# Integral points of bounded height
on a certain toric variety

Florian Wilsch  Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuburg, Austria<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We determine an asymptotic formula for the number of integral points of
bounded height on a certain toric variety, which is incompatible with part of
a preprint by Chambert-Loir and Tschinkel. We provide an alternative
interpretation of the asymptotic formula we get. To do so, we construct an
analogue of Peyre’s constant $\alpha$ and describe its relation to a new
obstruction to the Zariski density of integral points in certain regions of
varieties.

###### 2020 Mathematics Subject Classification:

Primary 11D45; Secondary 11G35, 14G05

###### Contents

  1. 1 Introduction
  2. 2 Geometric framework
    1. 2.1 Metrics, heights, Tamagawa measures, and Clemens complexes
    2. 2.2 A divisor group and the constants $\alpha_{\boldsymbol{A}}$
    3. 2.3 An obstruction
    4. 2.4 Relating the obstruction to the constants $\alpha_{\boldsymbol{A}}$
    5. 2.5 Asymptotic formulas
  3. 3 Integral points on a toric threefold
    1. 3.1 Passage to a universal torsor
    2. 3.2 Counting
    3. 3.3 Interpretation of the result

##  1\. Introduction

A classical problem in Diophantine geometry is to understand the number of
rational points on a given algebraic variety. For Fano varieties—that is,
smooth projective varieties whose anticanonical bundle is ample—the set of
rational points is expected to be Zariski dense (thus in particular infinite)
as soon as it is nonempty. A conjecture of Manin’s [FMT89] makes a more
precise quantitative prediction for this setting: Given a Fano variety $X$
over a number field $K$, one orders the set of rational points $X(K)$ by an
_anticanonical height_ $H\colon X(K)\to\mathbb{R}_{>0}$. There might be a
closed subvariety $Z\subset X$, or, more generally, a _thin_ subset $Z\subset
X(K)$ of the rational points on $X$ that dominates the number of rational
points of bounded height, and one should count points on its complement
$V=X(K)\setminus Z$. In its current form, the conjecture predicts that the
number

$N_{V,H}(B)=\\#\\{x\in V\mid H(x)\leq B\\}$

of rational points of height at most $B$ conforms to the asymptotic formula

(1) $N_{V,H}(B)\sim cB(\log B)^{\operatorname{rk}\operatorname{Pic}X-1},$

where the exponent of $\log B$ is the rank of the Picard group of $X$ and a
conjectural explicit description of the leading constant $c$ has been proposed
by Peyre [Pey03].

A related problem is the quantitative study of _integral points_. On
projective varieties, rational and integral points can be seen to coincide by
clearing denominators in the solution to a homogeneous equation or, more
formally, as a consequence of the valuative criterion for properness. A
problem analogous to the one treated by Manin’s conjecture, asking about
integral points on a quasiprojective variety, is the following: Let $X$ be a
smooth, projective variety over a number field $K$ and $D$ be a reduced,
effective divisor with strict normal crossings such that the log anticanonical
bundle $\omega_{X}(D)^{\vee}$ is at least big. Let $\mathfrak{U}$ be an
integral model of $U=X\setminus D$, and let $H$ be a _log anticanonical height
function_. The number of integral points on $\mathfrak{U}$ of bounded height
might be dominated by points lying on an _accumulating_ thin set $Z\subset
X(K)$, which should be excluded. What is the asymptotic behavior of the number

(2) $\\#\\{x\in\mathfrak{U}(\mathfrak{o}_{K})\cap V\mid H(x)\leq B\\}$

of integral points of bounded height that (as rational points) belong to the
complement $V=X(K)\setminus Z$ of accumulating subsets?

For complete intersections of low degree compared to their dimension, this
kind of problem can be studied using the circle method (e.g. [Bir62, Sch85]).
Methods such as harmonic analysis exploiting a group action can be used to
study linear algebraic groups and their homogeneous spaces (e.g. [DRS93, EM93,
EMS96, Mau07, GOS09]) as well as partial equivariant compactifications thereof
([CLT12, TBT13] and the incomplete [CLT10b]); the asymptotic formulas in the
latter cases are interpreted in a way that is similar to the formula (1)
proposed by Manin and Peyre, building on the framework set out in [CLT10a].

Universal torsors were defined by Colliot-Thélène and Sansuc [CTS87] and were
first used by Salberger [Sal98] to count rational points on toric varieties.
This method can be adapted to count integral points [Wil19], and in the
present paper, we use it to count integral points of bounded height on the
toric variety defined as follows. Let
$X_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a product of
projective lines with coordinate pairs $(a_{0}\mathbin{\mathchar
58\relax}\nobreak a_{1})$, $(b_{0}\mathbin{\mathchar 58\relax}\nobreak
b_{1})$, and $(c_{0}\mathbin{\mathchar 58\relax}\nobreak c_{1})$, and consider
the two lines $l_{1}=V(a_{1},b_{1})$ and $l_{2}=V(a_{1},c_{1})$. Let
$\pi\colon X\to X_{0}$ be the toric variety obtained by blowing up
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ in $l_{1}$ and then
blowing up the resulting variety in $l_{2}$. Denote by $T_{0}=X_{0}\setminus
V(a_{0}a_{1}b_{0}b_{1}c_{0}c_{1})$ the open torus in $X_{0}$ and by
$T=\pi^{-1}(T_{0})$ the open torus in $X$. Denote by $E_{1}$ and $E_{2}$ the
two exceptional divisors above the lines $l_{1}$ and $l_{2}$ and by $M$ the
preimage of the plane $V(a_{0})$ parallel to them. Let

$D=E_{1}+E_{2}+M,\quad U=X\setminus D,\quad\text{and}\quad
U_{0}=X_{0}\setminus(V(a_{0})\cup l_{1}\cup l_{2}),$

the latter subvariety being isomorphic to $U$. Consider the integral model

$\mathfrak{U}=\mathbb{P}^{1}_{\mathbb{Z}}\times\mathbb{P}^{1}_{\mathbb{Z}}\times\mathbb{P}^{1}_{\mathbb{Z}}\
\setminus\ \overline{V(a_{0})\cup l_{1}\cup l_{2}}$

of $U_{0}\cong U$. An integral point on
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ can be represented by
three pairs $(a_{0},a_{1})$, $(b_{0},b_{1})$, $(c_{0},c_{1})$ of coprime
integers, uniquely up to three choices of sign. Such a point lies in the
complement of $V(a_{0})$ if $a_{0}$ does not vanish modulo any prime, that is,
if $a_{0}$ is a unit. Similarly, the point lies in the complement of $l_{1}$
if $a_{1}$ and $b_{1}$ do not simultaneously vanish modulo any prime, that is,
if $\gcd(a_{1},b_{1})=1$, and analogously for $l_{2}$. The set of integral
points on $\mathfrak{U}$ is thus

(3) $\mathfrak{U}(\mathbb{Z})=\left\\{\left((a_{0}\mathbin{\mathchar
58\relax}\nobreak a_{1}),\,(b_{0}\mathbin{\mathchar 58\relax}\nobreak
b_{1}),\,(c_{0}\mathbin{\mathchar 58\relax}\nobreak c_{1})\right)\in
X_{0}(\mathbb{Q})\
\mathrel{}\middle|\mathrel{}\begin{subarray}{c}a_{0},\dots,c_{1}\in\mathbb{Z},\quad
a_{0}\in\\{\pm 1\\},\\\
\gcd(a_{1},b_{1})=\gcd(a_{1},c_{1})=\gcd(b_{0},b_{1})=\gcd(c_{0},c_{1})=1\end{subarray}\right\\}.$

A point $P\in\mathfrak{U}(\mathbb{Z})$ represented as in (3) lies in
$T_{0}(\mathbb{Q})$ (resp. in $T$ when interpreted as a point on $X$) if and
only if all coordinates are nonzero, and for any such point, we set

$H(P)=\left\lvert a_{0}\right\rvert\max\\{\left\lvert
b_{0}\right\rvert,\left\lvert b_{1}\right\rvert\\}^{2}\max\\{\left\lvert
c_{0}\right\rvert\left\lvert c_{1}\right\rvert\\}^{2},$

which will turn out to be a log anticanonical height function on the pair
$(X,D)$ (Lemma 3.1.2).

######  Theorem 1.0.1.

The number

$N(B)=\\{P\in\mathfrak{U}(\mathbb{Z})\cap T(\mathbb{Q})\mid H(P)\leq B\\}$

of integral points of bounded height satisfies

(4) $N(B)=cB(\log B)^{2}+O(B\log B(\log\log B)^{3})\text{,}$

where

$c=4\prod_{p}\left(\left(1-\frac{1}{p}\right)^{2}\left(1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}\right)\right)\text{.}$

Thanks to the machinery in [Der09], the proof is very straightforward
(Sections 3.1 and 3.2). The main interest of the theorem lies in the shape of
the asymptotic formula: it contradicts part of the unpublished preprint
[CLT10b] by Chambert-Loir and Tschinkel, exemplifying a gap in a proof of
which they were already aware and due to which they no longer believed in the
correctness of their result (see Remark 3.3.2 for more details on this issue).
Conceptually, the deviation from the asymptotic formula in op. cit. can be
explained by an obstruction to the existence of integral points in a region
that was expected to dominate the number of integral points. More precisely,
the maximal number of components of $D$ that have a common real point is part
of the exponent of $\log B$ because integral points in arbitrarily small real
neighborhoods of such intersections normally constitute 100% of the total
asymptotic number.

Figure 1. Integral points of height at most $9$ in
$\mathfrak{U}(\mathbb{Z})\cap T(\mathbb{Q})$, viewed as a subset of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. The two lines
$l_{1}$ and $l_{2}$ blown up are shown in red. By [CLT10b], one expects
arbitrarily small neighborhoods of the intersection of the two red lines to
dominate the counting function—but in fact, any sufficiently small such
neighborhood contains no points counted by $N$ at all: as $a_{1}/a_{0}$ is an
integer for all integral points, all integral points lie on “sheets”, and all
these sheets have distance $\geq 1$ from the intersection point, which
corresponds to the unique _maximal dimensional face of the Clemens complex_.
(The plane $a_{1}/a_{0}=0$ defined by both lines contains integral points on
$\mathfrak{U}$, which are not shown as they are not in $T(\mathbb{Q})$; in
fact, it contains infinitely many points, all of height $1$, hence has to be
discarded as an accumulating subvariety to achieve a well-defined counting
function.)

In the case the toric variety $X\setminus D$, this does not hold: The function
$f=a_{1}/a_{0}$ (in fact, a character of $T_{0}$) is regular on $U_{0}$ and
spreads out to $\mathfrak{U}$; hence, it is an integer on every integral point
$P\in\mathfrak{U}(\mathbb{Z})$. If $P\in T_{0}(\mathbb{Q})$, then $f(P)\neq
0$. So $\left\lvert f(P)\right\rvert\geq 1$ for
$P\in\mathfrak{U}(\mathbb{Z})\cap T_{0}(\mathbb{Q})$, that is, for every point
$P$ counted by $N$ (Figure 1). The set $V=\pi^{-1}\\{\left\lvert
f\right\rvert<1\\}$ is a neighborhood of $E_{1}(\mathbb{R})\cap
E_{2}(\mathbb{R})$, and in fact even of $E_{1}(\mathbb{R})\cup
E_{2}(\mathbb{R})$, in the analytic topology, as $f$ vanishes on these
sets—but $V$ does not meet $\mathfrak{U}(\mathbb{Z})\cap T(\mathbb{Q})$,
whence cannot contribute to the counting function $N$. This leaves only the
irreducible component $M$ of $D$ to contribute to the number $N(B)$ of points
of bounded height. We interpret the asymptotic formula in a way that only
takes this component into account (Theorem 3.0.1), explaining the smaller
power of $\log B$. Section 3.3 deals with the details of this interpretation
and the comparison with loc. cit.

This phenomenon is an instance of a more general obstruction and related to
the construction of a factor in the leading constant. After fixing notation
and recalling the relevant definitions on Clemens complexes and the kind of
Tamagawa measures appearing in the context of integral points (Section 2.1),
we describe factors $\alpha_{\boldsymbol{A}}$ associated with each _maximal
face_ ${\boldsymbol{A}}$ of the Clemens complex analogous to Peyre’s constant
$\alpha$, slightly generalizing a construction in [CLT10b] to nontoric
varieties (Section 2.2, in particular Definition 2.2.8).

In (9), we associate an open subvariety $U_{\boldsymbol{A}}$ with every
_maximal face ${\boldsymbol{A}}$ of the Clemens complex_. As soon as this open
subvariety admits nonconstant regular functions, the set of integral points in
a certain region of the variety associated with ${\boldsymbol{A}}$ fails to be
Zariski dense, and we say that there is an _obstruction to the Zariski density
of integral points near ${\boldsymbol{A}}$_; in such a case, the maximal face
${\boldsymbol{A}}$ cannot contribute to the counting function (2). This
obstruction is described and studied in detail in Section 2.3. It turns out
that it is closely related to an obstruction described by Jahnel and
Schindler: under some assumptions on $X$ and $D$, including that they be
defined over a field $K$ with only one archimedean place $\infty$, an
obstruction at _every_ face of the archimedean Clemens complex implies that
the open subvariety $X\setminus D$ is _weakly obstructed at $\infty$_ [JS17,
Def. 2.2 (ii–iii)] (Lemma 2.3.7); in particular, its set of integral points is
not Zariski dense. This allows a generalization of [JS17, Thm. 2.6] to
arbitrary number fields: the set of integral points on $U$ is not Zariski
dense if all maximal faces of the Clemens complex are obstructed, without
restrictions on $K$ (Corollary 2.3.6) or, more generally, if a condition
similar to being obstructed at infinity in the sense of op. cit.,
simultaneously involving all infinite places, holds (Theorem 2.3.11).

The relation between this obstruction and the construction of
$\alpha_{\boldsymbol{A}}$ is explored in Section 2.4: whenever this constant
vanishes or some pathologies appear in its construction, the Zariski density
of the corresponding set of integral points is obstructed (Theorem 2.4.1),
providing a geometric reason for the face ${\boldsymbol{A}}$ to be discarded.
Finally, we briefly sketch how to take this obstruction into account when
interpreting asymptotic formulas for the number of integral points of bounded
height (Section 2.5), Theorem 3.0.1 providing an example for this kind of
interpretation.

### Acknowledgements

Part of this work was conducted as a guest at the Institut de Mathématiques de
Jussieu–Paris Rive Gauche invited by Antoine Chambert-Loir and funded by DAAD.
During this time, I had interesting and fruitful discussions on the
interpretation of the result for the toric variety discussed in Section 3 with
Antoine Chambert-Loir. I wish to thank him for these opportunities and for his
useful remarks on earlier versions of this article. This work was partly
funded by FWF grant P 32428-N35.

