ such that $I_{\sigma^{\prime}}=I_{\sigma}$,
$(i,j_{i})\in\mathcal{I}_{\sigma^{\prime}}$ for all $i\in I_{\sigma}$, and
$\alpha_{i,\sigma^{\prime}}=\alpha_{i,\sigma}$ for all $i\in\\{1,\dots,s\\}$.
Then

$|x^{L(\sigma^{\prime})}|_{\nu}=\prod_{i\in
I_{\sigma^{\prime}}}|x_{i,j_{i}}|_{\nu}^{\alpha_{i,\sigma^{\prime}}}=\prod_{i\in
I_{\sigma}}\sup_{1\leq j\leq
n_{i}}|x_{i,j}|_{\nu}^{\alpha_{i,\sigma}}=\prod_{i=1}^{s}\sup_{1\leq j\leq
n_{i}}|x_{i,j}|_{\nu}^{\alpha_{i,\sigma}}.\qed$

Thus
$H_{L}(\mathbf{x})=\prod_{\nu\in\Omega_{\mathbb{K}}}\sup_{\sigma\in\Sigma_{\max}}\prod_{i=1}^{s}\sup_{1\leq
j\leq n_{i}}|x_{i,j}|_{\nu}^{\alpha_{i,\sigma}}$ for all
$\mathbf{x}\in\mathscr{Y}(\mathbb{K})$.

###  2.3. Coprimality conditions

We now rewrite the coprimality condition (2.1) in terms of the notation
introduced in this paper.

######  Lemma 2.3.

For all
$\mathbf{x}\in\bigoplus_{(i,j)\in{\mathcal{I}}}\mathfrak{c}^{[D_{i,j}]}$,

$\sum_{\sigma\in\Sigma_{\max}}\mathbf{x}^{D_{\sigma}}\mathfrak{c}^{-[D_{\sigma}]}=\sum_{\sigma\in\Sigma_{\max}}\prod_{i\in
I_{\sigma}}(x_{i,1},\dots,x_{i,n_{i}})\mathfrak{c}^{-{\mathbf{D}}_{i}}.$

###### Proof.

For $\sigma\in\Sigma_{\max}$, let $X_{\sigma}=\\{\prod_{i\in
I_{\sigma}}x_{i,j_{i}}:j_{i}\in\\{1,\dots,n_{i}\\}\ \forall
i\in\\{1,\dots,s\\}\\}$. The inclusion $\subseteq$ is clear as
$\mathbf{x}^{D_{\sigma}}\in X_{\sigma}$ and
$\mathfrak{c}^{-[D_{\sigma}]}=\prod_{i\in
I_{\sigma}}\mathfrak{c}^{-{\mathbf{D}}_{i}}$ for all $\sigma\in\Sigma_{\max}$.
For the converse inclusion, fix $\sigma\in\Sigma_{\max}$ and $x\in
X_{\sigma}$. For every $i\in I_{\sigma}$, let $j_{i}\in\\{1,\dots,n_{i}\\}$
such that $x=\prod_{i\in I_{\sigma}}x_{i,j_{i}}$. By Lemma 2.1(iv) there is
$\sigma^{\prime}\in\Sigma_{\max}$ such that $I_{\sigma^{\prime}}=I_{\sigma}$
and $(i,j_{i})\in\mathcal{I}_{\sigma^{\prime}}$ for $i\in I_{\sigma}$ . Then
$\mathbf{x}^{D_{\sigma^{\prime}}}=x$. ∎

###  2.4. Möbius function

Let ${\mathcal{I}_{\mathbb{K}}}$ be the set of nonzero ideals of
$\mathcal{O}_{\mathbb{K}}$. Let
$\chi:{\mathcal{I}_{\mathbb{K}}}^{s}\to\\{0,1\\}$ be the characteristic
function of the subset

$\left\\{\mathfrak{b}\in{\mathcal{I}_{\mathbb{K}}}^{s}:\sum_{\sigma\in\Sigma_{\max}}\prod_{i\in
I_{\sigma}}\mathfrak{b}_{i}=\mathcal{O}_{\mathbb{K}}\right\\}.$ (2.2)

For every $\mathfrak{d}\in{\mathcal{I}_{\mathbb{K}}}^{s}$, let
$\chi_{\mathfrak{d}}:{\mathcal{I}_{\mathbb{K}}}^{s}\to\\{0,1\\}$ be the
characteristic function of the subset

$\left\\{\mathfrak{b}\in{\mathcal{I}_{\mathbb{K}}}^{s}:\mathfrak{b}_{i}\subseteq\mathfrak{d}_{i}\
\forall i\in\\{1,\dots,s\\}\right\\}.$

As in [Pey95, Lemme 8.5.1] there exists a unique multiplicative function
$\mu:{\mathcal{I}_{\mathbb{K}}}^{s}\to\mathbb{Z}$ such that

$\chi=\sum_{\mathfrak{d}\in{\mathcal{I}_{\mathbb{K}}}^{s}}\mu(\mathfrak{d})\chi_{\mathfrak{d}}.$

Note that if $X=\mathbb{P}^{n}_{\mathbb{Q}}$, the function $\mu$ defined above
coincides with the classical Möbius function.

######  Remark 2.4.

Let $\mathfrak{p}\in\mathcal{I}_{\mathbb{K}}$ be a prime ideal. The function
$\mu$ is defined recursively by the formula
$\mu(\mathfrak{b})=\chi(\mathfrak{b})-\sum_{\mathfrak{b}\subsetneq\mathfrak{d}}\mu(\mathfrak{d})$
for every $\mathfrak{b}\in{\mathcal{I}_{\mathbb{K}}}^{s}$, and satisfies the
following properties.

  1. (i)

$\mu(\mathbf{1})=\chi(\mathbf{1})=1$.

  2. (ii)

If $e_{i}\geq 2$ for some $i\in\\{1,\dots,s\\}$, then
$\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=0$, as in that case
$\chi(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=\chi(\mathfrak{p}^{e^{\prime}_{1}},\dots,\mathfrak{p}^{e^{\prime}_{s}})$
for $e^{\prime}_{i}=e_{i}-1$ and $e^{\prime}_{l}=e_{l}$ for all $l\neq i$.

  3. (iii)

By induction one shows that
$\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=0$ whenever
$(e_{1},\dots,e_{s})\neq\mathbf{0}$ and there is $\sigma\in\Sigma_{\max}$ such
that $e_{i}=0$ for all $i\in I_{\sigma}$, as
$\chi(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=1$ if and only if there
is $\sigma\in\Sigma_{\max}$ such that $e_{i}=0$ for all $i\in I_{\sigma}$.

  4. (iv)

Let

$\tilde{f}:=\min\left\\{\sharp J:J\subseteq\\{1,\dots,s\\},J\cap
I_{\sigma}\neq\emptyset\ \forall\sigma\in\Sigma_{\max}\right\\}.$

By property (iii), if
$\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\neq 0$, then there are
at least $\tilde{f}$ indices $i$ with $e_{i}=1$. Let
$J\subseteq\\{1,\dots,s\\}$ be smallest with respect to inclusion and such
that $J\cap I_{\sigma}\neq\emptyset$ for all $\sigma\in\Sigma_{\max}$. Let
$J^{\prime}=J\smallsetminus\\{j\\}$ for some $j\in J$. Let $e_{i}=1$ for $i\in
J$ and $e_{i}=0$ for $i\notin J$. Let $e_{i}^{\prime}=e_{i}$ for $i\neq j$ and
$e_{j}^{\prime}=0$. Then
$\chi(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=0$ and
$\chi(\mathfrak{p}^{e^{\prime}_{1}},\dots,\mathfrak{p}^{e^{\prime}_{s}})=1$ by
minimality of $J$. Thus
$\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})=-1\neq 0$. Hence,

$\tilde{f}=\min\left\\{\sum_{i=1}^{s}e_{i}:(e_{1},\dots,e_{s})\neq\mathbf{0},\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\neq
0\right\\}.$ (2.3)

For $\beta=(\beta_{1},\dots,\beta_{s})\in\mathbb{R}_{\geq 0}^{s}$, let

$f_{\beta}:=\min\left\\{\sum_{i=1}^{s}\beta_{i}e_{i}:(e_{1},\dots,e_{s})\neq\mathbf{0},\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\neq
0\right\\}.$

######  Lemma 2.5.

  1. (i)

The series

$\sum_{\mathfrak{d}\in\mathcal{I}_{\mathbb{K}}^{s}}\frac{\mu(\mathfrak{d})}{\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{d}_{i})^{\beta_{i}}}$

converges absolutely if $f_{\beta}>1$.

  2. (ii)

If $f_{\beta}>1$ and $\beta_{1},\dots,\beta_{s}\in\mathbb{Z}_{>0}$, then
$\sum_{\mathfrak{d}\in\mathcal{I}_{\mathbb{K}}^{s}}\frac{\mu(\mathfrak{d})}{\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{d}_{i})^{\beta_{i}}}>0$.

###### Proof.

For part (i) we follow the proof of [Sal98, Lemma 11.15] and [Pie16,
Proposition 4]. For $\mathfrak{p}\in\mathcal{I}_{\mathbb{K}}$ prime ideal, let
$S(\mathfrak{p})=\sum_{(e_{1},\dots,e_{s})\in\mathbb{Z}_{\geq
0}^{s}}\frac{|\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})|}{\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{p})^{\beta_{i}e_{i}}}$.
As in [Sal98, Lemma 11.15] and [Pie16, Proposition 4],

$\lim_{b\to\infty}\sum_{\begin{subarray}{c}\mathfrak{d}\in\mathcal{I}_{\mathbb{K}}^{s}\\\
\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{d}_{i})\leq
b\end{subarray}}\frac{|\mu(\mathfrak{d})|}{\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{d}_{i})^{\beta_{i}}}=\prod_{\mathfrak{p}}S(\mathfrak{p}).$

By Remark 2.4(ii) the sum $S(\mathfrak{p})$ is finite. By definition of
$f_{\beta}$, if $\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\neq 0$
and $(e_{1},\dots,e_{s})\neq\mathbf{0}$, then
${f_{\beta}}\leq\sum_{i=1}^{s}\beta_{i}e_{i}$. Thus

$\displaystyle
S(\mathfrak{p})=1+\frac{1}{\mathfrak{N}(\mathfrak{p})^{f_{\beta}}}Q\left(\frac{1}{\mathfrak{N}(\mathfrak{p})}\right),$

where $Q:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is a monotone increasing
function. Since $\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})$ is
independent of the choice of $\mathfrak{p}$, the function $Q$ is independent
of the choice of $\mathfrak{p}$. Thus