##  2\. Geometric framework

Throughout this section, let $K$ be a number field, $\mathfrak{o}_{K}$ its
ring of integers, ${\overline{K}}$ an algebraic closure, $K_{v}$ the
completion at a place $v$, and $k_{v}$ the residue field at a finite place
$v$. Equip the completions with the absolute values
$\left\lvert\cdot\right\rvert_{v}$ normalized such that

$\left\lvert x\right\rvert_{v}=\left\lvert
N_{K_{v}/\mathbb{Q}_{w}}(x)\right\rvert_{w}$

at a place $v$ lying above a place $w$ of $\mathbb{Q}$, such that $\left\lvert
p\right\rvert_{p}=1/p$ on $\mathbb{Q}_{p}$, and with the usual absolute value
on $\mathbb{R}$. Moreover, equip each of the local fields with a Haar measure
$\mu_{v}$ satisfying $\mu_{v}(\mathfrak{o}_{K_{v}})=1$ at finite places, the
usual Lebesgue measure $\mathop{}\\!\mathrm{d}\mu_{v}=\mathop{}\\!\mathrm{d}x$
at real places, and
$\mathop{}\\!\mathrm{d}\mu_{v}=i\mathop{}\\!\mathrm{d}z\mathop{}\\!\mathrm{d}\overline{z}=2\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
at complex places.

We consider pairs $(X,D)$ as follows. Let $X$ be a smooth, projective,
geometrically integral $K$-variety, and let $D$ be a reduced, effective
divisor with strict normal crossings. Let $U=X\setminus D$, and let
$\mathfrak{U}$ be an integral model, that is, a flat and separated
$\mathfrak{o}_{K}$-scheme of finite type together with an isomorphism between
its generic fiber $\mathfrak{U}\times_{\mathfrak{o}_{K}}K$ and $U$.
Throughout, similarly to [Pey03, Déf 3.1, Hyp. 3.3], we assume that

  1. (1)

$H^{1}(X,\mathcal{O}_{X})=H^{2}(X,\mathcal{O}_{X})=0$,

  2. (2)

the geometric Picard group $\operatorname{Pic}(X_{\overline{K}})$ is torsion
free,

  3. (3)

there is a finite number of effective divisors $D_{1},\dots,D_{n}$ that
generate the pseudoeffective cone
$\overline{\operatorname{Eff}}_{X}=\overline{\\{\sum a_{i}D_{i}\mid
a_{i}\in\mathbb{R}_{\geq 0}\\}}\subset{\operatorname{Pic}(X)}_{\mathbb{R}}$
(which we shall also simply call the _effective cone_ for that reason), and

  4. (4)

the log anticanonical bundle ${\omega_{X}(D)}^{\vee}$ is big, that is, it is
in the interior of the pseudoeffective cone.

In particular, the anticanonical bundle $\omega_{X}^{\vee}$ is also big, and
$X$ is _almost Fano_ in the sense of [Pey03, Déf 3.1].

For simplicity, we will assume some form of _splitness_ of the pair $(X,D)$:
that the canonical homomorphism
$\operatorname{Pic}(X)\to\operatorname{Pic}(X_{\overline{K}})$ is an
isomorphism and that all irreducible components of $D_{\overline{K}}$ are
defined over $K$. This assumption on $D$ is weaker than the pair $(X,D)$ being
split in the sense of [Har17].

To fix further notation, for an open subvariety $V\subset X$, we let

$E(V)=\mathcal{O}_{X}(V)^{\times}/K^{\times}$

be the finitely generated abelian group of invertible regular functions on $V$
up to constants.

###  2.1. Metrics, heights, Tamagawa measures, and Clemens complexes

To make this section self-contained, we begin by briefly recalling several
definitions needed for the geometric interpretation of asymptotic formulas, as
found for example in [Pey03, CLT10a].

####  2.1.1. Adelic metrics

Several of the invariants appearing in the interpretation of asymptotics
formulas often depend on a choice of an _adelic metric_ on one or several line
bundles $\mathcal{L}$, that is, a family of norm functions
$\left\lVert\cdot\right\rVert_{v}\colon\mathcal{L}(x_{v})\to\mathbb{R}_{\geq
0}$ on the fibers $\mathcal{L}(x_{v})=x_{v}^{*}\mathcal{L}$ above every local
point $x_{v}\in X(K_{v})$ that varies continuously with the points $x_{v}$ and
is _induced by a model_ at almost all places [Pey03, Ex. 2.2, Déf. 2.3].

In an application of the torsor method, we shall make use of the following
standard construction of an adelic metric on a base point free bundle
$\mathcal{L}$: a set $(s_{0},\dots,s_{n})$ of sections of $\mathcal{L}$
without a common base point defines a morphism $f\colon X\to\mathbb{P}^{n}$,
$x\mapsto(s_{0}(x):\dots:s_{n}(x))$, and we set

$\left\lVert
s(x)\right\rVert_{v}=\min\left\\{\left\lvert\frac{s(x)}{s_{0}(x)}\right\rvert_{v},\dots,\left\lvert\frac{s(x)}{s_{n}(x)}\right\rvert_{v}\right\\}$

for a $K_{v}$-point $x=(x_{0}:\dots:x_{n})$ and a local section $s$ of
$\mathcal{L}$. $\mathcal{O}_{\mathbb{P}^{n}}(1)$. Furthermore, adelic metrics
$\left\lVert\cdot\right\rVert_{\mathcal{L}_{1}},\left\lVert\cdot\right\rVert_{\mathcal{L}_{2}}$
on two line bundles $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ induce an adelic
metric on their quotient $\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{\vee}$ by
setting

$\left\lVert g(x)\right\rVert_{v}=\frac{\left\lVert(g\otimes
f_{2})(x)\right\rVert_{\mathcal{L}_{1},v}}{\left\lVert
f_{2}(x)\right\rVert_{\mathcal{L}_{2},v}}$

for a local section $g$ of $\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{\vee}$,
which is independent of a choice of a nonvanishing local section $f_{2}$ of
$\mathcal{L}_{2}$.

####  2.1.2. Heights

Such an adelic metric $\left\lVert\cdot\right\rVert$ induces a height function

$H_{\mathcal{L},\left\lVert\cdot\right\rVert}\colon X(K)\to\mathbb{R}_{\geq
0},\quad x\mapsto\prod_{v}\left\lVert s(x)\right\rVert^{-1}_{v}\text{,}$

where $s$ is an arbitrary section that does not vanish in $x$ [Pey03, Déf
2.3]. If $\mathcal{L}$ is ample, the number $\\#\\{x\in X(K)\mid
H_{\mathcal{L}}(x)\leq B\\}$ of rational points of bounded height is finite
for any $B\geq 0$. This still holds outside a closed subvariety if
$\mathcal{L}$ is big.

######  Example 2.1.1.

In order to construct a height associated with an arbitrary (not necessarily
base point free) line bundle $\mathcal{L}$, something that we shall need in
the second section, we can write the bundle as a quotient
$\mathcal{L}=\mathcal{A}\otimes\mathcal{B}^{-1}$ of base point free bundles.
Choosing sets $a_{0},\dots,a_{r}$ and $b_{0},\dots,b_{s}$ of global sections
of $A$ and $B$ without a common base point induces morphisms $f_{1}\colon
X\to\mathbb{P}^{r}$ and $f_{2}\colon X\to\mathbb{P}^{s}$, respectively, as
well as metrics on the two bundles. Then

$H_{\mathcal{L}}=\prod_{v}\frac{\max\\{\left\lvert a_{i}\right\rvert_{v}\mid
i=0,\dots,r\\}}{\max\\{\left\lvert b_{j}\right\rvert_{v}\mid
j=0,\dots,s\\}}=\frac{H_{\mathbb{P}^{r}}\circ f_{1}}{H_{\mathbb{P}^{s}}\circ
f_{2}},$

where $H_{\mathbb{P}^{r}}=\prod_{v}\max\\{\left\lvert
x_{0}\right\rvert_{v},\dots,\left\lvert x_{r}\right\rvert_{v}\\}$ is the
standard height on $\mathbb{P}^{r}$, using that $\prod_{v}\left\lvert
s(x)\right\rvert_{v}=1$ for any section $s$ not vanishing in the respective
image of $x$.

If $\mathcal{L}$ is not base point free, it does not suffice to take global
sections of $\mathcal{L}$ and consider the maximum of their absolute values.
There is, however, an inequality: If $\mathcal{L}$ has global sections, take a
basis $s_{0},\dots,s_{n}$ of them. After completing $\\{s_{i}b_{j}\\}_{i,j}$
to a basis of the global sections of $\mathcal{A}$, we get

$H(x)\geq\prod_{v}\max_{i=0,\dots,n}\\{\left\lvert s_{i}\right\rvert_{v}\\}$

for the height function $H$ induced by this choice of basis. If $x$ is not
contained in the base locus, the right-hand side is $H_{\mathbb{P}^{n}}(g(x))$
for the rational map $g\colon X\dashrightarrow\mathbb{P}^{n}$ associated with
$\mathcal{L}$. From this, we can recover the above fact: Assume that
$\mathcal{L}$ is big. After passing to an appropriate multiple (and taking the
corresponding root of all resulting metrics and height functions), we can
write $\mathcal{L}=\mathcal{A}\otimes\mathcal{E}$, where $\mathcal{A}$ is very
ample and $\mathcal{E}=\mathcal{O}_{X}(E)$ is effective. Endow $\mathcal{A}$
and $\mathcal{E}$ with metrics according to the above constructions and
$\mathcal{L}$ with the product metric. Let $f\colon
X\hookrightarrow\mathbb{P}^{n}$ be the resulting closed immersion
corresponding to $\mathcal{A}$ and $g\colon X\dashrightarrow\mathbb{P}^{m}$ be
the resulting rational map corresponding to $\mathcal{E}$. If $x\in(X\setminus
E)(\mathbb{Q})$, then $g(x)$ is defined, and in particular
$H_{\mathcal{E}}(x)\geq H_{\mathbb{P}^{m}}(g(x))\geq 1$; if moreover
$H_{\mathcal{L}}(x)\leq B$, then

$H_{\mathbb{P}^{n}}(f(x))=\frac{H_{\mathcal{L}}(x)}{H_{\mathcal{E}}(x)}\leq
B,$

making the number of rational points on $X\setminus E$ of height at most $B$
finite.

####  2.1.3. Tamagawa measures

A Tamagawa measure, appearing in asymptotic formulas for the number of
rational points of bounded height, is a Borel measure $\tau_{X,v}$ on
$X(K_{v})$ for a place $v$, induced by an adelic metric on the canonical
bundle $\omega_{X}$. In local coordinates $x_{1},\dots,x_{n}$, it is defined
as

$\mathop{}\\!\mathrm{d}\tau_{X,v}=\frac{\mathop{}\\!\mathrm{d}x_{1}\cdots\mathop{}\\!\mathrm{d}x_{n}}{\left\lVert\mathop{}\\!\mathrm{d}x_{1}\wedge\dots\wedge\mathop{}\\!\mathrm{d}x_{n}\right\rVert_{v}},$

with respect to the fixed Haar measure on $K_{v}^{n}$ [Pey03, Not. 4.3]. This
is independent of the choice of local coordinates by the change of variables
formula.

When counting integral points, variants of this measure induced by a metric on
the log anticanonical bundle $\omega_{X}(D)$ take its place:

(5)
$\mathop{}\\!\mathrm{d}\tau_{(X,D),v}=\frac{\mathop{}\\!\mathrm{d}x_{1}\cdots\mathop{}\\!\mathrm{d}x_{n}}{\left\lVert
1_{D}\otimes\mathop{}\\!\mathrm{d}x_{1}\wedge\dots\wedge\mathop{}\\!\mathrm{d}x_{n}\right\rVert_{v}}$

and its restriction to $U(K_{v})$ [CLT10a, § 2.1.9]. Here, $1_{D}$ denotes the
canonical section of $\mathcal{O}_{X}(D)$, corresponding to $1$ under the
canonical embedding $\mathcal{O}_{X}(D)\to\mathcal{K}_{X}$. For a finite place
$v<\infty$, the set $\mathfrak{U}(\mathfrak{o}_{v})$ is compact, so the norm
$\left\lVert 1_{D}\right\rVert^{-1}_{v}$ is bounded on it, and its volume is
finite. Moreover, these volumes verify

$\tau_{(X,D),v}(\mathfrak{U}(\mathfrak{o}_{v}))=\frac{\\#\mathfrak{U}(k_{v})}{(\\#k_{v})^{\dim
X}}$

at almost all places ([CLT10a, § 2.4.1 with § 2.4.3] going back to Weil
[Wei82]). These measures are multiplied with convergence factors associated
with the Galois representations $\operatorname{Pic}(U_{{\overline{K}}})$ and
$E(U_{\overline{K}})$ [CLT10a, Def. 2.2]. Our splitness assumption implies
that the Galois action on both modules is trivial, and we get the powers

$\left(1-\frac{1}{\\#k_{v}}\right)^{\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)}$

of the local factors at $s=1$ of the Dedekind zeta function $\zeta_{K}$ of
$K$. These make the product

$\prod_{v<\infty}\left(1-\frac{1}{\\#k_{v}}\right)^{\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)}\tau_{(X,D),v}(\mathfrak{U}(\mathfrak{o}_{v}))$

absolutely convergent [CLT10a, Thm. 2.5]. Finally, this product is multiplied
with the principal value of the corresponding $L$-function [CLT10a, Def. 2.8],
in this case
$\rho_{K}^{\operatorname{rk}\operatorname{Pic}U-\operatorname{rk}E(U)}$, where

$\rho_{K}=\frac{2^{r}(2\pi)^{s}\operatorname{Reg}_{K}h_{k}}{\\#\mu_{K}\sqrt{\left\lvert
d_{K}\right\rvert}}$

is the principal value of the Dedekind zeta function, with the numbers $r$ and
$s$ of real and complex places, the regulator $\operatorname{Reg}_{K}$, the
class number $h_{K}$, the group $\mu_{K}$ of roots of unity, and the
discriminant $d_{K}$ of $K$.

####  2.1.4. Clemens complexes

Often, there are simple topological reasons for which integral points cannot
evenly distribute along $U(K_{v})$ for archimedean places $v$. For instance,
if $K=\mathbb{Q}$ and $U$ is affine, its integral points are lattice points,
which form a discrete subset of $U(\mathbb{R})$; however, regarded as a subset
of the compact set $X(\mathbb{R})$, the set of integral points must have
accumulation points as soon as it is infinite, which consequently lie on the
boundary $D(\mathbb{R})$. Phenomena like this can be observed in many cases:
integral points tend to accumulate near the boundary, with “more” points lying
near intersections of several components of the boundary divisor. For this
reason, combinatorial data on the boundary, encoded in Clemens complexes
[CLT10a, § 3.1], appears in asymptotic formulas for the number of integral
points of bounded height.