$\displaystyle\sum_{\mathfrak{p}}\frac{1}{\mathfrak{N}(\mathfrak{p})^{f_{\beta}}}Q\left(\frac{1}{\mathfrak{N}(\mathfrak{p})}\right)\leq[\mathbb{K}:\mathbb{Q}]Q(1)\sum_{n\in\mathbb{Z}_{>0}}\frac{1}{n^{f_{\beta}}}.$

In part (ii) the series is absolutely convergent by part (i), hence it
suffices to show that each factor of its Euler product
$\prod_{\mathfrak{p}}S_{\mathfrak{p}}$ is positive. For a prime ideal
$\mathfrak{p}\in\mathcal{I}_{\mathbb{K}}$, let $\mathcal{O}_{\mathfrak{p}}$ be
the ring of integers of the completion $\mathbb{K}_{\mathfrak{p}}$ of
$\mathbb{K}$ at the valuation $v_{\mathfrak{p}}$ defined by $\mathfrak{p}$.
Endow $\mathbb{K}_{\mathfrak{p}}$ with the Haar measure normalized such that
$\mathcal{O}_{\mathfrak{p}}$ has volume $1$. Then
$\int_{\mathfrak{p}^{j}\mathcal{O}_{\mathfrak{p}}}dy=\mathfrak{N}(\mathfrak{p})^{-j}$
for all $j\geq 0$ by [CLNS18, §1.1.13] and [Neu99, Proposition II.4.3]. We
denote by $\chi$ the characteristic function of (2.2) where ideals of
$\mathcal{O}_{\mathbb{K}}$ are replaced by ideals of
$\mathcal{O}_{\mathfrak{p}}$. By Remark 2.4(ii),

$\displaystyle
S_{\mathfrak{p}}=\sum_{\boldsymbol{e}\in\\{0,1\\}^{s}}\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\prod_{i=1}^{s}\mathfrak{N}(\mathfrak{p})^{-e_{i}\beta_{i}}$
$\displaystyle=\sum_{\boldsymbol{e}\in\\{0,1\\}^{s}}\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\prod_{i=1}^{s}\prod_{j=1}^{\beta_{i}}\int_{\mathfrak{p}^{e_{i}}}\mathrm{d}y_{i,j}$
$\displaystyle=\int_{\mathcal{O}_{\mathfrak{p}}^{\sum_{i=1}^{s}\beta_{i}}}\chi((y_{1,1},\dots,y_{1,\beta_{1}}),\dots,(y_{s,1},\dots,y_{s,\beta_{s}}))\prod_{i=1}^{s}\prod_{j=1}^{\beta_{i}}\mathrm{d}y_{i,j}$
$\displaystyle\geq\int_{(\mathcal{O}_{\mathfrak{p}}^{\times})^{\sum_{i=1}^{s}\beta_{i}}}\prod_{i=1}^{s}\prod_{j=1}^{\beta_{i}}\mathrm{d}y_{i,j}=\left(1-\frac{1}{\mathfrak{N}(\mathfrak{p})}\right)^{\sum_{i=1}^{s}\beta_{i}}>0,$

as $\chi$ is a nonnegative function with
$\chi(\mathcal{O}_{\mathfrak{p}},\dots,\mathcal{O}_{\mathfrak{p}})=1$. ∎

######  Definition 2.6.

A function $A:\mathbb{Z}^{s}_{>0}\to\mathbb{R}$ is compatible with Möbius
inversion on $X$ if there are $\beta_{1},\dots,\beta_{s}\in\mathbb{R}^{s}$
such that $A(\mathbf{d})\ll\prod_{i=1}^{s}d_{i}^{-\beta_{i}}$ with
$f_{(\beta_{1},\dots,\beta_{s})}>1$.

######  Remark 2.7.

  1. (i)

The inequality $f_{\beta}>1$ holds whenever $\beta_{1},\dots,\beta_{s}>1$.

  2. (ii)

If $\beta_{1}=\dots=\beta_{s}=1$, then $f_{\beta}=\tilde{f}$ by (2.3).

  3. (iii)

(Case $\beta_{1}=n_{1},\dots,\beta_{s}=n_{s}$) As in [Sal98, Lemma 11.15(d)],
let $f$ be the smallest positive integer such that there are $f$ rays of the
fan $\Sigma$ that are not contained in a maximal cone. Then $f\geq 2$, as $X$
is proper. Moreover,

$f=\min\left\\{\sharp\mathcal{J}:\mathcal{J}\subseteq\mathcal{I},\mathcal{J}\cap\mathcal{I}_{\sigma}\neq\emptyset\
\forall\sigma\in\Sigma_{\max}\right\\},$

and Remark 2.4 combined with Lemma 2.1(vi) gives

$\displaystyle f$ $\displaystyle=\min\left\\{\sum_{i\in
J}n_{i}:J\subseteq\\{1,\dots,s\\},J\cap I_{\sigma}\neq\emptyset\
\forall\sigma\in\Sigma_{\max},\sharp J=\tilde{f}\right\\}$
$\displaystyle=\min\left\\{\sum_{i=1}^{s}n_{i}e_{i}:(e_{1},\dots,e_{s})\neq\mathbf{0},\mu(\mathfrak{p}^{e_{1}},\dots,\mathfrak{p}^{e_{s}})\neq
0\right\\}.$

##  3\. Subvarieties

Here we want to count rational points or Campana points of bounded height in
subvarieties of toric varieties.

From now on $\mathbb{K}=\mathbb{Q}$. Let $X$ be a complete smooth split toric
variety as in Section 2. Assume that
$\operatorname{rk}\operatorname{Pic}(X)\geq 2$, that is $X$ is not a
projective space. Let $L$ be a semiample toric invariant $\mathbb{Q}$-divisor
on $X$ that satisfies [PS24, Assumption 6.3]. The latter holds, for example,
if $L$ is ample.

Let $g_{1},\dots,g_{t}\in R$ be $\operatorname{Pic}(X)$-homogeneous elements.
Let $V\subseteq X$ be the schematic intersection of the $t$ hypersurfaces
defined by $g_{1},\dots,g_{t}$. Let $T\subseteq X$ be the torus. Without loss
of generality, we can assume that $V\cap T\neq\emptyset$. Otherwise, $V$ is
contained in a complete smooth split toric subvariety $X^{\prime}$ of $X$, and
we can replace $X$ by $X^{\prime}$. Fix $m_{i,j}\in\mathbb{Z}_{\geq 1}$ for
each $(i,j)\in{\mathcal{I}}$. Let
$\mathbf{m}=(m_{i,j})_{(i,j)\in{\mathcal{I}}}$, and
$\mathscr{D}_{\mathbf{m}}=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\left(1-\frac{1}{m_{i,j}}\right)\mathscr{D}_{i,j}$.
Let $\mathscr{V}$ be the Zariski closure of $V$ in $\mathscr{X}$. Let
$\left(\mathscr{V},\mathscr{D}_{\mathbf{m}}|_{\mathscr{V}}\right)(\mathbb{Z})$
be the set of Campana $\mathbb{Z}$-points on the Campana orbifold
$\left(\mathscr{V},\mathscr{D}_{\mathbf{m}}|_{\mathscr{V}}\right)$ as in
[PSTVA21, Definition 3.4], where if $V$ is singular, the intersection
multiplicity of a point
$\mathbf{x}:\operatorname{Spec}\mathcal{O}_{\mathbb{K}}\to\mathscr{V}$ with
$\mathscr{D}_{i}|_{\mathscr{V}}$ at a place $v$ of $\mathbb{K}$ is defined as
the colength of the ideal of the fiber product of
$\operatorname{Spec}\mathcal{O}_{\mathbb{K}}\times_{\mathscr{V}}\mathscr{D}_{i}|_{\mathscr{V}}$
after base change to the completion of $\mathcal{O}_{\mathbb{K}}$ at $v$.

Let $N_{V}(B)$ be the number of points in
$\left(\mathscr{V},\mathscr{D}_{\mathbf{m}}|_{\mathscr{V}}\right)(\mathbb{Z})\cap
T(\mathbb{Q})$ of height $H_{L}$ at most $B$. If $m_{i,j}=1$ for all
$(i,j)\in{\mathcal{I}}$, then $N_{V}(B)$ is the set of $\mathbb{Q}$-rational
points on $V\cap T$ of height $H_{L}$ at most $B$.

For $i\in\\{1,\dots,s\\}$ and $\mathbf{x}\in\mathscr{Y}(\mathbb{Z})$, let
$y_{i}=\sup_{1\leq j\leq n_{i}}|x_{i,j}|$. For $\sigma\in\Sigma_{\max}$, write
$L(\sigma)=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}D_{i,j}$ and
$\alpha_{i,\sigma}=\sum_{j=1}^{n_{i}}\alpha_{i,j,\sigma}$ for all
$i\in\\{1,\dots,s\\}$. Then by [PS24, Proposition 6.10] and Lemma 2.2,

$H_{L}(\mathbf{x})=\sup_{\sigma\in\Sigma_{\max}}\prod_{i=1}^{s}y_{i}^{\alpha_{i,\sigma}}.$

By construction,
$\left(\mathscr{V},\mathscr{D}_{\mathbf{m}}|_{\mathscr{V}}\right)(\mathbb{Z})=\left(\mathscr{X},\mathscr{D}_{\mathbf{m}}\right)(\mathbb{Z})\cap
V(\mathbb{Q})$. We use the torsor parameterization of
$\left(\mathscr{X},\mathscr{D}_{\mathbf{m}}\right)(\mathbb{Z})$ from [PS24,
§6.4]. For $B>0$ and $\mathbf{d}\in(\mathbb{Z}_{>0})^{s}$, let
$A(B,\mathbf{d})$ be the set of points $\mathbf{x}=(x_{i,j})_{1\leq i\leq
s,1\leq j\leq n_{i}}\in(\mathbb{Z}_{\neq 0})^{{\mathcal{I}}}$ such that

$\displaystyle H(\mathbf{x})\leq B,$ (3.1) $\displaystyle d_{i}\mid x_{i,j}\
\forall i\in\\{1,\dots,s\\},\forall j\in\\{1,\dots,n_{i}\\},$ (3.2)
$\displaystyle x_{i,j}\text{ is }m_{i,j}\text{-full}\ \forall
i\in\\{1,\dots,s\\},\forall j\in\\{1,\dots,n_{i}\\},$ (3.3) $\displaystyle
g_{1}=\dots=g_{t}=0.$ (3.4)

We observe that $A(B,\mathbf{d})$ is a finite set by [PS24, Lemma 6.11]. Then

$N_{V}(B)=\frac{1}{2^{r}}\sum_{\mathbf{d}\in(\mathbb{Z}_{>0})^{s}}\mu(d)\sharp
A(B,\mathbf{d})$ (3.5)

by Lemma 2.3 and the definition of $\mu$ in Section 2.4.

Write

$\sharp
A(B,\mathbf{d})=\sum_{\begin{subarray}{c}y_{1},\dots,y_{s}\in\mathbb{Z}_{>0}\\\
\prod_{i=1}^{s}y_{i}^{\alpha_{i,\sigma}}\leq B,\
\forall\sigma\in\Sigma_{\max}\end{subarray}}f_{\mathbf{d}}(y_{1},\dots,y_{s}),$

where

$\displaystyle
f_{\mathbf{d}}(y_{1},\dots,y_{s})=\sharp\\{\mathbf{x}\in(\mathbb{Z}_{\neq
0})^{{\mathcal{I}}}:\eqref{cond:divisibility},\eqref{cond:m-full},\eqref{cond:equations},y_{i}=\sup_{1\leq
j\leq n_{i}}|x_{i,j}|\ \forall i\in\\{1,\dots,s\\}\\}.$

Let

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=\sum_{1\leq y_{i}\leq B_{i},1\leq i\leq
s}f_{\mathbf{d}}(y_{1},\dots,y_{s}).$

######  Lemma 3.1.

Assume that

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=C_{M,\mathbf{d}}\prod_{i=1}^{s}B_{i}^{\varpi_{i}}+O\left(C_{E,\mathbf{d}}\left(\min_{1\leq
i\leq s}B_{i}\right)^{-\epsilon}\prod_{i=1}^{s}B_{i}^{\varpi_{i}}\right)$
(3.6)

with
$C_{M,\mathbf{d}},C_{E,\mathbf{d}},\varpi_{1},\dots,\varpi_{s},\epsilon>0$
such that $C_{M,\mathbf{d}}$, $C_{E,\mathbf{d}}$ are compatible with Möbius
inversion on $X$ as functions of the variables $\mathbf{d}$.

Let $a$ be the maximal value of $\sum_{i=1}^{s}\varpi_{i}u_{i}$ on the
polytope $\mathcal{P}\subseteq\mathbb{R}^{s}$ defined by

$\sum_{i=1}^{s}\alpha_{i,\sigma}u_{i}\leq 1\
\forall\sigma\in\Sigma_{\max},\qquad u_{i}\geq 0\ \forall
i\in\\{1,\dots,s\\}.$

Let $F$ be the face of $\mathcal{P}$ where $\sum_{i=1}^{s}\varpi_{i}u_{i}=a$.
Let $k$ be the dimension of $F$.

  1. (i)

If $F$ is not contained in a coordinate hyperplane of $\mathbb{R}^{s}$, then

$N_{V}(B)=cB^{a}(\log B)^{k}+O(B^{a}(\log B)^{k-1}(\log\log B)^{s}),$

where $k$ is the dimension of $F$, and

$c=(s-1-k)!c_{\mathcal{P}}2^{-r}\sum_{\mathbf{d}\in\mathbb{Z}_{>0}^{s}}\mu(\mathbf{d})C_{M,\mathbf{d}}.$
(3.7)

Here, $c_{\mathcal{P}}=\lim_{\delta\to
0}\delta^{k+1-s}\operatorname{meas}_{s-1}(H_{\delta}\cap\mathcal{P})$, where
$H_{\delta}\subseteq\mathbb{R}^{s}$ is the hyperplane defined by
$\sum_{i=1}^{s}\varpi_{i}u_{i}=a-\delta$, and $\operatorname{meas}_{s-1}$ is
the $(s-1)$-dimensional measure on $H_{\delta}$ given by $\prod_{1\leq i\leq
s,i\neq\tilde{i}}(\varpi_{i}\mathrm{d}u_{i})$ for any choice of
$\tilde{i}\in\\{1,\dots,s\\}$.

  2. (ii)

If $L$ is ample, then

$a=\inf\left\\{t\in\mathbb{R}:t[L]-\left[\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}\right]\text{
is effective}\right\\},$

and $k+1$ is the codimension of the minimal face of the effective cone of $X$
containing $a[L]-[\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}]$.

  3. (iii)

If $[L]=\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}$ is ample, then the face $F$
is not contained in a coordinate hyperplane, $a=1$ and
$k=\operatorname{rk}\operatorname{Pic}(X)-1$.

###### Proof.

  1. (i)

Let $t_{i}=\varpi_{i}u_{i}$ for all $i\in\\{1,\dots,s\\}$. By the assumptions
on $L$, the polytope $\mathcal{P}$ is bounded and nondegenerate by [PS24,
Remark 6.2]. Applying [PS24, Theorem 1.1] to $\sharp A(B,\mathbf{d})$ gives

$N_{V}(B)=cB^{a}(\log B)^{k}+O\left(B^{a}(\log B)^{k-1}(\log\log
B)^{s}\sum_{\mathbf{d}\in(\mathbb{Z}_{>0})^{s}}\mu(d)C_{E,\mathbf{d}}\right).$

The sums $\sum_{\mathbf{d}\in(\mathbb{Z}_{>0})^{s}}$ in the leading constant
$c$ and in the error term converge absolutely by Lemma 2.5 as
$C_{M,\mathbf{d}}$, $C_{E,\mathbf{d}}$ are compatible with Möbius inversion on
$X$.

  2. (ii)

Let

$\mathbb{R}^{r}\hookrightarrow\mathbb{R}^{s}\hookrightarrow\mathbb{R}^{{\mathcal{I}}}$

be the sequence of injective linear maps dual to

$d:\bigoplus_{(i,j)\in{\mathcal{I}}}D_{i,j}\mathbb{Z}\twoheadrightarrow\bigoplus_{i=1}^{s}{\mathbf{D}}_{i}\mathbb{Z}\twoheadrightarrow\operatorname{Pic}(X).$

Here,

$\mathbb{R}^{s}\hookrightarrow\mathbb{R}^{{\mathcal{I}}},\qquad\sum_{i=1}^{s}u_{i}e_{i}\mapsto\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}u_{i}e_{i,j},$

where $\\{e_{1},\dots,e_{s}\\}$ denotes the dual basis to
$\\{{\mathbf{D}}_{1},\dots,{\mathbf{D}}_{s}\\}$, and
$\\{e_{i,j}:(i,j)\in{\mathcal{I}}\\}$ denotes the dual basis to
$\\{D_{i,j}:(i,j)\in{\mathcal{I}}\\}$. Let $\widetilde{P}$ be the polytope
defined by

$\sum_{(i,j)\in{\mathcal{I}}}\alpha_{i,j,\sigma}u_{i,j}\leq
1\quad\forall\sigma\in\Sigma_{\max},\qquad u_{i,j}\geq
0\quad\forall(i,j)\in{\mathcal{I}}.$

Then $\widetilde{P}\cap\mathbb{R}^{s}=P$ and
$\left.\sum_{(i,j)\in{\mathcal{I}}}\frac{\varpi_{i}}{n_{i}}u_{i,j}\right|_{P}=\sum_{i=1}^{s}\left(\sum_{j=1}^{n_{i}}\frac{\varpi_{i}}{n_{i}}\right)u_{i}$.
By [PS24, Lemma 6.7], the face $F$ of $\widetilde{P}$ where the maximal value
of the function

$\sum_{(i,j)\in{\mathcal{I}}}\frac{\varpi_{i}}{n_{i}}u_{i,j}$ (3.8)

is attained is contained in $\widetilde{P}\cap\mathbb{R}^{r}$, and hence also
in $P$. Then $a$ is the maximal value of the function (3.8) on $P$. The dual
linear programming problem is given by minimizing
$\sum_{\sigma\in\Sigma_{\max}}\lambda_{\sigma}$ on the polytope given by

$\sum_{\sigma\in\Sigma_{\max}}\alpha_{i,j,\sigma}\lambda_{\sigma}\geq\frac{\varpi_{i}}{n_{i}}\quad\forall(i,j)\in{\mathcal{I}},\qquad\lambda_{\sigma}\geq
0,\ \forall\sigma\in\Sigma_{\max}.$

The arguments in [PS24, §6.5.1] show that $a$ is the smallest real number such
that
$a[L]-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\frac{\varpi_{i}}{n_{i}}{\mathbf{D}}_{i}$
is effective. As in [PS24, Proposition 6.13], the smallest face of
$\operatorname{Eff}(X)$ that contains
$a[L]-\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}$ is dual to the cone generated
by $F$ in $\mathbb{R}^{r}$, and the latter is defined by
$a\sum_{i=1}^{s}\alpha_{i,\sigma}u_{i}-\sum_{i=1}^{s}\varpi_{i}u_{i}=0$ for
any $\sigma\in\Sigma_{\max}$ such that
$F\subseteq\\{\sum_{i=1}^{s}\alpha_{i,j,\sigma}u_{i,j}=1\\}$. Thus the minimal
face of $\operatorname{Eff}(X)$ containing
$a[L]-[\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}]$ has codimension $k+1$.

  3. (iii)

We argue as in the proof of [PS24, Lemma 6.7(ii)]. Let
$\widetilde{H}\subseteq\mathbb{R}^{s}$ be the inclusion dual to the surjection
$\bigoplus_{i=1}^{s}\mathbb{R}{\mathbf{D}}_{i}\to\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$.
Then $\sum_{i=1}^{s}\alpha_{i,\sigma}u_{i}=\sum_{i=1}^{s}\varpi_{i}u_{i}$ for
all $\mathbf{u}\in\widetilde{H}$ and all $\sigma\in\Sigma_{\max}$. Thus
$\mathcal{P}\cap\widetilde{H}$ is the set of elements $\mathbf{u}$ of
$\widetilde{H}$ such that $u_{1},\dots,u_{s}\geq 0$ and
$\sum_{i=1}^{s}\varpi_{i}u_{i}\leq 1$. Since $F\subseteq\widetilde{H}$ by
[PS24, Lemma 6.7(i)], we have
$F=\widetilde{H}\cap\\{\sum_{i=1}^{s}\varpi_{i}u_{i}=1\\}$. As in in the proof
of [PS24, Lemma 6.7(ii)] we conclude that $F$ is not contained in a coordinate
hyperplane of $\mathbb{R}^{s}$.