The _geometric Clemens complex_ $\mathcal{C}_{\overline{K}}(D)$ is a partially
ordered set defined as follows: Let $\mathcal{A}$ be an index set for the set
of irreducible components of $D$ (which are the same as the irreducible
components of $D_{\overline{K}}$ by our assumptions); denote by $D_{\alpha}$
the irreducible component of $D$ corresponding to $\alpha\in\mathcal{A}$, and,
for any $A\subset\mathcal{A}$, by $Z_{A}$ the intersection $\bigcap_{\alpha\in
A}D_{\alpha}$. Then the geometric Clemens complex consists of all pairs
$(A,Z)$, such that $A$ is a subset of $\mathcal{A}$, and $Z$ is an irreducible
component of $(Z_{A})_{\overline{K}}$. Its ordering is given by
$(A,Z)\preceq(A^{\prime},Z^{\prime})$ if $A\subset A^{\prime}$ and $Z\supset
Z^{\prime}$. In other words, we add a vertex for every component of $D$; if
the intersection of a set of components is nonempty, we glue one simplex to
the corresponding set of vertices for every geometric component of the
intersection. In the following, we will often suppress $Z$ from the notation.

For an archimedean place $v$, we will also be interested in the _$K_{v}$
-analytic Clemens complex_ $\mathcal{C}^{\mathrm{an}}_{v}(D)$. It is the
subset of $\mathcal{C}_{\overline{K}}(D)$ consisting of all pairs $(A,Z)$ such
that $Z$ is defined over $K_{v}$ and has a $K_{v}$-rational point. (Note that
this depends on $v$ and not just on the isomorphism class of $K_{v}$.) By the
splitness assumption in the beginning of this section, we have the following:

######  Lemma 2.1.2.

If a face $(A,Z)$ of the geometric Clemens complex is part of the
$K_{v}$-analytic Clemens complex $\mathcal{C}^{\mathrm{an}}_{v}(D)$, then so
are all of its subfaces $(A^{\prime},Z^{\prime})$.

###### Proof.

Such a subface is given by data $A^{\prime}=\\{D_{1},\dots,D_{r}\\}\subset A$
and an irreducible component $Z^{\prime}\subset Z_{A^{\prime}}$ with
$Z^{\prime}\supset Z$. Since $Z(K_{v})\neq\emptyset$, it contains a
$K_{v}$-point $P$, that is, a point $P$ invariant under the action of the
Galois group of $K_{v}$; since $Z^{\prime}\supset Z$, the point $P$ also lies
on $Z^{\prime}$. For contradiction, assume now that $Z^{\prime}$ is not
defined over $K_{v}$. Since the $D_{i}$ are all defined over $K$, they are
invariant under the Galois action, and thus all conjugates
${}^{\sigma}\\!Z^{\prime}$ of $Z^{\prime}$ under the action of the Galois
group are contained in all $Z_{A_{i}}$, hence also in $Z_{A^{\prime}}$. As $P$
is contained in the intersection of all conjugates (and there is more than one
by the assumption), $Z_{A^{\prime}}$ is singular in $P$, and $D$ cannot have
strict normal crossings. ∎

Observe that, since $Z_{A}$ is smooth for every $A\subset\mathcal{A}$, the set
$Z_{A}(K_{v})$ is a smooth $K_{v}$-manifold. We are interested in maximal
faces of the analytic Clemens complex with respect to the ordering (i.e.,
_facets_), corresponding to minimal strata of the boundary; such faces need
not be maximal-dimensional, that is, maximal with respect to their number of
vertices. Speaking geometrically, maximal faces are faces $(A,Z)$ such that
$Z(K_{v})$ intersects no other divisor component $D_{\alpha}(K_{v})$,
$\alpha\not\in A$. We denote the set of maximal faces by
$\mathcal{C}^{\mathrm{an,max}}_{v}(D)$; if the $K_{v}$-analytic Clemens
complex is empty at a place $v$—that is, if $D(K_{v})=\emptyset$—then the
empty set is its unique maximal face.

####  2.1.5. Archimedean analytic Clemens complexes

Finally, we shall often be interested in all archimedean places at the same
time. To this end, we define the _archimedean analytic Clemens complex_ to be
the product (or _join_)

$\mathcal{C}^{\mathrm{an}}_{\infty}(D)=\prod_{v\mid\infty}\mathcal{C}^{\mathrm{an}}_{v}(D)$

of all $K_{v}$-analytic Clemens complexes, that is, the product set with the
induced partial order. In particular, a face
${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ is a tuple
${\boldsymbol{A}}=(A_{v})_{v\mid\infty}$ of faces
$A_{v}\in\mathcal{C}^{\mathrm{an}}_{v}(D)$ and has dimension
$\sum_{v}\\#A_{v}-1$. Associated with such a face is the set

(6) $Z_{\boldsymbol{A}}=\prod_{v\mid\infty}Z_{A_{v}}(K_{v});$

it is a closed subset of $X(K_{\mathbb{R}})=\prod_{v\mid\infty}X(K_{v})$.

Finally, note that a face ${\boldsymbol{A}}$ is maximal if and only if all the
$A_{v}$ are; again, we denote by $\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$
the set of maximal faces.

######  Remark 2.1.3.

There are conflicting conventions on whether to include the empty face
$(\\{\emptyset\\},X)$ in such complexes. Here, we include it, as a face of
dimension $-1$. For instance, this convention is necessary to neatly deal with
the case $D=0$, in which studying the empty face of the Clemens complex
recovers Manin’s conjecture on rational points; moreover, it facilitates the
statement of some theorems that follow, in which the empty face is an
interesting special case.

####  2.1.6. The measure associated with a maximal face

Let $v$ be an archimedean place, and let
$A\in\mathcal{C}^{\mathrm{an,max}}_{v}(D)$ be a maximal face of the
$K_{v}$-analytic Clemens complex, that is, a maximal subset of the irreducible
components whose intersection $Z_{A}$ has a $K_{v}$-rational point. Let

$D_{A}=\sum_{\alpha\in A}D_{\alpha}\qquad\text{and}\qquad\Delta_{A}=D-D_{A}$

be the sum of divisors corresponding to $A$ and their “complement”,
respectively. We are interested in a measure $\tau_{Z_{A}}$ on $Z_{A}(K_{v})$
defined as follows [CLT10a, § 2.1.12]: A metric on $\omega_{X}(D_{A})$ defines
a metric on $\omega_{Z_{A}}$ and thus a Tamagawa measure $\tau$ on
$Z_{A}(K_{v})$ by repeated use of the adjunction isomorphism (since $D$ is
assumed to have strict normal crossings). We are interested in the modified
measure $\left\lVert
1_{\Delta_{A}}\right\rVert^{-1}_{\mathcal{O}(\Delta_{A}),v}\tau$. This measure
only depends on the metric on the log canonical bundle $\omega_{X}(D)$: this
metric induces one on
$\omega_{Z_{A}}\otimes\mathcal{O}_{X}(\Delta_{A})|_{Z_{A}}$ via the adjunction
isomorphism, and the above measure is equal to

$\left\lVert
1_{\Delta^{\prime}_{A}}\big{|}_{Z_{A}}\otimes\mathop{}\\!\mathrm{d}x_{1}\wedge\dots\wedge\mathop{}\\!\mathrm{d}x_{s}\right\rVert^{-1}_{\omega_{Z_{A}}\otimes\mathcal{O}_{X}(\Delta_{A})|_{Z_{A}},v}\mathop{}\\!\mathrm{d}x_{1}\cdots\mathop{}\\!\mathrm{d}x_{s},$

for local coordinates $x_{1},\dots,x_{s}$ on $Z_{A}(K_{v})$. Note that the
maximality of $A$ guarantees that $\left\lVert
1_{\Delta_{A}}\right\rVert_{v}^{-1}$ does not have a pole on $Z_{A}(K_{v})$,
and is thus bounded on the compact set $Z_{A}(K_{v})$. These measures are
further renormalized by a factor $c_{K_{v}}^{\\#A}$, where $c_{\mathbb{R}}=2$,
resp. $c_{\mathbb{C}}=2\pi$, is the volume of the unit ball in the archimedean
local field with respect to the Haar measure we are using. This results in the
_residue measure_

$\tau_{Z_{A}}=c_{K_{v}}^{\\#A}\left\lVert
1_{\Delta_{A}}\right\rVert^{-1}_{\mathcal{O}(\Delta_{A}),v}\tau$

on $Z_{A}(\mathbb{R})$. See [CLT10a, §§ 3.3.1, 4.1] for more details.

######  Remark 2.1.4.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$ be maximal
face of the archimedean analytic Clemens complex. The above constructions
furnish a finite measure

$\tau_{\boldsymbol{A}}=\prod_{v<\infty}\left(1-\frac{1}{\\#k_{v}}\right)^{\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)}\tau_{U,v}\times\prod_{v\mid\infty}\tau_{Z_{A_{v}},v}$

on the subset

$\prod_{v<\infty}\mathfrak{U}(\mathfrak{o}_{v})\times\prod_{v\mid\infty}Z_{A_{v}}(K_{v})\subset
X(\mathbb{A}_{K})$

of adelic points. Often, integral points of bounded height _equidistribute_
towards the sum of (suitably normalized) measures on a disjoint union of such
sets: usually that over all _maximal dimensional_ faces, but Sections 2.3 and
3 provide a reason for which this can fail. Chambert-Loir and Tschinkel
provide an abstract theorem that allows such an equidistribution theorem to be
deduced from sufficiently general counting theorems [CLT10a, Prop. 2.10],
generalizing work of Peyre’s [Pey03, Prop. 5.4] in the setting of rational
points.

###  2.2. A divisor group and the constants $\alpha_{\boldsymbol{A}}$

We associate some data with tuples of maximal faces analogous to groups
defined by Chambert-Loir and Tschinkel for toric varieties [CLT10b, § 3.5],
using the full set of divisors instead of invariant ones:

######  Definition 2.2.1.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$.

  1. (i)

An _${\boldsymbol{A}}$ -divisor_ $L$ is a tuple
$(L_{U},(L_{v})_{v\mid\infty})$, where $L\in\operatorname{Div}(U)$ is a
divisor on $U$ and $L_{v}\in\operatorname{Div}(X)$ are divisors supported on
$D_{A_{v}}$, that is, linear combinations of $D_{\alpha}$ with $\alpha\in
A_{v}$. Denote by

$\operatorname{Div}(U;{\boldsymbol{A}})\cong\operatorname{Div}(U)\oplus\bigoplus_{v\mid\infty}\mathbb{Z}^{A_{v}}$

the _group of ${\boldsymbol{A}}$-divisors_.

  2. (ii)

Let
$\operatorname{div}_{\boldsymbol{A}}\colon\mathcal{K}_{X}\to\operatorname{Div}(U;{\boldsymbol{A}})$
be the map associating with a rational function $f$ its corresponding
_principal ${\boldsymbol{A}}$-divisor_

$\operatorname{div}_{\boldsymbol{A}}(f)=\left(\operatorname{div}_{U}(f),\
\left(\sum_{\alpha\in
A_{v}}\operatorname{ord}_{D_{\alpha}}(f)D_{\alpha}\right)_{v}\,\right).$

  3. (iii)

Finally, denote by

$\operatorname{Pic}(U;{\boldsymbol{A}})=\operatorname{Div}(U;{\boldsymbol{A}})/\operatorname{im}(\operatorname{div}_{\boldsymbol{A}})$

the _group of ${\boldsymbol{A}}$-divisor classes_.

######  Notation 2.2.2.

Since $\operatorname{div}_{\boldsymbol{A}}$ is compatible with the standard
divisor function, there are canonical homomorphisms

(7) $\displaystyle\pi_{\boldsymbol{A}}$
$\displaystyle\colon\operatorname{Pic}(X)\to\operatorname{Pic}(U;{\boldsymbol{A}}),\quad$
$\displaystyle[L]$ $\displaystyle\mapsto[(L\cap U,(\delta_{L\in
A_{v}}L)_{v})]\qquad\text{(for irreducible $L$) and}$ (8)
$\displaystyle\sigma_{\boldsymbol{A}}$
$\displaystyle\colon\operatorname{Pic}(U;{\boldsymbol{A}})\to\operatorname{Pic}(U),$
$\displaystyle[L]$ $\displaystyle\mapsto L_{U}.$

Here, $\delta_{L\in A_{v}}$ is $1$ if $L$ is a component of $D$ and belongs to
the maximal face $A_{v}$ and $0$ otherwise. If $K$ has only one archimedean
place and $A\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$, these constructions
simplify to the Picard group
$\operatorname{Pic}(U;A)=\operatorname{Pic}(U_{A})$ of $U_{A}$ and the
pullback homomorphisms $\operatorname{Pic}(X)\to\operatorname{Pic}(U_{A})$ and
$\operatorname{Pic}(U_{A})\to\operatorname{Pic}(U)$. Moreover, there is an
isomorphism

$\displaystyle\operatorname{Pic}(U;{\boldsymbol{A}})$
$\displaystyle\to\left\\{(L_{v})_{v}\in\bigoplus_{v\mid\infty}\operatorname{Pic}(U_{A_{v}})\mathrel{}\middle|\mathrel{}L_{v}|_{U}\cong
L_{v^{\prime}}|_{U}\text{ for all $v,v^{\prime}\mid\infty$}\right\\}$
$\displaystyle[L,(L_{v})_{v}]$ $\displaystyle\mapsto([L+L_{v}])_{v}$

that allows an equivalent interpretation of a ${\boldsymbol{A}}$-divisor class
as a tuple of divisor classes (or line bundles) on $U_{A_{v}}$ whose
restrictions to $U$ coincide. In particular, $\sigma_{\boldsymbol{A}}$ factors
through a canonical homomorphism

$\sigma_{{\boldsymbol{A}},v}\colon\operatorname{Pic}(U;{\boldsymbol{A}})\to\operatorname{Pic}(U_{A_{v}})$

for each $v\mid\infty$.