∎

##  4\. Rational points on linear complete intersections

###### Proof of Theorem 1.1.

For $1\leq i\leq s$ and $1\leq l\leq t_{i}$, let $g_{i,l}\in R$ be a linear
polynomial defining $H_{i,j}$. Then

$g_{i,l}=\sum_{j=1}^{n_{i}}c_{i,j,l}x_{i,j},\quad l\in\\{1,\dots,t_{i}\\},$

with $c_{i,j,l}\in\mathbb{Z}$, and $g_{i,1},\dots,g_{i,t_{i}}$ are linearly
independent for all $i\in\\{1,\dots,s\\}$. Let $m_{i,j}=1$ for all
$(i,j)\in{\mathcal{I}}$. Then

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=\prod_{i=1}^{s}F_{i,d_{i}}(B_{i}),$

where for $i\in\\{1,\dots,s\\}$, $d\in\mathbb{Z}_{>0}$ and $B>0$,

$F_{i,d}(B)=\sharp\\{(x_{i,1},\dots,x_{i,n_{i}})\in(\mathbb{Z}_{\neq
0})^{n_{i}}:\sup_{1\leq j\leq n_{i}}|x_{i,j}|\leq B,\\\ d\mid x_{i,j}\ \forall
j\in\\{1,\dots,n_{i}\\},\ g_{i,1}=\dots=g_{i,t_{i}}=0\\}.$

For $i\in\\{1,\dots,s\\}$, let $W_{i}\subseteq\mathbb{R}^{n_{i}}$ be the
linear space defined by $g_{i,1}=\dots=g_{i,t_{i}}=0$, and let
$\Lambda_{i}\subseteq W_{i}$ be restriction of the standard lattice
$\mathbb{Z}^{n_{i}}\subseteq\mathbb{R}^{n_{i}}$ to $W_{i}$. Then by [BG06,
Lemma 11.10.15] for every $T\geq 1$,

$\displaystyle\sharp(\mathbb{Z}^{n_{i}}\cap[-T,T]^{n_{i}}\cap
W_{i})=\sharp(\Lambda_{i}\cap(T([-1,1]^{n}\cap W_{i}))$
$\displaystyle=T^{n_{i}-t_{i}}\frac{\operatorname{meas}_{n_{i}-t_{i}}([-1,1]^{n_{i}}\cap
W_{i})}{\det\Lambda_{i}}+O\left(T^{n_{i}-t_{i}-1}\right),$

where $\operatorname{meas}_{n_{i}-t_{i}}$ is the $(n_{i}-t_{i})$-dimensional
measure induced by the Lebesgue measure on $\mathbb{R}^{n_{i}}$. Let

$c_{i}=\frac{\operatorname{meas}_{n_{i}-t_{i}}([-1,1]^{n_{i}}\cap
W_{i})}{\det\Lambda_{i}}.$

Then applying this estimate with $T=B/d$ gives

$F_{i,d}(B)=c_{i}(B/d)^{n_{i}-t_{i}}+O((B/d)^{n_{i}-t_{i}-1})$

whenever $d\leq B$. If $d>B$, then $F_{i,d}(B)=0$ and the same estimate holds.
Hence, for $\delta>0$,

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=C_{M,\mathbf{d}}\prod_{i=1}^{s}B_{i}^{n_{i}-t_{i}}+O\left(C_{E,\mathbf{d}}\left(\prod_{i=1}^{s}B_{i}^{n_{i}-t_{i}}\right)\left(\min_{1\leq
i\leq s}B_{i}\right)^{-\delta}\right),$

where

$\displaystyle
C_{M,\mathbf{d}}=\prod_{i=1}^{s}\frac{c_{i}}{d_{i}^{n_{i}-t_{i}}},$ (4.1)
$\displaystyle C_{E,\mathbf{d}}=\prod_{i=1}^{s}d_{i}^{-(n_{i}-t_{i})+\delta}.$
(4.2)

We show that for $\delta>0$ sufficiently small, the assumptions of Lemma 3.1
are satisfied. Since $n_{i}-t_{i}\geq 2$ for all $i\in\\{1,\dots,s\\}$ such
that $t_{i}\neq 0$, if $f_{(n_{1}-t_{1},\dots,n_{s}-t_{s})}<2$, by Remark
2.4(iv) there is $\tilde{i}\in\\{1,\dots,s\\}$ such that $t_{\tilde{i}}=0$,
$\tilde{i}\in I_{\sigma}$ for all $\sigma\in\Sigma_{\max}$, and
$n_{\tilde{i}}=1$. Then $\rho_{\tilde{i},1}$ is not contained in any maximal
cone of $\Sigma$, contradicting the fact that $X$ is proper. Thus
$f_{(n_{1}-t_{1},\dots,n_{s}-t_{s})}\geq 2$. By definition and by Remark
2.4(ii), $f_{(n_{1}-t_{1}-\delta,\dots,n_{s}-t_{s}-\delta)}\geq
f_{(n_{1}-t_{1},\dots,n_{s}-t_{s})}-s\delta$.

Since $V$ is a smooth complete intersection of smooth divisors, by adjunction
[MR192, Proposition 16.4] we have
$K_{V}=K_{X}+\sum_{i=1}^{s}\sum_{l=1}^{t_{i}}[H_{i,l}]$. Since

$\sum_{i=1}^{s}(n_{i}-t_{i}){\mathbf{D}}_{i}=-[K_{X}]-\sum_{i=1}^{s}t_{i}{\mathbf{D}}_{i}=-[K_{X}]-\sum_{i=1}^{s}\sum_{l=1}^{t_{i}}[H_{i,l}],$

Lemma 3.1 gives

$N_{V}(B)=cB(\log B)^{b-1}+O(B(\log B)^{b-2}(\log\log B)^{s}),$

where $b=\operatorname{rk}\operatorname{Pic}(X)$, and $c$ is defined in (3.7)
with $k=b-1$, $C_{M,\mathbf{d}}$ given by (4.1) and $\varpi_{i}=n_{i}-t_{i}$
for $i\in\\{1,\dots,s\\}$. The restriction
$\operatorname{Pic}(X)\to\operatorname{Pic}(V)$ is an isomorphism as
$t_{i}\leq n_{i}-2$ for all $i\in\\{1,\dots,s\\}$. The leading constant $c$ is
positive by Lemma 2.5(ii). ∎

##  5\. Bihomogeneous hypersurfaces

###### Proof of Theorem 1.2.

In the setting of Theorem 1.2, the hypersurfaces $H_{1},\dots,H_{t}$ are
defined by bihomogeneous polynomials $g_{1},\dots,g_{t}$ of degree
$(e_{1},e_{2})$ in the two sets of variables $\\{x_{1,j}:1\leq j\leq n_{1}\\}$
and $\\{x_{2,j}:1\leq j\leq n_{2}\\}$. Let $m_{i,j}=1$ for all
$(i,j)\in{\mathcal{I}}$.

We will apply [Sch16, Theorem 4.4] with $R=t$, $F_{i}=g_{i}$,
$\mathscr{B}_{i}=[-1,1]^{n_{i}}$, $P_{i}=B_{i}/d_{i}$, $d_{i}=e_{i}$. In order
to apply [Sch16, Theorem 4.4] we need to restrict the points to an open set.
Let $U\subseteq\mathbb{A}^{n_{1}+n_{2}}$ be the open set in [Sch16, Theorem
4.4]. Since the complement of $U$ is the zero set of homogeneous polynomials
by [Sch16, Theorems 4.1, 4.2], the set $W:=\pi(\\{\mathbf{x}\in
Y:(x_{1,1},\dots,x_{1,n_{1}},x_{2,1},\dots,x_{2,n_{2}})\in U\\})$ is an open
subset of $X$. Then

$N_{V,W}(B)=\frac{1}{2^{r}}\sum_{\mathbf{d}\in(\mathbb{Z}_{>0})^{s}}\mu(\mathbf{d})\sharp
A^{W}(B,\mathbf{d}),$

with

$A^{W}(B,\mathbf{d})=\sum_{\begin{subarray}{c}y_{1},\dots,y_{s}\in\mathbb{Z}_{>0}\\\
\prod_{i=1}^{s}y_{i}^{\alpha_{i,\sigma}}\leq B,\
\forall\sigma\in\Sigma_{\max}\end{subarray}}f^{W}_{\mathbf{d}}(y_{1},\dots,y_{s})$

and

$f^{W}_{\mathbf{d}}(y_{1},\dots,y_{s})=\sharp\\{\mathbf{x}\in(\mathbb{Z}_{\neq
0})^{{\mathcal{I}}}:(x_{1,1},\dots,x_{1,n_{1}},x_{2,1},\dots,x_{2,n_{2}})\in
U(\mathbb{Q}),\\\
\eqref{cond:divisibility},\eqref{cond:m-full},\eqref{cond:equations},y_{i}=\sup_{1\leq
j\leq n_{i}}|x_{i,j}|\ \forall i\in\\{1,\dots,s\\}\\}.$

Let

$F^{W}_{\mathbf{d}}(B_{1},\dots,B_{s})=\sum_{1\leq y_{i}\leq B_{i},1\leq i\leq
s}f^{W}_{\mathbf{d}}(y_{1},\dots,y_{s}).$

Then

$F^{W}_{\mathbf{d}}(B_{1},\dots,B_{s})=\widetilde{F}^{W}_{d_{1},d_{2}}(B_{1},B_{2})\prod_{i=3}^{s}F_{i,d_{i}}(B_{i}),$

where

$\widetilde{F}_{d_{1},d_{2}}^{W}=\sharp\\{(x_{1,1},\dots,x_{1,n_{1}},x_{2,1},\dots,x_{2,n_{2}})\in(\mathbb{Z}_{\neq
0})^{n_{1}+n_{2}}\cap U(\mathbb{Q}):\\\ \sup_{1\leq j\leq n_{i}}|y_{i,j}|\leq
B_{i}/d_{i}\ \forall i\in\\{1,2\\},\ g_{1}=\dots=g_{t}=0\\}.$

and for $d\in\mathbb{Z}_{>0}$ and $B>0$,

$F_{i,d}(B)=\sharp\\{(x_{i,1},\dots,x_{i,n_{i}})\in(\mathbb{Z}_{\neq
0})^{n_{i}}:\sup_{1\leq j\leq n_{i}}|x_{i,j}|\leq B,\\\ d\mid x_{i,j}\ \forall
j\in\\{1,\dots,n_{i}\\}\\}.$

If $d\leq B_{i}$, then

$F_{i,d}(B)=2^{n_{i}}(B/d)^{n_{i}}+O((B/d)^{n_{i}-\delta})$

with $0<\delta\leq 1$. If $d>B$, then $F_{i,d}(B)=0$, and the same estimate
holds.

To compute $\widetilde{F}^{W}_{d_{1},d_{2}}(B_{1},B_{2})$, write
$x_{i,j}=d_{i}y_{i,j}$ for all $(i,j)\in{\mathcal{I}}$. Then

$\widetilde{F}^{W}_{d_{1},d_{2}}(B_{1},B_{2})=\sharp\\{(y_{1,1},\dots,y_{1,n_{1}},y_{2,1},\dots,y_{2,n_{2}})\in(\mathbb{Z}_{\neq
0})^{n_{1}+n_{2}}\cap U(\mathbb{Q}):\\\ \sup_{1\leq j\leq n_{i}}|y_{i,j}|\leq
B_{i}/d_{i}\ \forall i\in\\{1,2\\},\ g_{1}=\dots=g_{t}=0\\},$

as the complement of $U$ is the zero set of homogeneous polynomials by [Sch16,
Theorems 4.1, 4.2]. Let $V_{i}^{*}\subseteq\mathbb{A}^{n_{1}+n_{2}}$ be the
locus where the matrix $\left(\frac{\partial g_{l}}{\partial
x_{i,j}}\right)_{\begin{subarray}{c}1\leq l\leq t\\\ 1\leq j\leq
n_{i}\end{subarray}}$ does not have full rank. If $n_{1}+n_{2}>\dim
V_{1}^{*}+\dim V_{2}^{*}+3\cdot 2^{e_{1}+e_{2}}e_{1}e_{2}t^{3}$, then by
[Sch16, Theorem 4.4] there is $\delta>0$ such that

$\widetilde{F}^{W}_{d_{1},d_{2}}(B_{1},B_{2})=C\prod_{i=1}^{2}\left(B_{i}/d_{i}\right)^{n_{i}-te_{i}}+O\left(\left(\min_{i=1,2}B_{i}/d_{i}\right)^{-\delta}\prod_{i=1}^{2}(B_{i}/d_{i})^{n_{i}-te_{i}}\right)\\\
=C\prod_{i=1}^{2}\left(B_{i}/d_{i}\right)^{n_{i}-te_{i}}+O\left(\left(\prod_{i=1}^{2}d_{i}^{-(n_{i}-te_{i})+\delta}\right)\left(\min_{i=1,2}B_{i}\right)^{-\delta}\prod_{i=1}^{2}B_{i}^{n_{i}-te_{i}}\right)$

with $C\in\mathbb{R}_{\geq 0}$, and $C>0$ whenever $V$ has nonsingular
$\mathbb{Q}_{v}$-points for all places $v$ of $\mathbb{Q}$. Thus

$F^{W}_{\mathbf{d}}(B_{1},\dots,B_{s})=C_{M,\mathbf{d}}B_{1}^{n_{1}-te_{1}}B_{2}^{n_{2}-te_{2}}\prod_{i=3}^{s}B_{i}^{n_{i}}\\\
+O\left(C_{E,\mathbf{d}}\left(\min_{1\leq i\leq
s}B_{i}\right)^{-\delta}B_{1}^{n_{1}-te_{1}}B_{2}^{n_{2}-te_{2}}\prod_{i=3}^{s}B_{i}^{n_{i}}\right),$

where

$\displaystyle
C_{M,\mathbf{d}}=Cd_{1}^{-(n_{1}-te_{1})}d_{2}^{-(n_{2}-te_{2})}\prod_{i=3}^{s}d_{i}^{-n_{i}},$
(5.1) $\displaystyle
C_{E,\mathbf{d}}=d_{1}^{-(n_{1}-te_{1})+\delta}d_{2}^{-(n_{2}-te_{2})+\delta}\prod_{i=3}^{s}d_{i}^{-n_{i}+\delta}.$
(5.2)

Recall that $n_{i}-te_{i}\geq 2$ for $i\in\\{1,2\\}$. For $\delta>0$
sufficiently small, if

$f_{n_{1}-te_{1}-\delta,n_{2}-te_{2}-\delta,n_{3}-\delta,\dots,n_{s}-\delta}\leq
1,$

then by Remark 2.4(iv) there is $\tilde{i}\in\\{3,\dots,s\\}$ such that
$\tilde{i}\in I_{\sigma}$ for all $\sigma\in\Sigma_{\max}$ and
$n_{\tilde{i}}=1$. Then the ray $\rho_{\tilde{i},1}$ is contained in no
maximal cone of $\Sigma$, contradicting the fact that $X$ is proper.

Since $V$ is a smooth complete intersection, the adjunction formula [MR192,
Proposition 16.4] gives $K_{V}=K_{X}+H_{1}+\dots+H_{t}$. Let
$\varpi_{i}=n_{i}-te_{i}$ for $i\in\\{1,2\\}$, and $\varpi_{i}=n_{i}$ for
$i\in\\{3,\dots,s\\}$. Since

$\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}=-[K_{X}]-t(e_{1}{\mathbf{D}}_{1}+e_{2}{\mathbf{D}}_{2})=-[K_{X}]-[H_{1}+\dots+H_{t}],$

Lemma 3.1 applied to $F^{W}_{\mathbf{d}}(B_{1},\dots,B_{s})$ and $N_{V,W}(B)$
gives

$N_{V,W}(B)=cB(\log B)^{b-1}+O(B^{a}(\log B)^{b-2}(\log\log B)^{s})$

for $B>0$, where $b=\operatorname{rk}\operatorname{Pic}(X)$, and $c$ is
defined in (3.7) with $k=b-1$, $C_{M,\mathbf{d}}$ given by (5.1). Moreover,
the restriction $\operatorname{Pic}(X)\to\operatorname{Pic}(V)$ is an
isomorphism, as $t\leq\min\\{n_{1},n_{2}\\}-2$. By Lemma 2.5(ii) the leading
constant $c$ is positive if $V(\mathbb{Q}_{v})\neq\emptyset$ for all places
$v$ of $\mathbb{Q}$ as $C$ is positive under the same conditions by [Sch16,
Theorems 4.3 and 4.4]. ∎

##  6\. Campana points on certain diagonal complete intersections

###### Proof of Theorem 1.3.

In the setting of Theorem 1.3, the hypersurfaces $H_{1},\dots,H_{t}$ are
defined by homogeneous diagonal polynomials $g_{1},\dots,g_{t}\in R$ with
$\deg g_{i}=e_{i}{\mathbf{D}}_{i}$ in $\operatorname{Pic}(X)$ for all
$i\in\\{1,\dots,t\\}$. Then

$g_{i}=\sum_{j=1}^{n_{i}}c_{i,j}x_{i,j}^{e_{i}},$

with $c_{i,j}\in\mathbb{Z}_{\neq 0}$, and

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=\prod_{i=1}^{s}F_{i,d_{i}}(B_{i}),$

where for $i\leq t$,

$F_{i,d}(B)=\sharp\\{(x_{i,1},\dots,x_{i,n_{i}})\in(\mathbb{Z}_{\neq
0})^{n_{i}}:d\mid x_{i,j},\ x_{i,j}\text{ is }m_{i,j}\text{-full}\ \forall
j\in\\{1,\dots,n_{i}\\},\\\ \sup_{1\leq j\leq n_{i}}|x_{i,j}|\leq B,\
g_{i}=0\\},$

and for $i>t$,

$F_{i,d}(B)=\sharp\\{(x_{i,1},\dots,x_{i,n_{i}})\in(\mathbb{Z}_{\neq
0})^{n_{i}}:\sup_{1\leq j\leq n_{i}}|x_{i,j}|\leq B,\ d\mid x_{i,j},\\\
x_{i,j}\text{ is }m_{i,j}\text{-full}\ \forall j\in\\{1,\dots,n_{i}\\}\\}.$
(6.1)

For $i\leq t$, we estimate $F_{i,d}(B)$ via the following lemma.

######  Lemma 6.1.

Let $n,e,m_{1},\dots,m_{n}\in\mathbb{Z}_{>0}$. Let
$c_{1},\dots,c_{n}\in\mathbb{Z}_{\neq 0}$. Let $d$ be a squarefree positive
integer. Assume that $n\geq 2$ and $2\leq m_{1}\leq\dots\leq m_{n}$.

  1. (1)

If $e=1$, assume that $\sum_{j=1}^{n}\frac{1}{m_{j}}>3$, and
$\sum_{j=1}^{n-1}\frac{1}{em_{j}(em_{j}+1)}\geq 1$.

  2. (2)

If $e\geq 2$ assume that $\sum_{j=1}^{n}\frac{1}{em_{j}}>3$,
$\sum_{j=1}^{n}\frac{1}{2s_{0}(em_{j})}>1$, where

$s_{0}(m)=\min\left\\{2^{m-1},\frac{1}{2}m(m-1)+\lfloor\sqrt{2m+2}\rfloor\right\\},\
m\in\mathbb{Z}_{\geq 0}.$

For $B>0$, let

$F_{d}(B)=\sharp\\{(x_{1},\dots,x_{n})\in(\mathbb{Z}_{\neq 0})^{n}:d\mid
x_{j},\ x_{j}\text{ is }m_{j}\text{-full}\ \forall j\in\\{1,\dots,n\\},\\\
\sup_{1\leq j\leq n}|x_{j}|\leq B,\ \sum_{j=1}^{n}c_{j}x_{j}^{e}=0\\}.$

Then there is $\eta>0$ such that

$F_{d}(B)=c_{e,d}B^{\Gamma}+O(d^{-1-\eta}B^{\Gamma-\eta}),$

where $\Gamma=\sum_{j=1}^{n}\frac{1}{m_{j}}-e$ and $c_{e,d}$ is defined in
(6.5) and satisfies $0\leq c_{e,d}\ll d^{-1-\eta}$.

###### Proof.

For every $j\in\\{1,\dots,n\\}$ and $x_{j}\in\mathbb{Z}_{\neq 0}$ that is
$m_{j}$-full, there exist unique
$u_{j},v_{j,1},\dots,v_{j,m_{j}-1}\in\mathbb{Z}_{>0}$ such that

$\displaystyle|x_{j}|=u_{j}^{m_{j}}\prod_{r=1}^{m_{j}-1}v_{j,r}^{m_{j}+r},$
$\displaystyle\mu^{2}(v_{j,r})=1,\quad\gcd(v_{j,r},v_{j,r^{\prime}})=1\quad\forall
r,r^{\prime}\in\\{1,\dots,m_{j}-1\\},r\neq r^{\prime}.$

For every choice of $u_{j}$ and $v_{j,r}$ as above, if $d\mid x_{j}$ with
$d\in\mathbb{Z}_{>0}$ squarefree, then there exist unique
$s_{j},t_{j,1},\dots,t_{j,m_{j}-1}\in\mathbb{Z}_{>0}$ such that

$\displaystyle
d=s_{j}\prod_{r=1}^{m_{j}-1}t_{j,r},\quad\mu^{2}(s_{j})=\mu^{2}(t_{j,r})=1\
\forall r\in\\{1,\dots,m_{j}-1\\}$
$\displaystyle\gcd(s_{j},v_{j,r})=\gcd(s_{j},t_{j,r})=\gcd(t_{j,r},t_{j,r^{\prime}})=1\
\forall r,r^{\prime}\in\\{1,\dots,m_{j}-1\\},r\neq r^{\prime}$ $\displaystyle
s_{j}\mid u_{j},\ t_{j,r}\mid v_{j,r}\ \forall r\in\\{1,\dots,m_{j}-1\\}.$

Write $u_{j}=s_{j}\tilde{u}_{j}$ and $v_{j,r}=t_{j,r}\tilde{v}_{j,r}$ for all
$r\in\\{1,\dots,m_{j-1}\\}$. Write $\mathbf{s}=(s_{1},\dots,s_{n})$,
$\mathbf{t}=(t_{j,r})_{1\leq j\leq n,1\leq r\leq m_{j}-1}$. For
$j\in\\{1,\dots,n\\}$, write