Set

(9) $\Delta_{{\boldsymbol{A}}}=\sum_{\begin{subarray}{c}\alpha\not\in A_{v}\\\
\text{for all }v\mid\infty\end{subarray}}D_{\alpha}\qquad\text{and}\qquad
U_{\boldsymbol{A}}=X\setminus\Delta_{{\boldsymbol{A}}},$

so that $\Delta_{\boldsymbol{A}}\subset D$ is again the “complement” of
${\boldsymbol{A}}$. With $\Delta_{A_{v}}$ as before for the maximal face
$A_{v}$ at a place $v$ and $U_{A_{v}}=X\setminus\Delta_{A_{v}}$, there are
inclusions

$\Delta_{\boldsymbol{A}}\subset\Delta_{A_{v}}\subset D\qquad\text{and}\qquad
U\subset U_{A_{v}}\subset U_{\boldsymbol{A}}\subset X.$

######  Lemma 2.2.3.

The sequences

(10)
${0}$${E(U)}$${\operatorname{CH}^{0}(D)}$${\operatorname{Pic}(X)}$${\operatorname{Pic}(U)}$${0,}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{(i_{D})_{*}}$$\scriptstyle{i_{U}^{*}}$

where $i_{D}\colon D\to X$ and $i_{U}\colon U\to X$ are the inclusions, and

(11)
${0}$${E(U_{\boldsymbol{A}})}$${E(U)}$${\bigoplus_{v\mid\infty}\operatorname{CH}^{0}(D_{A_{v}})}$${\operatorname{Pic}(U;{\boldsymbol{A}})}$${\operatorname{Pic}(U)}$${0,}$$\scriptstyle{\cdot|_{U}}$$\scriptstyle{o}$$\scriptstyle{p}$$\scriptstyle{\sigma_{{\boldsymbol{A}}}}$

where $o=(\operatorname{ord}_{D_{\alpha}})_{\alpha,v}$ and $p$ is induced by
the projection
$\operatorname{Div}(U;{\boldsymbol{A}})\to\operatorname{Pic}(U;{\boldsymbol{A}})$,
are exact.

###### Proof.

The part on the right of (10) is the localization sequence for Chow groups.
Exactness on the left follows from the observation that a relation making a
divisor supported on $\operatorname{supp}(D)$ linearly trivial has to come
from a meromorphic section whose only zeroes and poles are on
$\operatorname{supp}(D)$, that is, an invertible regular function on $U$. The
only such functions mapping to $0$ in $\operatorname{CH}^{0}(D)$ are regular
and invertible on $X$, hence invertible constants, and we get $E(U)$ on the
left.

Turning to (11), the kernel of $\sigma_{\boldsymbol{A}}$ is generated by
${\boldsymbol{A}}$-divisors supported outside $U$, that is, on the
$D_{A_{v}}$. If such an ${\boldsymbol{A}}$-divisor $L=(0,(L_{v})_{v})$ is
linearly equivalent to $0$ in $\operatorname{Pic}(U;{\boldsymbol{A}})$, this
equivalence has to be induced by a section which has corresponding zeroes and
poles on $L$, but no zeroes and poles on $U$; again, we can exclude constants.
Finally, the invertible regular functions on $U$ not inducing such a relation,
thus mapped to $0$ by $o$, are those which do not have a zero or pole on any
${\boldsymbol{A}}$, that is, those that are regular and invertible on
$U_{\boldsymbol{A}}$. ∎

In the context of asymptotic formulas, we will be interested in the two
numbers

$\displaystyle b_{\boldsymbol{A}}$
$\displaystyle=\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)+\sum_{v\mid\infty}\\#A_{v}\quad\text{and}$
$\displaystyle b_{\boldsymbol{A}}^{\prime}$
$\displaystyle=\operatorname{rk}\operatorname{Pic}(U;{\boldsymbol{A}})$

connected to the exponent of $\log B$ and a factor of the leading constant
associated with $\operatorname{Pic}(U;{\boldsymbol{A}})$.

######  Lemma 2.2.4.

For every ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$, we
have

$\displaystyle b_{\boldsymbol{A}}$
$\displaystyle=\operatorname{rk}\operatorname{Pic}X-\\#\mathcal{A}+\sum_{v\mid\infty}\\#A_{v}\quad\text{and}$
$\displaystyle b_{\boldsymbol{A}}^{\prime}$
$\displaystyle=b_{\boldsymbol{A}}+\operatorname{rk}E(U_{\boldsymbol{A}}).$

###### Proof.

These equalities follow directly from Lemma 2.2.3, on noting that
$\operatorname{rk}\operatorname{CH}^{0}(D)=\\#\mathcal{A}$. ∎

######  Remark 2.2.5.

An equality $b_{\boldsymbol{A}}=b_{\boldsymbol{A}}^{\prime}$ always holds for
toric varieties [CLT10b, § 3.7.1 with the remark before Lem. 3.8.5] and
partial equivariant compactifications of vector groups and semisimple groups
(since their effective cones are simplicial and generated by invariant
divisors, so there cannot be an element in $E(U_{\boldsymbol{A}})$ that would
induce a relation between some of them). More generally, both numbers play a
role in asymptotic formulas, and we shall see in Theorem 2.4.1 (ii) that they
are equal whenever we can expect a tuple ${\boldsymbol{A}}$ of maximal faces
to contribute to an asymptotic formula.

######  Remark 2.2.6.

As a consequence of the splitness assumption, it does not matter whether we
work over $K$ or ${\overline{K}}$: for a group
$\operatorname{Pic}(U_{\overline{K}};{\boldsymbol{A}})$ similarly defined over
${\overline{K}}$, there would be a canonical isomorphism
$\operatorname{Pic}(U_{\overline{K}};{\boldsymbol{A}})\cong\operatorname{Pic}(U;{\boldsymbol{A}})$.
Indeed, consider the exact sequences in Lemma 2.2.3 over both $K$ and
${\overline{K}}$ together with the obvious homomorphisms from the former to
the latter. The splitness assumptions imply that the homomorphisms
$\operatorname{Pic}(X)\to\operatorname{Pic}(X_{\overline{K}})$ and
$\operatorname{CH}^{0}(D)\to\operatorname{CH}^{0}(D_{\overline{K}})$ are
isomorphisms, so using the five lemma three times yields
$\operatorname{Pic}(U_{\overline{K}};{\boldsymbol{A}})\cong\operatorname{Pic}(U;{\boldsymbol{A}})$.

In [CLT10b], Chambert-Loir and Tschinkel define a group
$\operatorname{Pic}(U;{\boldsymbol{A}})$ for (not necessarily split) toric
varieties starting with torus-invariant divisors. Their construction coincides
with the one above.

######  Lemma 2.2.7.

If $X$ is a split toric variety, $D$ is invariant under the action of the
torus, and ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$,
then

(12)
$\operatorname{Pic}(U;{\boldsymbol{A}})\cong\left(\operatorname{Pic}^{T}(U)\oplus\bigoplus_{v\mid\infty}\mathbb{Z}^{A_{v}}\right)/\operatorname{div}_{\boldsymbol{A}}(M),$

where $\operatorname{Pic}^{T}(U)$ is the group of torus invariant divisors and
$M\subset\mathcal{K}_{X}$ is the character group of the torus.

###### Proof.

The group on the right-hand side of (12) fits into the sequence (11) in place
of $\operatorname{Pic}(U;{\boldsymbol{A}})$ on noting that
$E(U),E_{{\boldsymbol{A}}}(U)\subset M$ and that $\operatorname{Pic}(U)$ is
generated by $\operatorname{Pic}^{T}(U)$. Then the five lemma implies that the
inclusion $\operatorname{Pic}^{T}(U)\to\operatorname{Div}(U)$ induces the
desired isomorphism. ∎

#### Effective cones and the $\alpha$-constant

If $V$ is a real vector space and $\Lambda\subset V$ a lattice, we can equip
the dual space $V^{\vee}$ with a Haar measure such that $\Lambda^{\vee}$ has
covolume $1$. Recall that if $C\subset V$ a convex cone, the _characteristic
function_ $\mathcal{X}_{C}$ of $C$ is defined to be

$\mathcal{X}_{C}\colon V\to\mathbb{R}_{\geq 0}\cup\\{\infty\\}\qquad
x\mapsto\int_{C^{\vee}}e^{-\langle x,t\rangle}\mathop{}\\!\mathrm{d}t.$

It is finite in the interior of $C$.

######  Definition 2.2.8.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$.

  1. (i)

Let $V_{\boldsymbol{A}}=\operatorname{Pic}(U;{\boldsymbol{A}})_{\mathbb{R}}$,
and define the _effective cone_ associated with ${\boldsymbol{A}}$ to be the
cone
$\operatorname{Eff}_{\boldsymbol{A}}\subset\operatorname{Pic}(U;{\boldsymbol{A}})_{\mathbb{R}}$
generated by the images of effective divisors $\operatorname{Div}_{\geq
0}(U)_{\mathbb{R}}\oplus\bigoplus_{v}\mathbb{R}_{\geq 0}^{A_{v}}$.

  2. (ii)

Assume that $\operatorname{Pic}(U;{\boldsymbol{A}})$ is torsion free; it is
thus a lattice in $V_{\boldsymbol{A}}$. Define

$\alpha_{{\boldsymbol{A}}}=\frac{1}{(b_{\boldsymbol{A}}^{\prime}-1)!}\mathcal{X}_{\operatorname{Eff}_{\boldsymbol{A}}}(\pi_{\boldsymbol{A}}(\omega_{X}(D)^{\vee})),$

where
$\pi_{\boldsymbol{A}}\colon\operatorname{Pic}(X)\to\operatorname{Pic}(U;{\boldsymbol{A}})$
is as in (7).

######  Remark 2.2.9.

  1. (i)

If $K$ has only one archimedean place and
$A\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$, then $\operatorname{Eff}_{A}$
is simply the effective cone $\overline{\operatorname{Eff}}_{U_{A}}$ of
$U_{A}$.

  2. (ii)

In Theorem 2.4.1 (iii), we shall see that the assumption in Definition 2.2.8
(ii) holds whenever we can expect ${\boldsymbol{A}}$ to contribute to an
asymptotic formula.

  3. (iii)

Since the log anticanonical bundle $\omega_{X}(D)^{\vee}$ is big, its image is
in the interior of $\operatorname{Eff}_{\boldsymbol{A}}$, making
$\alpha_{\boldsymbol{A}}$ finite.

This value is nonzero if and only if $\operatorname{Eff}_{\boldsymbol{A}}$ is
strictly convex. In Theorem 2.4.1 (i), we shall see that if this is not the
case, then there is an obstruction to the Zariski density of integral points
“near ${\boldsymbol{A}}$”, and the face ${\boldsymbol{A}}$ should not
contribute to an asymptotic formula.

  4. (iv)

The constant $\alpha_{\boldsymbol{A}}$ can alternatively be described as a
volume: Equip the hyperplanes $H_{a}=\\{t\in
V_{\boldsymbol{A}}^{\vee}\mid\langle\mathcal{L},t\rangle=a\\}$ with measures
$\nu_{H_{a}}$ normalized such that

$\int_{V_{\boldsymbol{A}}^{\vee}}f\mathop{}\\!\mathrm{d}\nu=\int_{\mathbb{R}}\left(\int_{H_{a}}f\mathop{}\\!\mathrm{d}\nu_{H_{a}}\right)\mathop{}\\!\mathrm{d}a$

for all functions $f$ on $V_{\boldsymbol{A}}^{\vee}$ with compact support.
Then (cf. [Vin63, Ch. 2, § 2] and [BT98, Prop. 5.3])

$\displaystyle\alpha_{\boldsymbol{A}}$
$\displaystyle=\nu_{H_{1}}\\{t\in\operatorname{Eff}_{\boldsymbol{A}}^{\vee}\mid\langle\omega_{X}(D)^{\vee},t\rangle=1\\}$
$\displaystyle=b_{\boldsymbol{A}}^{\prime}\,\nu\\{t\in\operatorname{Eff}_{\boldsymbol{A}}^{\vee}\mid\langle\omega_{X}(D)^{\vee},t\rangle\leq
1\\}.$

If the cone $\operatorname{Eff}_{\boldsymbol{A}}$ is _smooth_ , that is,
generated by a $\mathbb{Z}$-basis
$r_{1},\dots,r_{b_{\boldsymbol{A}}^{\prime}}$ of
$\operatorname{Pic}(U;{\boldsymbol{A}})$, this can be simplified further: If
$\pi(\omega(D)^{\vee})$ has the representation
$(a_{1},\dots,a_{b_{\boldsymbol{A}}^{\prime}})$ in this basis, we have

$\alpha_{\boldsymbol{A}}=\frac{1}{(b_{{\boldsymbol{A}}}^{\prime}-1)!}\,\prod_{1\leq
i\leq b_{\boldsymbol{A}}^{\prime}}\frac{1}{a_{i}}.$

More generally, if $\operatorname{Eff}_{\boldsymbol{A}}$ is generated by a
$\mathbb{Z}$-basis of a sublattice
$\Lambda\subset\operatorname{Pic}(U;{\boldsymbol{A}})$, the same is true after
dividing the right-hand side by the index
$[\operatorname{Pic}(U;{\boldsymbol{A}}):\Lambda]$.

###  2.3. An obstruction

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$ be a
maximal face of the archimedean analytic Clemens complex, and consider the
regular functions $\mathcal{O}_{X}(U_{\boldsymbol{A}})$ on
$U_{\boldsymbol{A}}$. If $\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$, that is,
if there are nonconstant functions on $U_{\boldsymbol{A}}$, then there are no
integral points that are simultaneously near all $Z_{A_{v}}$, except possibly
on a finite set of strict subvarieties (Corollary 2.3.2). If any such
subvariety were to contribute to the asymptotic behavior, we would have to
exclude it ass accumulating; hence, there cannot be a contribution of “points
near ${\boldsymbol{A}}$” to an asymptotic formula if
$\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$. In this case, we will say that
there is _an obstruction to the Zariski density of integral points near
${\boldsymbol{A}}$_.

The most general statement of this obstruction deals with not necessarily
maximal faces ${\boldsymbol{A}}$. In this setting, the existence of such a
function prevents the density of points “near ${\boldsymbol{A}}$”, except
possibly near strictly larger faces ${\boldsymbol{B}}$:

######  Proposition 2.3.1.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ be a (not
necessarily maximal) face such that $\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq
K$. For each $v\mid\infty$, let
$B_{v,1},\dots,B_{v,r_{v}}\in\mathcal{C}^{\mathrm{an}}_{v}(D)$ be the faces
strictly containing $A_{v}$. For each $v\mid\infty$ and $1\leq i\leq r_{v}$,
let $W_{v,i}$ be an arbitrary analytic neighborhood of $Z_{B_{v,i}}(K_{v})$.