$\sigma_{j}=s_{j}\prod_{r=1}^{m_{j}-1}t_{j,r},\qquad\tau_{j}=s_{j}^{m_{j}}\prod_{r=1}^{m_{j}-1}t_{j,r}^{m_{j}+r},\qquad
w_{j}=\prod_{r=1}^{m_{j}-1}\tilde{v}_{j,r}^{m_{j}+r}.$

Let $\mathcal{T}_{d}(B)$ be the set of pairs
$(\mathbf{s},\mathbf{t})\in\mathbb{Z}_{>0}^{n}\times\mathbb{Z}_{>0}^{\sum_{j=1}^{n}(m_{j}-1)}$
that satisfy

$\displaystyle\mu^{2}(\sigma_{j})=1,\quad d=\sigma_{j},\quad\tau_{j}\leq
B\quad\forall j\in\\{1,\dots,s\\}.$

Note that the first two conditions imply

$\sharp\mathcal{T}_{d}(B)\leq\prod_{j=1}^{n}m_{j}^{\omega(d)}\ll
d^{\epsilon},$ (6.2)

where $\omega(d)$ is the number of distinct prime divisors of $d$. Let
$\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)$ be the set of
$\tilde{\mathbf{v}}=(\tilde{v}_{j,r})_{1\leq j\leq n,1\leq r\leq
m_{j}-1}\in\mathbb{Z}_{>0}^{\sum_{j=1}^{n}(m_{j}-1)}$ such that

$\displaystyle\mu^{2}(t_{j,r}\tilde{v}_{j,r})=1,\
\gcd(s_{j},\tilde{v}_{j,r})=1\ \forall
j\in\\{1,\dots,n\\},r\in\\{1,\dots,m_{j}-1\\},$
$\displaystyle\gcd(t_{j,r}\tilde{v}_{j,r},t_{j,r^{\prime}}\tilde{v}_{j,r^{\prime}})=1\
\forall j\in\\{1,\dots,n\\},r,r^{\prime}\in\\{1,\dots,m_{j}-1\\},r\neq
r^{\prime},$ $\displaystyle\tau_{j}w_{j}\leq B\ \forall j\in\\{1,\dots,n\\}.$

Let $\mathcal{T}_{d}(\infty)=\bigcup_{B>0}\mathcal{T}_{d}(B)$ and
$\mathcal{V}_{\mathbf{s},\mathbf{t}}(\infty)=\bigcup_{B>0}\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)$.

Then

$F_{d}(B)=\begin{cases}\sum_{\boldsymbol{\varepsilon}\in\\{\pm
1\\}^{n}}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}M_{\boldsymbol{\varepsilon}\mathbf{c},\boldsymbol{\gamma}}(B^{e})&\text{
if $e$ is odd},\\\
2^{n}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}M_{\mathbf{c},\boldsymbol{\gamma}}(B^{e})&\text{
if $e$ is even},\\\ \end{cases}$ (6.3)

where $\mathbf{c}=(c_{1},\dots,c_{n})$,
$\boldsymbol{\varepsilon}=(\varepsilon_{1},\dots,\varepsilon_{n})$,
$\boldsymbol{\varepsilon}\mathbf{c}=(\varepsilon_{1}c_{1},\dots,\varepsilon_{n}c_{n})$,
$\boldsymbol{\gamma}=(\gamma_{1},\dots,\gamma_{n})$ with

$\gamma_{j}=s_{j}^{em_{j}}\prod_{r=1}^{m_{j}-1}t_{j,r}^{e(m_{j}+r)}\tilde{v}_{j,r}^{e(m_{j}+r)}\quad\forall
j\in\\{1,\dots,n\\},$

and

$M_{\boldsymbol{\varepsilon}\mathbf{c},\boldsymbol{\gamma}}(B^{e})=\sharp\left\\{(\tilde{u}_{1},\dots,\tilde{u}_{n})\in\mathbb{Z}_{>0}^{n}:\max_{1\leq
j\leq n}\gamma_{j}\tilde{u}_{j}^{em_{j}}\leq B^{e},\
\sum_{j=1}^{n}\varepsilon_{j}c_{j}\gamma_{j}\tilde{u}_{j}^{em_{j}}=0\right\\}.$

An estimate for
$M_{\boldsymbol{\varepsilon}\mathbf{c},\boldsymbol{\gamma}}(B^{e})$ is proven
in [BY21, Theorem 2.7] in the case where
$\sum_{j=1}^{n-1}\frac{1}{em_{j}(em_{j}+1)}\geq 1$. The subsequent paper
[BBK+23, Theorem 5.3] extends the range of applicability of [BY21, Theorem
2.7] to the case where $\sum_{j=1}^{n}\frac{1}{em_{j}}>3$,
$\sum_{j=1}^{n}\frac{1}{2s_{0}(em_{j})}>1$.

Let

$\Theta_{e}=\begin{cases}\frac{1}{m_{n}(m_{n}+1)}&\text{ if }e=1,\\\
\sum_{j=1}^{n}\frac{1}{2s_{0}(em_{j})}-1&\text{ if }e\geq 2.\\\ \end{cases}$

For $0<\delta<\frac{1}{(2(n-1)+5)em_{n}(em_{n}+1)}$ and $\epsilon>0$, [BY21,
Theorem 2.7] and [BBK+23, Theorem 5.3] give

$\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}M_{\boldsymbol{\varepsilon}\mathbf{c},\boldsymbol{\gamma}}(B^{e})=\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\frac{\mathfrak{S}_{\boldsymbol{\varepsilon}\mathbf{c},\mathbf{\boldsymbol{\gamma}}}\mathfrak{I}_{\boldsymbol{\varepsilon}\mathbf{c}}}{\prod_{j=1}^{n}\gamma_{j}^{\frac{1}{em_{j}}}}B^{\Gamma}\\\
+O\left(B^{\Gamma}(F_{1}+F_{2}+F_{3})\right),$ (6.4)

where

$\displaystyle\mathfrak{S}_{\boldsymbol{\varepsilon}\mathbf{c},\boldsymbol{\gamma}}=\sum_{q=1}^{\infty}\frac{1}{q^{n}}\sum_{\begin{subarray}{c}a(\text{mod
}q)\\\ \gcd(a,q)=1\end{subarray}}\prod_{j=1}^{n}\sum_{r=1}^{q}\exp(2\pi
ia\varepsilon_{j}c_{j}\gamma_{j}r^{em_{j}}/q),$
$\displaystyle\mathfrak{I}_{\boldsymbol{\varepsilon}\mathbf{c}}=\int_{-\infty}^{\infty}\prod_{j=1}^{n}\left(\int_{0}^{1}\exp(2\pi
i\lambda\varepsilon_{j}c_{j}\xi^{em_{j}})\mathrm{d}\xi\right)\mathrm{d}\lambda,$
$\displaystyle
F_{1}=B^{e((2(n-1)+5)\delta-1)-\Gamma}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\left(\prod_{j=1}^{n}\frac{B^{\frac{1}{m_{j}}}}{\gamma_{j}^{\frac{1}{em_{j}}}}\right)\sum_{l=1}^{n}\frac{\gamma_{l}^{\frac{1}{em_{l}}}}{B^{\frac{1}{m_{l}}}},$
$\displaystyle
F_{2}=B^{-e\delta}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\sum_{q=1}^{\infty}q^{1-\Gamma/e+\epsilon}\prod_{j=1}^{n}\gcd(\gamma_{j},q)^{\frac{1}{em_{j}}}\gamma_{j}^{-\frac{1}{em_{j}}},$
$\displaystyle
F_{3}=\begin{cases}B^{-e\delta\Theta_{e}+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\prod_{j=1}^{n}\gamma_{j}^{-\frac{1}{m_{j}+1}}&\text{
if }e=1,\\\
B^{-e\delta\Theta_{e}+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\prod_{j=1}^{n}\gamma_{j}^{-\frac{1}{em_{j}}+\frac{1}{2s_{0}(em_{j})}}&\text{
if }e\geq 2.\end{cases}$

Let

$\displaystyle c_{e,d}=\begin{cases}\sum_{\boldsymbol{\varepsilon}\in\\{\pm
1\\}^{n}}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(\infty)}\frac{\mathfrak{S}_{\boldsymbol{\varepsilon}\mathbf{c},\mathbf{\boldsymbol{\gamma}}}\mathfrak{I}_{\boldsymbol{\varepsilon}\mathbf{c}}}{\prod_{j=1}^{n}\gamma_{j}^{\frac{1}{em_{j}}}}&\text{
if $e$ is odd},\\\
2^{n}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(\infty)}\frac{\mathfrak{S}_{\mathbf{c},\mathbf{\boldsymbol{\gamma}}}\mathfrak{I}_{\mathbf{c}}}{\prod_{j=1}^{n}\gamma_{j}^{\frac{1}{em_{j}}}}&\text{
if $e$ is even}.\\\ \end{cases}$ (6.5)

For $T>0$, let

$\displaystyle
f_{1}(q)=\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q)}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}},$
$\displaystyle
f_{2}(q)=\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{1},\mathbf{1}}(\infty)}\prod_{j=1}^{n}\left(\frac{\gcd(w_{j}^{e},q)}{w_{j}^{e}}\right)^{\frac{1}{em_{j}}},$

and

$\displaystyle
f_{2}(q,T,\mathbf{s},\mathbf{t})=\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(\infty)\smallsetminus\mathcal{V}_{\mathbf{s},\mathbf{t}}(T)}\prod_{j=1}^{n}\left(\frac{\gcd(w_{j}^{e},q)}{w_{j}^{e}}\right)^{\frac{1}{em_{j}}}.$

Note that for $\sum_{j=1}^{n}\frac{1}{em_{j}}>1$ we have

$\left|\mathfrak{I}_{\boldsymbol{\varepsilon}\mathbf{c}}\right|\ll 1.$

Similarly as in [BY21, (2.8), (2.9), (2.12)], the difference between
$c_{e,d}B^{\Gamma}$ and the main term obtained combining (6.3) and (6.4) is
bounded by

$\displaystyle
B^{\Gamma}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(\infty)\smallsetminus\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\sum_{q=1}^{\infty}q^{1-\sum_{j=1}^{n}\frac{1}{em_{j}}}\prod_{j=1}^{n}\gamma_{j}^{-\frac{1}{em_{j}}}\gcd(\gamma_{j},q)^{\frac{1}{em_{j}}}$
(6.6) $\displaystyle\ll
B^{\Gamma}\sum_{q=1}^{\infty}q^{-\Gamma/e+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q)}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}}f_{2}(q,B,\mathbf{s},\mathbf{t}),$
(6.7)

and $c_{e,d}\ll\sum_{q=1}^{\infty}q^{-\Gamma/e+\epsilon}f_{1}(q)f_{2}(q)$.

By [BY21, (3.10)] and the arguments used to prove [BY21, (3.9)], we have

$f_{2}(q)\ll q^{\epsilon},$ (6.8)

and

$\displaystyle
f_{2}(q,T,\mathbf{s},\mathbf{t})\ll\sum_{i_{0}=1}^{n}\sum_{\begin{subarray}{c}\tilde{v}_{i,r},\
1\leq i\leq n,\ 1\leq r\leq m_{i}-1\\\
\prod_{r=1}^{m_{i}-1}\tilde{v}_{i,r}^{m_{i}+r}>\frac{T}{\tau_{i}}\ \text{if
}i=i_{0}\end{subarray}}\prod_{i=1}^{n}\prod_{r=1}^{m_{i}-1}\frac{\mu^{2}(\tilde{v}_{i,r})\gcd(\tilde{v}_{i,r}^{e(m_{i}+r)},q)^{\frac{1}{em_{i}}}}{\tilde{v}_{i,r}^{(m_{i}+r)/m_{i}}}$
$\displaystyle\ll
q^{\epsilon}\sum_{i=1}^{n}\sum_{\begin{subarray}{c}\tilde{v}_{i,r},\ 1\leq
r\leq m_{i}-1\\\
\prod_{r=1}^{m_{i}-1}\tilde{v}_{i,r}^{m_{i}+r}>\frac{T}{\tau_{i}}\end{subarray}}\prod_{r=1}^{m_{i}-1}\frac{\mu^{2}(\tilde{v}_{i,r})\gcd(\tilde{v}_{i,r}^{e(m_{i}+r)},q)^{\frac{1}{em_{i}}}}{\tilde{v}_{i,r}^{(m_{i}+r)/m_{i}}}.$

Our next goal is to provide an upper bound for sums of the type occurring in
this estimate for $f_{2}(q,T,\mathbf{s},\mathbf{t})$.

######  Lemma 6.2.

Let $m\in\mathbb{N}_{\geq 2}$, $e\in\mathbb{N}$ and $A>0$ a real parameter.
Then, for every $0<\epsilon<\frac{1}{m(m+1)}$ we have

$\sum_{\begin{subarray}{c}v_{r}\in\mathbb{N},\ 1\leq r\leq m-1\\\
\prod_{r=1}^{m-1}v_{r}^{m+r}>A\end{subarray}}\prod_{r=1}^{m-1}\frac{\mu^{2}(v_{r})\gcd(v_{r}^{e(m+r)},q)^{\frac{1}{em}}}{v_{r}^{(m+r)/m}}\ll_{m,\epsilon}A^{-\frac{1}{m(m+1)}+\epsilon}q^{\frac{m-1}{em(m+1)}+\epsilon}.$

###### Proof.

We first consider the sum

$S_{1}:=\sum_{\begin{subarray}{c}v_{r}\in\mathbb{N},\ 1\leq r\leq m-1\\\
\prod_{r=1}^{m-1}v_{r}^{m+r}>A\end{subarray}}\prod_{r=1}^{m-1}\frac{1}{v_{r}^{(m+r)/m}}$

for $A>1$. A dyadic decomposition for each of the variables $v_{r}$, $1\leq
r\leq m-1$, leads us to the upper bound

$\displaystyle
S_{1}\ll\sum_{\begin{subarray}{c}l_{1},\ldots,l_{m-1}\in\mathbb{N}\\\
2^{(m+1)l_{1}+\ldots+(2m-1)l_{m-1}}>A\end{subarray}}2^{-\frac{1}{m}l_{1}-\ldots-\frac{m-1}{m}l_{m-1}}$

Note that for each $k\in\frac{1}{m}\mathbb{N}$ we have

$\sharp\\{l_{1},\ldots,l_{m-1}\in\mathbb{N}:\
\frac{1}{m}l_{1}+\ldots+\frac{m-1}{m}l_{m-1}=k\\}\ll_{m}k^{m-2}.$

We deduce that

$\displaystyle
S_{1}\ll_{m}\sum_{\begin{subarray}{c}k\in\frac{1}{m}\mathbb{N}\\\
r(k)>0\end{subarray}}k^{m-2}2^{-k}$

where $r(k)$ is the number of $(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{m-1}$ such
that both

$\frac{1}{m}l_{1}+\ldots+\frac{m-1}{m}l_{m-1}=k\quad\mbox{and}\quad
2^{(m+1)l_{1}+\ldots+(2m-1)l_{m-1}}>A$

hold. Observe that if $r(k)>0$, then there exists
$(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{m-1}$ with
$\frac{1}{m}l_{1}+\ldots+\frac{m-1}{m}l_{m-1}=k$ and

$\displaystyle m(m+1)k=(m+1)\left(l_{1}+\ldots+(m-1)l_{m-1}\right)$
$\displaystyle\geq(m+1)l_{1}+\frac{m+2}{2}2l_{2}+\ldots+\frac{2m-1}{m-1}(m-1)l_{m-1}>\log
A/\log 2,$

i.e.

$S_{1}\ll_{m}\sum_{\begin{subarray}{c}k\in\frac{1}{m}\mathbb{N}\\\
m(m+1)k>\log A/\log
2\end{subarray}}k^{m-2}2^{-k}\ll_{m,\epsilon}A^{-\frac{1}{m(m+1)}+\epsilon}.$

Note that the upper bound for $S_{1}$ also holds for $A\leq 1$ and
$\epsilon<\frac{1}{m(m+1)}$.

We now turn to the sum in the statement of the lemma. If $v_{r}$ is a square-
free natural number and $d_{r}=\gcd(v_{r}^{e(m+r)},q)$, then we can write

$d_{r}=d_{r,1}d_{r,2}^{2}\ldots
d_{r,e(m+r)}^{e(m+r)},\quad\mu^{2}(d_{r,j})=1,\ \forall 1\leq j\leq
e(m+r),\quad\gcd(d_{r,j},d_{r,j^{\prime}})=1,\ \forall j\neq j^{\prime}.$

Writing $v_{r}=v_{r}^{\prime}\prod_{j=1}^{e(m+r)}d_{r,j}$ and
$d_{r}^{\prime}=\prod_{j=1}^{e(m+r)}d_{r,j}$ we find that

$\displaystyle S_{2}:=\sum_{\begin{subarray}{c}v_{r}\in\mathbb{N},\ 1\leq
r\leq m-1\\\
\prod_{r=1}^{m-1}v_{r}^{m+r}>A\end{subarray}}\prod_{r=1}^{m-1}\frac{\mu^{2}(v_{r})\gcd(v_{r}^{e(m+r)},q)^{\frac{1}{em}}}{v_{r}^{(m+r)/m}}$
$\displaystyle\ll\sum_{d_{r,1}d_{r,2}^{2}\ldots d_{r,e(m+r)}^{e(m+r)}\mid q,\
1\leq r\leq m-1}\sum_{\begin{subarray}{c}v^{\prime}_{r}\in\mathbb{N},\ 1\leq
r\leq m-1\\\
\prod_{r=1}^{m-1}(d^{\prime}_{r}v^{\prime}_{r})^{m+r}>A\end{subarray}}\prod_{r=1}^{m-1}\frac{d_{r}^{\frac{1}{em}}}{(d^{\prime}_{r}v^{\prime}_{r})^{(m+r)/m}}$
$\displaystyle\ll\sum_{d_{r,1}d_{r,2}^{2}\ldots d_{r,e(m+r)}^{e(m+r)}\mid q,\
1\leq r\leq
m-1}\prod_{r=1}^{m-1}\left(\frac{d_{r}^{\frac{1}{em}}}{d_{r}^{\prime\frac{m+r}{m}}}\right)\sum_{\begin{subarray}{c}v^{\prime}_{r}\in\mathbb{N},\
1\leq r\leq m-1\\\
\prod_{r=1}^{m-1}(d^{\prime}_{r}v^{\prime}_{r})^{m+r}>A\end{subarray}}\prod_{r=1}^{m-1}\frac{1}{(v^{\prime}_{r})^{(m+r)/m}}$

By using the upper bound for $S_{1}$ we find that for $\epsilon>0$
sufficiently small

$\displaystyle S_{2}\ll_{\epsilon,m}\sum_{d_{r,1}d_{r,2}^{2}\ldots
d_{r,e(m+r)}^{e(m+r)}\mid q,\ 1\leq r\leq
m-1}\prod_{r=1}^{m-1}\left(\frac{d_{r}^{\frac{1}{em}}}{d_{r}^{\prime\frac{m+r}{m}}}\right)A^{-\frac{1}{m(m+1)}+\epsilon}\left(\prod_{r=1}^{m-1}(d_{r}^{\prime})^{m+r}\right)^{\frac{1}{m(m+1)}}$
$\displaystyle\ll_{\epsilon,m}A^{-\frac{1}{m(m+1)}+\epsilon}\sum_{d_{r,1}d_{r,2}^{2}\ldots
d_{r,e(m+r)}^{e(m+r)}\mid q,\ 1\leq r\leq
m-1}\prod_{r=1}^{m-1}\left(d_{r}^{\frac{1}{em}}(d_{r}^{\prime})^{-\frac{m+r}{m+1}}\right)$
$\displaystyle\ll_{\epsilon,m}A^{-\frac{1}{m(m+1)}+\epsilon}\sum_{d_{r,1}d_{r,2}^{2}\ldots
d_{r,e(m+r)}^{e(m+r)}\mid q,\ 1\leq r\leq
m-1}\prod_{r=1}^{m-1}d_{r}^{\frac{1}{em}-\frac{m+r}{e(m+r)(m+1)}}$
$\displaystyle\ll_{\epsilon,m}A^{-\frac{1}{m(m+1)}+\epsilon}\sum_{d_{r}\mid
q,\ 1\leq r\leq
m-1}\prod_{r=1}^{m-1}d_{r}^{\frac{1}{em(m+1)}}\ll_{\epsilon,m}A^{-\frac{1}{m(m+1)}+\epsilon}q^{\frac{m-1}{em(m+1)}+\epsilon}.\qed$

Lemma 6.2 shows that we can bound $f_{2}(q,T,\mathbf{s},\mathbf{t})$ by

$\displaystyle
f_{2}(q,T,\mathbf{s},\mathbf{t})\ll\sum_{i=1}^{n}\left(\frac{T}{\tau_{i}}\right)^{-\frac{1}{m_{i}(m_{i}+1)}+\epsilon}q^{\frac{m_{i}-1}{em_{i}(m_{i}+1)}+\epsilon}.$

In the following we write $\Delta_{i}=\frac{m_{i}-1}{em_{i}(m_{i}+1)}$. Then
(6.7) is bounded by