Then there exists an analytic neighborhood $U_{v}$ of

$Z_{A_{v}}(K_{v})\setminus\bigcup_{i=1}^{r_{v}}W_{v,i}$

for each $v\mid\infty$ and a Zariski dense open subvariety $V\subset X$ such
that

$\\{\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)\mid x\in U_{v}\text{ for all
}v\mid\infty\\}=\emptyset.$

###### Proof.

Let $s$ be a nonconstant regular function on $U_{\boldsymbol{A}}$. After
multiplying with a suitable constant, we can assume it is regular on the
integral model. Let $v$ be an infinite place. As the poles of $s$ are
contained in $\Delta_{A_{v}}$, its only poles on $Z_{A_{v}}(K_{v})$ are
contained in $Z_{A_{v}}\cap\Delta_{A_{v}}=\bigcup_{i}Z_{B_{v,i}}$. Hence,
$\left\lvert s\right\rvert_{v}$ is continuous on the compact set
$Z=Z_{A_{v}^{\prime}}\setminus\bigcup_{i}W_{B_{v,i}}$ and attains its maximum
$M_{v}$. It follows that

$U_{v}=\\{x\in X(K_{K})\mid\left\lvert s(x)\right\rvert_{v}<2M_{v}\\},$

is a neighborhood of $Z$. Since $s(x)\in\mathfrak{o}_{K}$ for integral points
$x\in\mathfrak{U}(\mathfrak{o}_{K})$, it can only attain the finitely many
integral values $\alpha$ in the box in $\prod_{v\mid\infty}K_{v}$ defined by
the $M_{v}$. Every integral point lying in all the $U_{v}$ must thus lie on
one of the finitely many subvarieties $\overline{V(s-\alpha)}\subset X$. ∎

######  Corollary 2.3.2.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$ be a
maximal face such that $\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$. Then there
is a nonempty Zariski open subset $V\subset X$ and an analytic neighborhood
$U_{v}$ of $Z_{A_{v}}$ in $X(K_{v})$ for every archimedean place $v$ such that

$\\{x\in\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)\mid x\in U_{v}\text{ for all
}v\mid\infty\\}=\emptyset.$

###### Proof.

This is the special case of Proposition 2.3.1 where ${\boldsymbol{A}}$ is
maximal. Then for each $v\mid\infty$, there is no strictly larger $B_{v}\succ
A_{v}$, and $U_{v}$ is simply a neighborhood of $Z_{A_{v}}(K_{v})$. ∎

The obstruction can be rephrased using the notation (6):

######  Corollary 2.3.3.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$. For each
strictly larger ${\boldsymbol{B}}\succ{\boldsymbol{A}}$, let
$W_{\boldsymbol{B}}$ be an open neighborhood of $Z_{\boldsymbol{B}}$. Then
there exists an open neighborhood $U$ of
$Z_{\boldsymbol{A}}\setminus\bigcup_{{\boldsymbol{B}}\succ{\boldsymbol{A}}}W_{\boldsymbol{B}}$
and a nonempty Zariski open $V\subset X$ such that

$\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)\cap U=\emptyset,$

where the intersection is to be understood in $X(K_{\mathbb{R}})$. In
particular, if ${\boldsymbol{A}}$ is maximal, then

$\overline{\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)}\cap
Z_{{\boldsymbol{A}}}=\emptyset.$

######  Remark 2.3.4.

If there is if there is an obstruction to the Zariski density of integral
points near a maximal face ${\boldsymbol{A}}$ (in the sense of Corollary
2.3.2), Proposition 2.3.1 implies that the density of points near
$Z_{{\boldsymbol{A}}^{\prime}}$ for subfaces ${\boldsymbol{A}}^{\prime}$ is
similarly obstructed—except possibly near a larger face
${\boldsymbol{B}}\succ{\boldsymbol{A}}^{\prime}$; indeed, any nonconstant
regular section on $U_{\boldsymbol{A}}$ is also regular on
$U_{{\boldsymbol{A}}^{\prime}}\subset U_{\boldsymbol{A}}$. In such a case, we
would expect that the number of such points is described by invariants
attached to the maximal faces ${\boldsymbol{B}}$ containing
${\boldsymbol{A}}^{\prime}$. In other words, Corollary 2.3.2 provides no
reason for which nonmaximal faces might have to be studied when counting
integral points.

As a consequence, if all maximal faces ${\boldsymbol{B}}$ containing a face
${\boldsymbol{A}}$ are obstructed, there are no integral points simultaneously
near all $Z_{A_{v}}(K_{v})$:

######  Corollary 2.3.5.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ be a (not
necessarily maximal) face of the archimedean analytic Clemens complex such
that $\mathcal{O}_{X}(U_{\boldsymbol{B}})\neq K$ for all maximal
${\boldsymbol{B}}\succeq{\boldsymbol{A}}$. Then there is a nonempty Zariski
open subset $V\subset X$ and a neighborhood $U$ of $Z_{\boldsymbol{A}}$ in
$X(K_{\mathbb{R}})$ such that

$\\{x\in\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)\mid x\in U_{v}\text{ for all
}v\mid\infty\\}=\emptyset.$

###### Proof.

This follows using induction on the codimension $c$ of ${\boldsymbol{A}}$. The
case $c=0$ is Corollary 2.3.2. For general ${\boldsymbol{A}}$, if
${\boldsymbol{B}}\succ{\boldsymbol{A}}$, then the induction hypothesis implies
the existence of a neighborhood $U_{\boldsymbol{B}}$ of $Z_{\boldsymbol{B}}$
and a nonempty Zariski open $V_{\boldsymbol{B}}$ such that
$\mathfrak{U}(\mathfrak{o}_{K})\cap V_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}}=\emptyset$. Corollary 2.3.3 guarantees the existence of a
neighborhood $U_{\boldsymbol{A}}$ of
$Z_{\boldsymbol{A}}\setminus\bigcup_{{\boldsymbol{B}}_{\succ}{\boldsymbol{A}}}U_{\boldsymbol{B}}$
and a nonempty Zariski open $V_{\boldsymbol{A}}$ such that
$\mathfrak{U}(\mathfrak{o}_{K})\cap V_{\boldsymbol{A}}\cap
U_{\boldsymbol{A}}=\emptyset$. The statement follows with
$U=U_{\boldsymbol{A}}\cup\bigcup_{{\boldsymbol{B}}\succ{\boldsymbol{A}}}U_{\boldsymbol{B}}$
and
$V=V_{\boldsymbol{A}}\cap\bigcap_{{\boldsymbol{B}}\succ{\boldsymbol{A}}}V_{\boldsymbol{B}}$.
∎

A notable special case is the empty face $(\emptyset,\dots,\emptyset)$, with
corresponding stratum $Z_{\emptyset}=X$:

######  Corollary 2.3.6.

Assume that $\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$ for all maximal faces
${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$. Then
$\mathfrak{U}(\mathfrak{o}_{K})$ is not Zariski dense for any integral model
$\mathfrak{U}$ of $U$.

This obstruction is very similar to the notion of a _weak obstruction at
infinity_ developed by Jahnel and Schindler [JS17, Def. 2.2]. For an
archimedean place $v$, the complement $U$ of a very ample divisor $D$ is
called _weakly obstructed at $v$_ if there is a connected component
$U^{\prime}$ of $U(K_{v})$, a constant $c>0$, an integer $d>0$, and a finite
set of rational functions of the form $s_{i}=f_{i}/1_{D}^{d}$ with $f_{i}\in
H^{0}(X,\mathcal{O}_{X}(D)^{\otimes d})$ not multiples of $1_{D}^{d}$ (that
is, nonconstant regular functions $s_{i}$ on $U$) such that, for every point
$x\in U^{\prime}$, there is at least one $s_{i}$ with $\left\lvert
s_{i}\right\rvert_{v}<c$.

######  Lemma 2.3.7.

Let $v$ be an archimedean place of $K$ and assume that $U(K_{v})$ is
connected. If $\mathcal{O}_{X}(U_{A})\neq K$ for all maximal faces $A$ of the
$K_{v}$-analytic Clemens complex, then $U$ is weakly obstructed at $v$.

###### Proof.

Take a nontrivial $s_{A}\in\mathcal{O}_{X}(U_{A})$ for all maximal faces $A$.
For every point $x$ on the boundary, at least one of the $s_{A}$ is regular in
$x$. Moreover, all of them are regular on $U$, whence $\\{\left\lvert
s_{i}\right\rvert<c\\}_{i,c}$ covers the compact set $X(K_{v})$, and there is
a finite subcover. We can then take $c$ as the maximal constant used in this
subcover. ∎

######  Remark 2.3.8.

Over fields with only one infinite place, integral points are not Zariski
dense if $U$ is weakly obstructed at $\infty$ by [JS17, Thm. 2.6]. In a more
general setting, this does not need to be the case, even if $U$ is obstructed
at every archimedean place (Example 2.3.10). However, the above Corollary
2.3.6 and the following, more general Theorem 2.3.11 generalize the
obstruction to arbitrary number fields.

This obstruction always vanishes after a suitable base change:

######  Lemma 2.3.9.

There is a finite extension $L\supset K$ such that there is a maximal face
${\boldsymbol{A}}=(A_{w})_{w}$ of the archimedean analytic Clemens complex
$\mathcal{C}^{\mathrm{an}}_{\infty}(D_{L})$ with
$\mathcal{O}_{X_{L}}((U_{L})_{\boldsymbol{A}})\neq L$.

###### Proof.

Let $A_{1},\dots,A_{n}$ be the maximal faces of the geometric Clemens complex
$\mathcal{C}_{\overline{K}}(D)$, and let $L\supset K$ be an extension with at
least $n$ complex places $w_{1},\dots,w_{n}$ (i.e., nonconjugate embeddings
into $\mathbb{C}$ whose image is not contained in $\mathbb{R}$). Then
$\mathcal{C}^{\mathrm{an}}_{L_{w_{i}}}(D_{L})=\mathcal{C}_{\overline{K}}(D)$
for these places as all $Z_{A_{i}}(L_{w_{i}})$ are nonempty, and we can take
the face ${\boldsymbol{A}}=(A_{w})_{w}$ with $A_{w_{i}}=A_{i}$ for these $n$
complex places and $A_{w}$ an arbitrary maximal face for all other places.
Since every $D_{i}$ belongs to at least one maximal face of the geometric
Clemens complex, we have $(U_{L})_{\boldsymbol{A}}=X_{L}$, hence
$\mathcal{O}_{X_{L}}((U_{L})_{\boldsymbol{A}})=L$. ∎

Figure 2. Integral points on $\mathbb{G}_{\mathrm{m}}$ over
$K=\mathbb{Q}(\sqrt{5})$—that is, units of the ring of integers—of small
height. Those of norm $1$ are shown in black, those of norm $-1$ in gray. They
are embedded into
$\mathbb{G}_{\mathrm{m}}(\mathbb{R})\times\mathbb{G}_{\mathrm{m}}(\mathbb{R})=\mathbb{R}^{\times}\times\mathbb{R}^{\times}$
along its two places $v_{1}$ and $v_{2}$; a chart showing both $0$ and
$\infty$ is used, and the complement of the multiplicative group is designated
by dashed lines. The maximal faces $(0,0)$ and $(\infty,\infty)$ of
$\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ are obstructed, and no points are near
them; the other two maximal faces $(0,\infty)$ and $(\infty,0)$ are not, and
integral points accumulate near them.

######  Example 2.3.10.

Let $X=\mathbb{P}^{1}=\operatorname{Proj}K[t_{0},t_{1}]$, $D=\\{0,\infty\\}$,
and $U=\mathbb{P}^{1}\setminus D=\mathbb{G}_{\mathrm{m}}$ with integral model
$\mathfrak{U}=\mathbb{G}_{\mathrm{m},\mathfrak{o}_{K}}$. (Its log
anticanonical bundle is trivial, violating the assumptions at the beginning of
the section.) For every archimedean place $v$, the analytic Clemens complex
consists of the two vertices $0$ and $\infty$, which are maximal faces.

Assume for now that $K$ has only one infinite place; that is, $K$ is the field
of rational numbers or imaginary quadratic. The two open subvarieties
associated with the two maximal faces are
$U_{0}=\mathbb{P}^{1}\setminus\\{\infty\\}$ and
$U_{\infty}=\mathbb{P}^{1}\setminus\\{0\\}$. Considering
$t=t_{1}/t_{0}\in\mathcal{O}_{\mathbb{P}^{1}}(U_{0})$ and
$t^{-1}\in\mathcal{O}_{\mathbb{P}^{1}}(U_{\infty})$, both maximal faces are
obstructed, and indeed,
$\mathbb{G}_{\mathrm{m},\mathfrak{o}_{K}}(\mathfrak{o}_{K})=\mathfrak{o}_{K}^{\times}$
is well known to be finite, whence not Zariski dense.

If $K$ has more than one place, there are more maximal faces to consider.
Choosing the maximal face $0$ at every place, we again have
$U_{(0,\dots,0)}=\mathbb{P}^{1}\setminus\\{\infty\\}$, and
${\boldsymbol{A}}=(0,\dots,0)$ is obstructed by $t$. Analogously,
$(\infty,\dots,\infty)$ is obstructed by $t^{-1}$. However, all remaining
tuples ${\boldsymbol{B}}$ of faces (e.g. ${\boldsymbol{B}}=(0,\infty,\dots)$)
satisfy $U_{\boldsymbol{B}}=\mathbb{P}^{1}$, whence
$\mathcal{O}_{\mathbb{P}^{1}}(U_{\boldsymbol{B}})=K$, and those tuples are
unobstructed. Indeed, in this case, the set
$\mathbb{G}_{\mathrm{m},\mathfrak{o}_{K}}(\mathfrak{o}_{K})=\mathfrak{o}_{K}^{\times}$
is well known to be infinite, whence Zariski dense (and can be checked to be
“dense near” all these ${\boldsymbol{B}}$ as a consequence of Dirichlet’s unit
theorem). See Figure 2 for an example involving a real quadratic field.

Finally, Corollary 2.3.6 can be generalized and stated in a setting without
assumptions on $U$, in a way and with a proof that is very similar to [JS17]:

######  Theorem 2.3.11.

Let $U$ be a $K$-variety. Assume that

  1. (O)

there are nonconstant regular functions
$s_{1},\dots,s_{n}\in\mathcal{O}_{U}(U)\setminus K$ and a constant $C>0$ such
that, for every point $(x_{v})_{v}\in\prod_{v\mid\infty}U(K_{v})$, there is a
$1\leq i\leq n$ with $\left\lvert s_{i}(x_{v})\right\rvert_{v}<C$
simultaneously for all $v\mid\infty$.

Then $\mathfrak{U}(\mathfrak{o}_{K})$ is not Zariski dense for any integral
model $\mathfrak{U}$ of $U$.