$\displaystyle
S_{3}:=B^{\Gamma}\sum_{i=1}^{n}\sum_{q=1}^{\infty}q^{-\Gamma/e+\Delta_{i}+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q)}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}}\left(\frac{B}{\tau_{i}}\right)^{-\frac{1}{m_{i}(m_{i}+1)}+\epsilon}$
$\displaystyle\ll
B^{\Gamma}\sum_{i=1}^{n}B^{-\frac{1}{m_{i}(m_{i}+1)}+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{q=1}^{\infty}q^{-\Gamma/e+\Delta_{i}+\epsilon}\tau_{i}^{\frac{1}{m_{i}(m_{i}+1)}}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q)}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}}.$

As we will encounter similar expressions in our further analysis, we introduce
for $E,D>0$ and $d$ squarefree the following sum

$S_{d}(D,E):=d^{E}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{q=1}^{\infty}q^{-\Gamma/e+D+\epsilon}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q)}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}}.$

We write $q=q_{1}q_{2}$ with $\gcd(q_{1},d)=1$ and such that all prime
divisors of $q_{2}$ divide $d$. With this we obtain that

$\displaystyle S_{d}(D,E)\ll
d^{E}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}\sum_{q_{1}=1}^{\infty}q_{1}^{-\Gamma/e+D+\epsilon}\sum_{\begin{subarray}{c}q_{2}=1\\\
p\mid q_{2}\Rightarrow p\mid
d\end{subarray}}^{\infty}q_{2}^{-\Gamma/e+D+\epsilon}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j}^{e},q_{2})}{\tau_{j}^{e}}\right)^{\frac{1}{em_{j}}}.$

If we assume $-\Gamma/e+D<-1$, then the sum over $q_{1}$ is absolutely
convergent. For a given vector
$(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)$ and a prime $p$ we write
$\tau_{j,p}$ for the power of $p$ which exactly divides $\tau_{j}$. We find
that

$\displaystyle
S_{d}(D,E)\ll\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}d^{E}\prod_{p\mid
d}\left(\sum_{l=0}^{\infty}p^{l(-\Gamma/e+D+\epsilon)}\prod_{j=1}^{n}\left(\frac{\gcd(\tau_{j,p}^{e},p^{l})}{\tau_{j,p}^{e}}\right)^{\frac{1}{em_{j}}}\right).$

We now split the summation over $l$ into the term $l=0$, where we use the
inequality $\tau_{j,p}\geq p^{m_{j}}$, and we bound the rest by a geometric
sum for $l\geq 1$ using $\gcd(\tau_{j,p}^{e},p^{l})\leq\tau_{j,p}^{e}$,

$\displaystyle
S_{d}(D,E)\ll_{D}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(\infty)}d^{E+\epsilon}\prod_{p\mid
d}\left(p^{-n}+p^{-\Gamma/e+D+\epsilon}\right)$
$\displaystyle\ll_{D}d^{\epsilon}\prod_{p\mid
d}\left(p^{E-n}+p^{-\Gamma/e+D+E+\epsilon}\right)$

If $-\Gamma/e+D+E<-1$, then we deduce that

$S_{d}(D,E)\ll_{D}d^{-1-\eta}$ (6.9)

for some $\eta>0$.

Applying equation (6.9) to $S_{3}$ with $D=\Delta_{i}$ and
$E=\frac{2m_{i}-1}{m_{i}(m_{i}+1)}$ we obtain $S_{3}\ll
B^{\Gamma-\eta}d^{-1-\eta}$, for some $\eta>0$. Hence,

$F_{d}(B)=c_{e,d}B^{\Gamma}+O(B^{\Gamma}(d^{-1-\eta}B^{-\eta}+F_{1}+F_{2}+F_{3})).$

We use the bound in (6.8) and apply equation (6.9) with $D=E=0$ to get
$c_{e,d}\ll d^{-1-\eta}$.

It remains to estimate the error terms $F_{1},F_{2},F_{3}$. We rewrite $F_{1}$
as follows:

$F_{1}=B^{e\delta(2(n-1)+5)}\sum_{l=1}^{n}B^{-\frac{1}{m_{l}}}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\sum_{\tilde{\mathbf{v}}\in\mathcal{V}_{\mathbf{s},\mathbf{t}}(B)}\prod_{\begin{subarray}{c}1\leq
j\leq n\\\ j\neq l\end{subarray}}\gamma_{j}^{-\frac{1}{em_{j}}}.$

As in [BY21, §3] and [BBK+23, §6], we have

$\displaystyle F_{1}\ll
B^{-\frac{1}{m_{n}(m_{n}+1)}+e\delta(2(n-1)+5)}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\prod_{j=1}^{n}\tau_{j}^{-\frac{1}{m_{j}+1}}$
$\displaystyle\ll
d^{-\sum_{j=1}^{n}\frac{m_{j}}{m_{j}+1}+\varepsilon}B^{-\frac{1}{m_{n}(m_{n}+1)}+e\delta(2(n-1)+5)},$

where the last estimate follows from

$\displaystyle\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\prod_{j=1}^{n}\tau_{j}^{-\frac{1}{m_{j}+1}}\leq\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\prod_{j=1}^{n}\sigma_{j}^{-\frac{m_{j}}{m_{j}+1}}$
$\displaystyle\leq
d^{-\sum_{j=1}^{n}\frac{m_{j}}{m_{j}+1}}\sharp\mathcal{T}_{d}(B)\ll
d^{-\sum_{j=1}^{n}\frac{m_{j}}{m_{j}+1}+\varepsilon}.$

by (6.2). Combining the arguments for $F_{3}$ in [BY21, §3] and in [BBK+23,
§6] and the estimate above we have

$\displaystyle F_{3}\ll
B^{-e\delta\Theta_{e}+\epsilon}\sum_{(\mathbf{s},\mathbf{t})\in\mathcal{T}_{d}(B)}\prod_{j=1}^{n}\tau_{j}^{-\frac{1}{m_{j}+1}}\ll
d^{-\sum_{j=1}^{n}\frac{m_{j}}{m_{j}+1}+\epsilon}B^{-e\delta\Theta_{e}+\epsilon}.$

Since $\sum_{j=1}^{n}\frac{m_{j}}{m_{j}+1}\geq\frac{2}{3}n>1$ is satisfied for
$n\geq 2$, we have $F_{1},F_{3}\ll d^{-1-\eta}B^{-\eta}$ for a suitable
$\eta>0$. Since

$F_{2}=B^{-e\delta}\sum_{q=1}^{\infty}q^{1-\Gamma/e+\epsilon}f_{1}(q)f_{2}(q),$

the estimate (6.8) combined with (6.9) for $D=1$ and $E=0$ yields $F_{2}\ll
d^{-1-\eta}B^{-e\delta}$, as $\Gamma/e>2$.∎

By Lemma 6.1 and [PS24, Lemma 5.6]

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=\prod_{i=1}^{s}\left(c_{M,i}B_{i}^{\varpi_{i}}+O(d_{i}^{\nu_{i}+\varepsilon}B_{i}^{\varpi_{i}-\delta})\right)$

where

$\displaystyle\varpi_{i}=\begin{cases}\sum_{j=1}^{n_{i}}\frac{1}{m_{i,j}}-e_{i}&\text{
if }i\leq t,\\\ \sum_{j=1}^{n_{i}}\frac{1}{m_{i,j}}&\text{ if }i>t,\\\
\end{cases}$ (6.10)

$\nu_{i}<-1$ for $i\leq t$, $\nu_{i}=-\frac{2}{3}n_{i}$ if $i>t$, $c_{M,i}$ is
the constant $c_{e_{i},d_{i}}$ defined in (6.5) if $i\leq t$, and
$c_{M,i}=2^{n_{i}}\left(\prod_{j=1}^{n_{i}}c_{m_{i,j},d_{i}}\right)$, where
$c_{m_{i,j},d_{i}}$ is the constant defined in [PS24, (5.11)].

Thus

$F_{\mathbf{d}}(B_{1},\dots,B_{s})=C_{M,\mathbf{d}}\prod_{i=1}^{s}B_{i}^{\varpi_{i}}+O\left(C_{E,\mathbf{d}}(\min_{1\leq
i\leq s}B_{i})^{-\delta}\prod_{i=1}^{s}B_{i}^{\varpi_{i}}\right),$

where

$C_{M,\mathbf{d}}=\prod_{i=1}^{s}c_{M,i}.$ (6.11)

Lemma 6.1 and [PS24, (5.14), (5.15)] give
$C_{M,\mathbf{d}},C_{E,\mathbf{d}}\ll\prod_{i=1}^{s}d_{i}^{-\beta_{i}}$ with
$\beta_{i}>1$ whenever $n_{i}\geq 2$, and $\beta_{i}>\frac{2}{3}-\varepsilon$
otherwise. For $\varepsilon>0$ sufficiently small, $\beta_{i}+\beta_{j}>1$ for
every $i,j\in\\{1,\dots,s\\}$. Thus, if $f_{\beta_{1},\dots,\beta_{s}}\leq 1$,
by Remark 2.4(iv) then there exists an index $\tilde{i}\in\\{1,\dots,s\\}$
such that $\tilde{i}\in I_{\sigma}$ for all $\sigma\in\Sigma_{\max}$ and
$n_{\tilde{i}}=1$. Then the ray $\rho_{\tilde{i},1}$ is contained in no
maximal cone of $\Sigma$, contradicting the fact that $X$ is proper.

Since $c_{i,j}\neq 0$ for all $i\in\\{1,\dots,t\\}$,
$j\in\\{1,\dots,n_{i}\\}$, the adjunction formula [MR192, Proposition 16.4]
gives $K_{V}=(K_{X}+H_{1}+\dots+H_{t})|_{V}$. Since

$\sum_{i=1}^{s}\varpi_{i}{\mathbf{D}}_{i}=-K_{X}-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}(1-\frac{1}{m_{i,j}}){\mathbf{D}}_{i}+\sum_{i=1}^{t}e_{i}{\mathbf{D}}_{i}=-(K_{X}+[\mathscr{D}_{\mathbf{m}}|_{X}]+[H_{1}+\dots+H_{t}]),$

Lemma 3.1 gives

$N_{V}(B)=cB(\log B)^{b-1}+O(B(\log B)^{b-2}(\log\log B)^{s}),$

where $b=\operatorname{rk}\operatorname{Pic}(X)$, and $c$ is defined in (3.7)
with $k=b-1$, $C_{M,\mathbf{d}}$ given by (6.11), and
$\varpi_{1},\dots,\varpi_{s}$ given by (6.10). Moreover, the restriction
$\operatorname{Pic}(X)\to\operatorname{Pic}(V)$ is an isomorphism as
$n_{i}\geq 3$ for $1\leq i\leq t$. ∎

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