###### Proof.

After multiplying the $s_{i}$ and $C$ with a suitable constant, we can assume
that they are regular on $\mathfrak{U}$. Let $\alpha_{1},\dots,\alpha_{s}$ be
the finitely many integers in $\mathfrak{o}_{K}$ such that
$\left\lvert\alpha_{j}\right\rvert_{v}<C$ for all $v\mid\infty$. Then each
point $P\in\mathfrak{U}(\mathfrak{o}_{K})$ lies on one of the finitely many
subvarieties $V(s_{i}-\alpha_{j})\subset U$. ∎

Returning to the setting with $(X,D)$ as in the beginning of this section
(assumptions that are stronger than necessary for what follows), Lemma 2.3.9
and the following Lemma 2.3.12 will imply that this obstruction, too, vanishes
after a suitable finite base change.

######  Lemma 2.3.12.

If xv holds, then for all
${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ such that each
$A_{v}$ is of dimension $0$ (i.e., $A_{v}=\\{D_{v}\\}$ for some component
$D_{v}$ of $D$), the group $\mathcal{O}_{X}(U_{{\boldsymbol{A}}})$ is
nontrivial.

###### Proof.

For each $s_{i}$, consider the set $U_{i}=\\{\left\lvert
s_{i}(x_{v})\right\rvert_{v}<C\text{ for all $v$}\\}\subset
X(K_{\mathbb{R}})$. By assumption, these sets cover $U(K_{\mathbb{R}})$, and
as this cover is finite, their closures $F_{i}$ cover $X(K_{\mathbb{R}})$.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D)$ be such that
each $A_{v}$ is of dimension $0$. If an $s_{i}$ is not in
$\mathcal{O}_{X}(U_{{\boldsymbol{A}}})$, that is, if it is singular along at
least one of the $Z_{A_{v}}=D_{v}$, then for that $v\mid\infty$, the set
$U_{i,v}=\\{\left\lvert s_{i}(x_{v})\right\rvert_{v}<C\\}$ is disjoint from
$D_{v}(K_{v})$ in $X(K_{v})$. As its closure $F_{i,v}$ can only meet
$D_{v}(K_{v})$ along $\overline{V(s_{i})}$, it is disjoint from the nonempty
set

$D_{v}(K_{v})\setminus\bigcup_{\begin{subarray}{c}\alpha\in\mathcal{A},\\\
D_{\alpha}\neq D_{v}\end{subarray}}D_{\alpha}(K_{v}).$

It follows that the set $U_{i}$ is disjoint from $Z_{\boldsymbol{A}}$ and its
closure $F_{i}$ is disjoint from the nonempty

$Z=Z_{\boldsymbol{A}}\setminus\bigcup_{{\boldsymbol{B}}\succ{\boldsymbol{A}}}Z_{\boldsymbol{B}}.$

In particular, if none of the $s_{i}$ were in
$\mathcal{O}_{X_{L}}((U_{L})_{{\boldsymbol{A}}})$, then
$\bigcup_{i}F_{i}=X(K_{\mathbb{R}})$ would be disjoint from $Z$, a
contradiction. ∎

######  Lemma 2.3.13.

There is a finite field extension $L/K$ such that xv does not hold for
$U_{L}$.

###### Proof.

By Lemma 2.3.9, there is a field extension such that
$\mathcal{O}_{X_{L}}((U_{L})_{{\boldsymbol{B}}})\neq L$ for at least one
maximal face ${\boldsymbol{B}}\in\mathcal{C}^{\mathrm{an}}_{\infty}(D_{L})$.
If $D=0$, then $U=X$ is proper and $\mathcal{O}_{X}(X)=K$, so xv cannot hold.
Otherwise, possibly enlarging $L$ to a totally imaginary field to make all
$K_{v}$-analytic Clemens complexes coincide with the geometric one,
${\boldsymbol{B}}$ contains a subface ${\boldsymbol{A}}$ such that all $A_{v}$
have dimension $0$, and
$\mathcal{O}_{X_{L}}((U_{L})_{\boldsymbol{A}})\supset\mathcal{O}_{X_{L}}((U_{L})_{\boldsymbol{B}})$
is still nontrivial. Now the statement follows from Lemma 2.3.12. ∎

###  2.4. Relating the obstruction to the constants $\alpha_{\boldsymbol{A}}$

This obstruction can be triggered if some of the objects defined in the
previous section behave pathologically.

######  Theorem 2.4.1.

Let ${\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)$ be a
maximal face such that one of the following holds:

  1. (i)

the effective cone $\operatorname{Eff}_{\boldsymbol{A}}$ is not strictly
convex (that is, $\alpha_{\boldsymbol{A}}=0$),

  2. (ii)

$b_{\boldsymbol{A}}\neq b^{\prime}_{\boldsymbol{A}}$, or

  3. (iii)

the group $\operatorname{Pic}(U;{\boldsymbol{A}})$ of
${\boldsymbol{A}}$-divisor classes is not torsion free.

Then $\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$, and there is an obstruction
to the Zariski density of points near ${\boldsymbol{A}}$. Moreover, if $K$ has
only one infinite place and (ii) holds, then $\mathfrak{U}(\mathfrak{o}_{K})$
is not Zariski dense for any integral model $\mathfrak{U}$ of $U$.

###### Proof.

Case (i). That $\operatorname{Eff}_{\boldsymbol{A}}$ is not strictly convex
means that it contains a line through $0$, that is, we can find two nonzero
effective divisors $(E,(E_{v})_{v})$ and
$(E^{\prime},(E^{\prime}_{v})_{v})\in\operatorname{Div}(U;{\boldsymbol{A}})$
with $E+E^{\prime}\sim 0$. Hence, there exists a rational function which
vanishes on all $E,E_{v},E^{\prime},E_{v}^{\prime}$ (and thus is nonconstant),
and whose only poles are outside $U_{\boldsymbol{A}}$.

Case (ii). We have seen in Lemma 2.2.4 that $b_{\boldsymbol{A}}\neq
b^{\prime}_{\boldsymbol{A}}$ if and only if there is a nonconstant invertible
regular function $s\in E(U_{\boldsymbol{A}})$; so, in particular,
$\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$ in this case.

Next, assume that $K$ has only one infinite place, that is, that the group of
units $\mathfrak{o}_{K}^{\times}$ is finite, and let $s\in
E(U_{\boldsymbol{A}})$ be such an invertible regular function. After
multiplying $s$ and $s^{-1}$ with appropriate constants, we get regular
sections $s$ and $s^{\prime}$ on $\mathfrak{U}$ such that
$ss^{\prime}=a\in\mathfrak{o}_{K}$. For a rational point
$x\in\mathfrak{U}(\mathfrak{o}_{K})$, the value $s(x)$ then has to be a
divisor of $a$, of which there are only finitely many. The integral point $x$
must thus lie on one of the finitely many subvarieties
$\overline{V(s-\alpha)}_{\alpha\mid a}$ of $X$.

Case (iii). Consider the embedding

$\mathbb{Z}^{\mathcal{A}\,\setminus\,\operatorname{supp}{\Delta_{\boldsymbol{A}}}}\to\bigoplus_{v}\mathbb{Z}^{A_{v}},\quad
L\mapsto(L|_{D_{A_{v}}})_{v};$

let $M$ be the quotient, and observe that it is torsion free. Let
$\phi\colon\operatorname{Pic}(U;{\boldsymbol{A}})\to M$ map the class of
$(L,(L_{v})_{v})$ to the class of $(L_{v})_{v}$; indeed, this is well-defined
as any divisor of the form $\operatorname{div}_{\boldsymbol{A}}(f)$ maps to

$\left(\sum_{\alpha\not\in\operatorname{supp}\Delta_{\boldsymbol{A}}}\operatorname{ord}_{D_{\alpha}}(f)D_{\alpha}\bigg{|}_{D_{A_{v}}}\right)_{v},$

which has trivial class in $M$. Then the sequence

(13)
${\operatorname{CH}^{0}(\Delta_{\boldsymbol{A}})}$${\operatorname{Pic}(X)}$${\operatorname{Pic}(U;{\boldsymbol{A}})}$${M}$${0,}$$\scriptstyle{(i_{\Delta_{\boldsymbol{A}}})_{*}}$$\scriptstyle{\pi_{\boldsymbol{A}}}$$\scriptstyle{\phi}$

where $i_{\Delta_{\boldsymbol{A}}}\colon\Delta_{\boldsymbol{A}}\to X$ is the
inclusion, is exact. Indeed, the kernel of $\pi_{\boldsymbol{A}}$ is generated
by divisors supported on $\Delta_{\boldsymbol{A}}$, hence by the image of the
pushforward map
$\operatorname{CH}^{0}(\Delta_{\boldsymbol{A}})\to\operatorname{Pic}(X)$; for
exactness on the right, note that
$\operatorname{Pic}(X)=(\operatorname{Div}(U)\oplus\mathbb{Z}^{\mathcal{A}})/\operatorname{im}(\operatorname{div}_{X})$,
so the cokernel of $\pi_{\boldsymbol{A}}$ is indeed
$\bigoplus_{v}\mathbb{Z}^{A_{v}}/\mathbb{Z}^{\mathcal{A}\,\setminus\,\operatorname{supp}\Delta_{\boldsymbol{A}}}$,
after omitting the part
$\mathbb{Z}^{\operatorname{supp}\Delta_{\boldsymbol{A}}}$ mapped to $0$ by
$\pi_{\boldsymbol{A}}$.

It follows that every nonzero torsion element
$T\in\operatorname{Pic}(U;{\boldsymbol{A}})$ has to be the image of a
(nonzero) element $\widetilde{T}\in\operatorname{Pic}(X)$ such that
$n\widetilde{T}\in\operatorname{im}(\mathbb{Z}^{\operatorname{supp}{\Delta_{\boldsymbol{A}}}})$;
that is, there are $b_{\alpha}\in\mathbb{Z}$ such that $n\widetilde{T}+\sum
b_{\alpha}D_{\alpha}\sim 0$. Consider

$\widetilde{T}^{\prime}=T+\sum\left\lceil\frac{b_{\alpha}}{n}\right\rceil
D_{\alpha}.$

The divisor $\widetilde{T}^{\prime}$ is nonzero and in the pseudoeffective
cone, so, using our assumptions on $X$, it is represented by an effective
$\mathbb{Q}$-divisor $E$. The image of $\widetilde{T}^{\prime}=[E]$ is still
$T$, so the image of $[nE]$ is trivial. Working with a suitable multiple of
$nE$ that is integral, this means that there is a rational function $s$ that
vanishes on the support of $E$ and can only have poles on
$\Delta_{\boldsymbol{A}}$. Since the image of $[E]$ in
$\operatorname{Pic}(U;{\boldsymbol{A}})$ is nonzero, the support of $E$ cannot
be contained in the support of $\Delta_{\boldsymbol{A}}$. Hence, $s$ is
nonconstant and regular on $U_{\boldsymbol{A}}$, and
$\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$. ∎

The toric variety studied in Section 3 furnishes an example for Theorem 2.4.1
(i). For the other two cases, we have the following:

######  Example 2.4.2.

Consider $\mathbb{P}^{n}$ and the three hyperplanes $V(x_{0})$, $V(x_{1})$,
and $V(x_{0}+x_{1})$. Their sum does not have strict normal crossings, a
situation that can be remedied by blowing up $V(x_{0},x_{1})$. Call the
resulting variety $X$, and consider the pair $(X,D)$ with
$D=H_{1}+H_{2}+H_{3}+E$, where the $H_{i}$ are the strict transforms of the
three hyperplanes and $E$ is the exceptional divisor. For $n\geq 3$, the log
anticanonical bundle is big (though never nef), and we have

$U=X\setminus D\cong\mathbb{A}^{n}\setminus(V(x_{1})\cup V(x_{1}+1)).$

The geometric and every $K_{v}$-analytic Clemens complex is a “star”, with the
vertex corresponding to $E$ connected to the other three vertices $H_{i}$. If
we take ${\boldsymbol{A}}=(A_{v})_{v}$ with the same maximal face
$A_{v}=A=\\{E,H_{i}\\}$ (for some fixed $i$) for all infinite places, we have
$U_{\boldsymbol{A}}\cong\mathbb{A}^{n-1}\times\mathbb{G}_{\mathrm{m}}$. Hence,

$b_{\boldsymbol{A}}=\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)+\sum_{v\mid\infty}\\#A=0-2+2(r+s)=2(r+s)-2.$

On the other hand, using (11), the sequence

$0\to E(U_{\boldsymbol{A}})\to
E(U)\to(\mathbb{Z}^{A})^{\oplus(r+s)}\to\operatorname{Pic}(U;{\boldsymbol{A}})\to
0$

is exact, with the groups to the left having ranks $1$, $2$, and $2(r+s)$,
respectively, so $b^{\prime}_{\boldsymbol{A}}=2(r+s)-1$, and there is an
obstruction by Theorem 2.4.1 (ii). In fact, the set of integral points is not
Zariski dense: every integral point lies on one of the subvarieties
$\\{ax_{0}-bx_{1}=0\\}$ parametrized by the finitely many solutions
$a,b\in\mathfrak{o}_{K}^{\times}$ of the unit equation $a+b=1$. (Note that by
Lemma 2.3.9, there are unobstructed faces over sufficiently large fields, so
this failure of Zariski density is not explained by Corollary 2.3.6 in
general.)

######  Example 2.4.3.

Consider $\mathbb{P}^{n}$ together with a divisor $D$ having two components:
the quadric hypersurface $Q=\\{x_{0}^{2}=\sum_{i=1}^{n}x_{i}^{2}\\}$ and the
hyperplane $H=\\{x_{0}=0\\}$. If $n\geq 3$, the log anticanonical bundle is
ample. The intersection $Q\cap H$ does not contain any $\mathbb{R}$-points;
so, if $K$ is a totally real field, every $K_{v}$-analytic Clemens complex
consists of two isolated vertices. Consider the face
${\boldsymbol{A}}=(H,\dots,H)\in\mathcal{C}^{\mathrm{an,max}}_{v}(D)$. Since
the Picard group of $U=\mathbb{P}^{n}\setminus D$ is trivial, (11) allows us
to compute

$\operatorname{Pic}(U;{\boldsymbol{A}})\cong\mathbb{Z}^{r}/(2,\dots,2)\cong\mathbb{Z}^{r-1}\oplus\mathbb{Z}/2\mathbb{Z},$

and Theorem 2.4.1 (iii) applies. Note that if $K=\mathbb{Q}$, there are only
finitely many points corresponding to the solutions of
$x_{1}^{2}+\cdots+x_{n}^{2}=2$, while for larger fields, we get the sets of
solutions of $x_{1}^{2}+\cdots+x_{n}^{2}=1+u$ for units
$u\in\mathfrak{o}_{K}^{\times}$.

###  2.5. Asymptotic formulas

These definitions allow the interpretation of asymptotic formulas. Keep all
the assumptions on $(X,D)$ from the beginning of this chapter, which included
$X$ and $D$ being _split_. Let $\mathfrak{U}$ be an integral model of $U$, and
assume that $\mathfrak{U}(\mathbb{Z})$ is Zariski dense. Let $H$ be the height
function associated with a metric on the log anticanonical bundle
$\omega_{X}(D)^{\vee}$. We are interested in the asymptotic behavior of the
number

$N(B)=\\{x\in\mathfrak{U}(\mathfrak{o}_{K})\cap V(K)\mid H(x)\leq B\\}$

of integral points of bounded height whose generic point is on a suitable
subset $V$ of $X$. If strong approximation holds (using the set of connected
components at archimedean places, cf. e.g. [CTWX20]), asymptotic expansions
for $N(B)$ tend to be similar to

(14) $c_{\infty}c_{\mathrm{fin}}B(\log B)^{b-1}(1+o(1)),$

where

$\displaystyle c_{\infty}$ $\displaystyle=\frac{1}{\left\lvert
d_{K}\right\rvert^{\dim
U/2}}\sum_{{\boldsymbol{A}}\in\mathcal{C}^{\mathrm{max},\circ}(D)}\alpha_{\boldsymbol{A}}\prod_{v\mid\infty}\tau_{Z_{A_{v}},v}\left(Z_{A_{v}}(K_{v})\right)\qquad\text{and}$
$\displaystyle c_{\mathrm{fin}}$
$\displaystyle=\rho_{K}^{\operatorname{rk}\operatorname{Pic}U-\operatorname{rk}E(U)}\prod_{v<\infty}\left(1-\frac{1}{\\#k_{v}}\right)^{\operatorname{rk}\operatorname{Pic}U-\operatorname{rk}E(U)}\tau_{U,v}(\mathfrak{U}(\mathfrak{o}_{K_{v}})).$

Here, the number $b$ in the exponent of $\log B$ is the maximal value of
$b_{\boldsymbol{A}}=b_{\boldsymbol{A}}^{\prime}$ attained on tuples
${\boldsymbol{A}}$ of maximal faces with
$\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq K$, i.e., on tuples without an
obstruction. The sum runs over the set

$\mathcal{C}^{\mathrm{max},\circ}(D)=\left\\{{\boldsymbol{A}}\in\mathcal{C}^{\mathrm{an,max}}_{\infty}(D)\mathrel{}\middle|\mathrel{}\mathcal{O}_{X}(U_{\boldsymbol{A}})\neq
K,\ b_{\boldsymbol{A}}=b\right\\}$

of faces ${\boldsymbol{A}}$ on which this maximum $b$ is attained, that is,
the set of maximal dimensional faces under those without an obstruction.
Corollary 2.3.6 guarantees that the sum does not run over the empty set, and
Theorem 2.4.1 (i) guarantees that the factors $\alpha_{\boldsymbol{A}}$ are
nonzero.

In a more general setting, the product of volumes has to be replaced by the
volume of a suitable subset of adelic points: those points in

$\bigcup_{{\boldsymbol{A}}\in\mathcal{C}^{\mathrm{max},\circ}(D)}\prod_{v<\infty}\mathfrak{U}(\mathfrak{o}_{K})\times\prod_{v}Z_{A_{v}}(K_{v})$

that are limit points of integral points (or, if there still are no such limit
points, a similarly defined set using maximal faces of smaller dimension
necessitating a further change of $b$), making necessary adjustments to the
Tagamawa volume on $Z_{A_{v}}(K_{v})$ if there are failures of strong
approximation involving some connected components of $U(K_{v})$ bordering on
$Z_{A_{v}}(K_{v})$ for archimedean $v$ to account for the fact that there are
fewer points near this stratum.

Moreover, the factor $\rho_{K}$ has to be replaced by the principal value of a
different $L$-function, and additional factors can appear in the constant,
related to failures of strong approximation, to nonsplitness, and to
cohomological invariants (similar to $\beta$ in the case of rational points).
It is unclear to the author what the shape of such a factor for arbitrary
$(X,D)$ should be and under which conditions one should expect it to be
different from $1$. Note that the Brauer group modulo constants, whose order
appears in Manin’s conjecture for rational points, might be nontrivial even
for split $U$.

We can compare (14) to results in the framework by Chambert-Loir and
Tschinkel. We note a difference in the case of toric varieties, and list the
additional factors appearing in these asymptotic formulas.

  * •

The formula above agrees with [CLT12, Thm. 3.5.6] on partial equivariant
compactifications of vector groups, since the obstruction never occurs in
these cases, and since the cones $\operatorname{Eff}_{\boldsymbol{A}}$ are all
smooth, satisfying

$\alpha_{\boldsymbol{A}}=\frac{1}{(b-1)!}\left(\prod_{\alpha\not\in\mathcal{A}}\frac{1}{\rho_{\alpha}}\right)\left(\prod_{v\mid\infty}\prod_{\alpha\in
A_{v}}\frac{1}{\rho_{\alpha}-1}\right)$

with the description
$-K_{X}=\sum_{\alpha\in\mathcal{D}}\rho_{\alpha}D_{\alpha}$ of the
anticanonical divisor as a sum of the boundary components
$\\{D_{\alpha}\\}_{\alpha\in\mathcal{D}}$.

  * •

Similarly, in the case of partial equivariant compactifications of split
semisimple groups $G$ [TBT13], the obstruction does not occur, and the cones
are smooth with a similar description of $\alpha_{\boldsymbol{A}}$, making the
formulas compatible. An additional factor is part of the asymptotic formula
(18) in op. cit.: the number $\left\lvert\chi_{S,D,\lambda}(G)\right\rvert$ of
certain automorphic characters of the underlying group $G$, related to strong
approximation on $G$.

  * •

The formula (14) is not compatible with [CLT10b, Thm. 3.11.5] on toric
varieties; it modifies the exponent $b-1$ of $\log B$ and the index set of the
sum. Our formula above agrees with the asymptotic formula we determine in
Section 3. The formula in loc. cit. contains additional factors

$\frac{\left\lvert A(T,U,K)^{*}\right\rvert}{\left\lvert
A(T)^{*}\right\rvert}\frac{\left\lvert
H^{1}(\Gamma,\operatorname{Pic}(X_{E}))\right\rvert}{\left\lvert
H^{1}(\Gamma,M_{E})\right\rvert}:$

two groups of automorphic characters, related to weak and strong approximation
on $T$, and cohomology groups from the action of the Galois group (which is
trivial in the split case). Moreover, the volume is taken on the subset of the
adelic points cut out by these automorphic characters.

  * •

The formula is compatible with [DW21], treating integral points of several
open subvarieties of the minimal desingularization of a singular quartic del
Pezzo surface. This variety is an example of a nontoric variety in which the
construction of $\alpha_{\boldsymbol{A}}$ does not lead to a simplicial cone.

##  3\. Integral points on a toric threefold

The aim of this section is to provide an asymptotic formula for the number
$N(B)$ of integral points of height at most $B$ on the toric variety $X$
defined in the introduction. Integral points on toric varieties are treated by
Chambert-Loir and Tschinkel in [CLT10b]; however, our result contradicts part
of this (unfinished) work. After parametrizing the set of integral points
using a universal torsor in Section 3.1, we determine an asymptotic formula in
Section 3.2, proving Theorem 1.0.1. The exponent of $\log B$ is $1$ less than
the one given in [CLT10b], which is explained by an obstruction to the
existence of integral points on a certain part of $X$: Chambert-Loir’s and
Tschinkel’s asymptotic formula is associated with the one-dimensional face
$\\{E_{1},E_{2}\\}$ of the Clemens complex. There is a function obstructing
the Zariski density of integral points near $E_{1}$ and $E_{2}$, which also
makes the leading constant of their asymptotic formula vanish. In Section 3.3,
we compare our formula to the one given by Chambert-Loir and Tschinkel in
greater detail and get a very similar geometric interpretation to theirs
(Theorem 3.0.1), associated with the maximal, but only zero-dimensional face
$M$ of the Clemens complex.

######  Theorem 3.0.1.

The number of integral points of bounded height satisfies the asymptotic
formula

$N(B)=c_{\infty}c_{\mathrm{fin}}B(\log B)^{b_{M}-1}(1+o(1)),$

with

$\displaystyle c_{\infty}$
$\displaystyle=\alpha_{M}\tau_{M,\infty}(M(\mathbb{R})),$ $\displaystyle
c_{\mathrm{fin}}$
$\displaystyle=\prod_{p}\left(1-\frac{1}{p}\right)^{\operatorname{rk}\operatorname{Pic}U}\tau_{U,p}(\mathfrak{U}(\mathbb{Z}_{p})),$

where all constants are associated with the maximal, but not maximal-
dimensional, face $M$ of the Clemens complex. More explicitly,

$N(B)=cB(\log B)^{2}+O(B\log B(\log\log B)^{3})\text{,}$

where

$c=4\prod_{p}\left(\left(1-\frac{1}{p}\right)^{2}\left(1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}\right)\right)\text{.}$

###  3.1. Passage to a universal torsor

The fan $\Sigma_{X}$ of $X$ can be obtained by starting with the fan of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$, then subdividing it
by adding the ray $\rho_{x}=\mathbb{R}(-1,-1,0)$ (corresponding to the
exceptional divisor $E_{1}$), then further subdividing it by adding the ray
$\rho_{y}=\mathbb{R}(-1,0,-1)$ (corresponding to $E_{2}$). The Picard group of
$X$ is

$\operatorname{Pic}(X)=\mathbb{Z}\pi^{*}[H_{1}]+\mathbb{Z}\pi^{*}[H_{2}]+\mathbb{Z}\pi^{*}[H_{3}]+\mathbb{Z}[E_{1}]+\mathbb{Z}[E_{2}]\cong\mathbb{Z}^{5}\text{,}$

where $H_{1}$, $H_{2}$, and $H_{3}$ are planes of degree $(1,0,0)$, $(0,1,0)$,
and $(0,0,1)$, respectively.

$a_{0}$$b_{0}$$c_{0}$$a_{1}$$b_{1}$$c_{1}$$x$$y$ Figure 3. The fan
$\Sigma_{X}$ of $X$, its rays labeled with the corresponding generators of the
Cox ring.

Aiming to count integral points by a parametrization using a universal torsor,
we start by determining the Cox ring; cf. for example [ADHL15, § 2.1.3] for
background on the following constructions. The Cox ring of $X$ is
$R_{X}=\mathbb{Q}[a_{0},a_{1},b_{0},b_{1},c_{0},c_{1},x,y]$, its generators
corresponding to the rays of $\Sigma_{X}$. It is graded by
$\operatorname{Pic}(X)$, the degree of each generator being the class of the
corresponding divisor in the Picard group. Under the above isomorphism
$\operatorname{Pic}(X)\cong\mathbb{Z}^{5}$, the grading is thus given by Table
1.

$a_{0}$ | $a_{1}$ | $b_{0}$ | $b_{1}$ | $c_{0}$ | $c_{1}$ | $x$ | $y$
---|---|---|---|---|---|---|---
$1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$0$ | $0$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $0$ | $0$
$0$ | $-1$ | $0$ | $-1$ | $0$ | $0$ | $1$ | $0$
$0$ | $-1$ | $0$ | $0$ | $0$ | $-1$ | $0$ | $1$
Table 1. The generators of $R_{X}$ and their degrees in
$\mathbb{Z}^{5}\cong\operatorname{Pic}(X)$.

The irrelevant ideal is generated by the set $\\{\prod
g\mid\rho_{g}\not\subset\sigma\\}_{\sigma\in\Sigma^{\text{(max)}}}$; it is
thus

$\displaystyle I_{\mathrm{irr}}=($ $\displaystyle
a_{1}b_{1}c_{1}xy,\,a_{1}b_{0}c_{1}xy,\,a_{1}b_{1}c_{0}xy,\,a_{1}b_{0}c_{0}xy,$
$\displaystyle
a_{0}b_{1}c_{1}xy,\,a_{0}b_{0}b_{1}c_{1}y,\,a_{0}a_{1}b_{0}c_{1}y,\,a_{0}b_{1}c_{0}c_{1}x,$
$\displaystyle
a_{0}a_{1}b_{1}c_{0}x,\,a_{0}a_{1}b_{0}c_{0}y,\,a_{0}b_{0}b_{1}c_{0}c_{1},\,a_{0}a_{1}b_{0}b_{1}c_{0}),$

and we get a universal torsor $Y=\operatorname{Spec}R_{X}\setminus
V(I_{\mathrm{irr}})\to X$. The image of a point

$(a_{0},a_{1},b_{0},b_{1},c_{0},c_{1},x,y)\in Y(\mathbb{Q})$

is denoted by $(a_{0}\mathbin{\mathchar 58\relax}\nobreak
a_{1}\mathbin{\mathchar 58\relax}\nobreak b_{0}\mathbin{\mathchar
58\relax}\nobreak b_{1}\mathbin{\mathchar 58\relax}\nobreak
c_{0}\mathbin{\mathchar 58\relax}\nobreak c_{1}\mathbin{\mathchar
58\relax}\nobreak x\mathbin{\mathchar 58\relax}\nobreak y)\in Y(\mathbb{Q})$
(expressed in _Cox coordinates_), and is further mapped to
$((a_{0}\mathbin{\mathchar 58\relax}\nobreak
a_{1}xy),\,(b_{0}\mathbin{\mathchar 58\relax}\nobreak
b_{1}x),\,(c_{0}\mathbin{\mathchar 58\relax}\nobreak c_{1}y))\in
X_{0}(\mathbb{Q})$ by the blow-up morphism $\pi$.

######  Lemma 3.1.1.

The log anticanonical bundle is big, i.e., in the interior of the effective
cone, but it is not nef. It has the description
$\omega_{X}(D)^{\vee}\cong\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{\vee}$ as a
quotient of base point free bundles, where the class of $\mathcal{L}_{1}$ is
$(2,2,2,-2,-2)$, and the class of $\mathcal{L}_{2}$ is $(1,0,0,0,0)$ under the
above isomorphism $\operatorname{Pic}(X)\cong\mathbb{Z}^{5}$.

###### Proof.

In Cox coordinates, the exceptional divisors are defined by $E_{1}=V(x)$ and
$E_{2}=V(y)$ and the third component of the boundary is $M=V(a_{0})$. The log
anticanonical class $\omega_{X}(D)$ corresponds to

$(1,2,2,-2,-2)=\sum_{g\text{ generator of
}R_{X}}\deg(g)-\deg(x)-\deg(y)-\deg(a_{0})$

under the above isomorphism $\operatorname{Pic}(X)\cong\mathbb{Z}^{5}$. It is
not base point free, since $b_{1}c_{1}$ divides all of its global sections.
Since the same holds for all its multiples, it is not semi-ample and, as a
consequence, not nef, since the two notions coincide on toric varieties. It
is, however, big: the effective cone is generated by the degrees of the
generators of the Cox ring, and

$(1,2,2,-2,-2)=\tfrac{\deg(a_{0})+3\deg(a_{1})+\deg(b_{0})+7\deg(b_{1})+\deg(c_{0})+7\deg(c_{1})+2\deg(x)+2\deg(y)}{4}$

is in its interior. We shall use its description $(2,2,2,-2,-2)-(1,0,0,0,0)$
as a difference of base point free classes to construct a corresponding height
function. The sets

(15)
$\\{a_{1}^{2}b_{0}^{2}c_{0}^{2},a_{1}^{2}b_{1}^{2}c_{0}^{2}x^{2},a_{1}^{2}b_{0}^{2}c_{1}^{2}y^{2},a_{1}^{2}b_{1}^{2}c_{1}^{2}x^{2}y^{2},a_{0}^{2}b_{1}^{2}c_{1}^{2}\\}\text{
and }\\{a_{0},a_{1}xy\\}$

of elements of the Cox ring correspond to global sections of $\mathcal{L}_{1}$
and $\mathcal{L}_{2}$, respectively. Neither of these sets can vanish
simultaneously, so both classes are indeed base point free. ∎

These choices of sections induce metrics on the bundles $\mathcal{L}_{1}$,
$\mathcal{L}_{2}$, and
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{\vee}\cong\omega_{X}(D)^{\vee}$, which
in turn induce a log anticanonical height function.

######  Lemma 3.1.2.

There is a $4$-to-$1$-correspondence between the set of integral points
$\mathfrak{U}(\mathbb{Z})\cap T(\mathbb{Q})$ and the set

$\\{(1,a_{1},b_{1},b_{2},c_{1},c_{2},1,1)\in\mathbb{Z}_{\neq{}0}^{8}\mid\text{\eqref{eq:gcd}
holds}\\}\subset Y(\mathbb{Q}),$

where

(16) $\gcd(a_{1}b_{0}c_{0},a_{1}b_{0}c_{1},a_{1}b_{1}c_{0},b_{1}c_{1})=1.$

The log anticanonical height of the image of a point
$(1,a_{1},b_{0},b_{1},c_{0},c_{1},1,1)$ in the above set is

(17) $H(a_{1},b_{0},b_{1},c_{0},c_{1})=\left\lvert
a_{1}\right\rvert\max\\{\left\lvert b_{0}^{2}\right\rvert,\left\lvert
b_{1}^{2}\right\rvert\\}\max\\{\left\lvert c_{0}^{2}\right\rvert,\left\lvert
c_{1}^{2}\right\rvert\\}\text{.}$

###### Proof.

Consider an integral point as in the description (3), that is, a point

$P=\left((a_{0}:a_{1}),\,(b_{0}:b_{1}),\,(c_{0}:c_{1})\right)\in\mathfrak{U}(\mathbb{Z})\subset
X_{0}(\mathbb{Q})$

with $a_{0}\in\\{\pm 1\\}$ satisfying coprimality conditions that can be
checked to be equivalent to (16). Multiplying the first pair with $a_{0}$
eliminates the choice of sign in $a_{0}$; but multiplying any of the latter
two pairs with a unit does not change the integral point $P$, resulting in the
claimed $4$-to-$1$-correspondence. The point $P$ is the image of

(18) $(1,a_{1},b_{0},b_{1},c_{0},c_{1},1,1)\in Y(\mathbb{Q})$

by the description of the torsor and blow-up morphism before Lemma 3.1.1. Such
a point is in the open torus $T(\mathbb{Q})$ if and only if
$a_{1}b_{0}b_{1}c_{0}c_{1}\neq 0$.

The choice (15) of global sections of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$
induces metrics on these line bundles and, consequently, on the line bundle
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{\vee}$ isomorphic to the log
anticanonical bundle. The latter metric then induces a log anticanonical
height function. Its value on the image of a point
$(a_{0},a_{1},b_{0},b_{1},c_{0},c_{1},x,y)\in Y(\mathbb{Q})$ is

$\prod_{v}\ \frac{\max\left\\{\left\lvert
a_{1}^{2}b_{0}^{2}c_{0}^{2}\right\rvert_{v},\left\lvert
a_{1}^{2}b_{1}^{2}c_{0}^{2}x^{2}\right\rvert_{v},\left\lvert
a_{1}^{2}b_{0}^{2}c_{1}^{2}y^{2}\right\rvert_{v},\left\lvert
a_{1}^{2}b_{1}^{2}c_{1}^{2}x^{2}y^{2}\right\rvert_{v},\left\lvert
a_{0}^{2}b_{1}^{2}c_{1}^{2}\right\rvert_{v}\right\\}}{\max\left\\{\left\lvert
a_{0}\right\rvert_{v},\left\lvert a_{1}xy\right\rvert_{v}\right\\}}\text{.}$

For a point as in (18) and a finite place $p$, the denominator is $\left\lvert
a_{0}\right\rvert_{p}=1$, while the numerator is $1$ as a consequence of the
coprimality condition (16); hence, the factors associated with finite primes
are $1$. Using that $\left\lvert a_{0}\right\rvert=\left\lvert
x\right\rvert=\left\lvert y\right\rvert=1$ and $\left\lvert
a_{1}\right\rvert\geq 1$, the factor at the archimedean place can be
simplified to (17). ∎

######  Remark 3.1.3.

More formally, the fan $\Sigma_{X}$ also induces a _toric $\mathbb{Z}$-scheme_
$\mathfrak{X}$ and a universal torsor $\mathfrak{Y}\to\mathfrak{X}$ (cf.
[Sal98, p. 187 and Rem. 8.6 (b)], building on [Dem70]). The fiber above each
integral point is a $\mathbb{G}_{\mathrm{m},\mathbb{Z}}^{5}$-torsor, the
isomorphism classes of which are parametrized by
$H^{1}_{\mathrm{fppf}}(\operatorname{Spec}\mathbb{Z};\mathbb{G}_{\mathrm{m},\mathbb{Z}}^{5})=\operatorname{Cl}(\mathbb{Q})^{5}$;
as this group is trivial, each fiber is isomorphic to
$\mathbb{G}_{\mathrm{m},\mathbb{Z}}^{5}$, which has $2^{5}$ integral points.
The set (18) of points in $Y(\mathbb{Q})$ then coincides with
$(\mathfrak{Y}\setminus V(a_{0}xy))(\mathbb{Z})$ up to fixing the three signs
$a_{0},x,y\in\\{\pm 1\\}$, yielding a $4$-to-$1$-correspondence to integral
points on the model $\mathfrak{X}\setminus\overline{D}$ of $U$, which can be
checked to coincide with $\mathfrak{U}$.

###  3.2. Counting

In other words, we now have a new description

$N(B)=\frac{1}{4}\\#\\{(a_{1},b_{0},b_{1},c_{0},c_{1})\in\mathbb{Z}_{\neq{}0}^{5}\mid
H(a_{1},b_{0},b_{1},c_{0},c_{1})\leq B\text{, (\ref{eq:gcd}) holds}\\}$

of the counting function, with the height function $H$ in (17).

######  Lemma 3.2.1.

We have

$N(B)=\prod_{p}\left(\left(1-\frac{1}{p}\right)^{2}\left(1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}\right)\right)V(B)+O(B\log
B(\log\log B)^{3})$

with

$V(B)=\frac{1}{4}\int_{\begin{subarray}{c}\left\lvert
a_{1}\right\rvert,\left\lvert b_{0}\right\rvert,\left\lvert
b_{1}\right\rvert,\left\lvert c_{0}\right\rvert,\left\lvert
c_{1}\right\rvert\geq 1,\\\ \left\lvert a_{1}\right\rvert\max\\{\left\lvert
b_{0}^{2}\right\rvert,\left\lvert b_{1}^{2}\right\rvert\\}\max\\{\left\lvert
c_{0}^{2}\right\rvert,\left\lvert c_{1}^{2}\right\rvert\\}\leq
B\end{subarray}}\mathop{}\\!\mathrm{d}a_{1}\mathop{}\\!\mathrm{d}b_{0}\mathop{}\\!\mathrm{d}b_{1}\mathop{}\\!\mathrm{d}c_{0}\mathop{}\\!\mathrm{d}c_{1}\text{.}$

###### Proof.

The counting problem can be rephrased as

$N(B)=\frac{1}{4}\sum_{\begin{subarray}{c}a_{1},b_{0},b_{1},c_{0},c_{1}\in\mathbb{Z}_{\neq{}0}\\\
H(a_{1},b_{0},b_{1},c_{0},c_{1})\leq
B\end{subarray}}\theta(a_{1},b_{0},b_{1},c_{0},c_{1})\text{,}$

where
$\theta=\delta_{\gcd(a_{1}b_{0}c_{0},a_{1}b_{0}c_{1},a_{1}b_{1}c_{0},b_{1}c_{1})=1}=\prod_{p}\theta^{(p)}$
with

$\theta^{(p)}(a_{1},b_{0},b_{1},c_{0},c_{1})=\begin{cases}0,&\text{if }p\mid
a_{1}b_{0}c_{0},a_{1}b_{0}c_{1},a_{1}b_{1}c_{0},b_{1}c_{1},\\\
1,&\text{else.}\end{cases}$

Aiming to first replace the sum over $b_{0}$ by an integral, observe that the
height conditions imply that

$\left\lvert a_{1}b_{0}^{2}c_{0}^{2}\right\rvert,\left\lvert
a_{1}b_{1}^{2}c_{0}^{2}\right\rvert,\left\lvert
a_{1}b_{0}b_{1}c_{1}^{2}\right\rvert\leq B\text{,}$

since the latter one is the geometric average of two terms in the height
function. We have

$\displaystyle 1=\frac{B}{\left\lvert
a_{1}b_{0}b_{1}c_{0}c_{1}\right\rvert}\left(\frac{B}{\left\lvert
a_{1}b_{0}^{2}c_{0}^{2}\right\rvert}\right)^{-1/4}\left(\frac{B}{\left\lvert
a_{1}b_{1}^{2}c_{0}^{2}\right\rvert}\right)^{-1/4}\left(\frac{B}{\left\lvert
a_{1}b_{0}b_{1}c_{1}^{2}\right\rvert}\right)^{-1/2},$

and note that the function $\theta$ satisfies Definition 7.9 in [Der09]. Using
[Der09, Prop. 3.9] with $r=1$ and $s=3$, we get

$N(B)=\sum_{\begin{subarray}{c}a_{1},b_{1},\\\
c_{0},c_{1}\in\mathbb{Z}_{\neq{}0}\end{subarray}}\theta_{1}(a_{1},b_{1},c_{0},c_{1})V_{1}(a_{1},b_{1},c_{0},c_{1};B)+O(B\log
B(\log\log B)^{3})\text{,}$

where
$V_{1}(a_{1},b_{1},c_{0},c_{1};B)=\frac{1}{4}\int_{\begin{subarray}{c}\left\lvert
b_{0}\right\rvert\geq 1\\\ H(a_{0},b_{0},b_{1},c_{0},c_{1})\leq
B\end{subarray}}\mathop{}\\!\mathrm{d}b_{0}$ and
$\theta_{1}=\prod_{p}\theta_{1}^{(p)}$ with

$\theta_{1}^{(p)}(a_{1},b_{1},c_{0},c_{1})=\begin{cases}0,&\text{ if }p\mid
a_{1}c_{0},a_{1}c_{1},b_{1}c_{1},\\\ 1-\frac{1}{p},&\text{ if }p\mid
b_{1},p\nmid a_{1}\text{ and }(p\nmid c_{0}\text{ or }p\nmid c_{1}),\\\
1,&\text{ if }p\nmid b_{1}\text{ and }(p\nmid c_{1}\text{ or }p\nmid
a_{1}c_{0})\text{.}\end{cases}$

Using the geometric average of the two height conditions involving $b_{0}$, we
can bound $V_{1}$ by

$V_{1}(a_{1},b_{1},c_{0},c_{1};B)\ll\sqrt{\frac{B}{\left\lvert
a_{1}c_{0}c_{1}\right\rvert}}=\frac{B}{\left\lvert
a_{1}b_{1}c_{0}c_{1}\right\rvert}\left(\frac{B}{\left\lvert
a_{1}b_{1}^{2}c_{0}^{2}\right\rvert}\right)^{-1/4}\left(\frac{B}{\left\lvert
a_{1}b_{1}^{2}c_{1}^{2}\right\rvert}\right)^{-1/4}\text{.}$

Since $\left\lvert a_{1}b_{1}^{2}c_{0}^{2}\right\rvert$ and $\left\lvert
a_{1}b_{1}^{2}c_{1}^{2}\right\rvert$ are bounded by $B$, applying [Der09,
Prop. 3.9] once more (with $r=1$, $s=2$) yields

$N(B)=\sum_{a_{1},b_{1},c_{1}\in\mathbb{Z}_{\neq{}0}}\theta_{2}(a_{1},b_{1},c_{1})V_{2}(a_{1},b_{1},c_{1};B)+O(B\log
B(\log\log B)^{3})\text{,}$

where
$V_{2}(a_{1},b_{1},c_{1};B)=\frac{1}{4}\int_{\begin{subarray}{c}\left\lvert
b_{0}\right\rvert,\left\lvert c_{0}\right\rvert\geq 1\\\
H(a_{0},b_{0},b_{1},c_{0},c_{1})\leq
B\end{subarray}}\mathop{}\\!\mathrm{d}(b_{0},c_{0})$ and
$\theta_{2}=\prod_{p}\theta_{2}^{(p)}$ with

$\theta_{2}^{(p)}(a_{1},b_{1},c_{0},c_{1})=\begin{cases}0,&\text{if }p\mid
a_{1},b_{1}c_{1}\\\ \left(1-\frac{1}{p}\right)^{2},&\text{if }p\mid
b_{1}c_{1}\text{, }p\nmid a_{1}\\\ 1-\frac{1}{p},&\text{if }p\mid b_{1}\text{,
}p\nmid a_{1}c_{1}\\\ 1-\frac{1}{p},&\text{if }p\mid c_{1}\text{, }p\nmid
a_{1}b_{1}\\\ 1,&\text{if }p\nmid b_{1}c_{1}\text{.}\end{cases}$

To complete the summations, we use the fact that the height conditions imply
$\left\lvert a_{1}b_{0}^{2}c_{0}c_{1}\right\rvert\leq B$, and get an upper
bound

$\displaystyle V_{2}(a_{1},b_{1},c_{1};B)$
$\displaystyle\ll\int_{\begin{subarray}{c}\left\lvert c_{0}\right\rvert\geq
1\\\ \left\lvert a_{1}b_{1}^{2}c_{0}^{2}\right\rvert\leq
B\end{subarray}}\sqrt{\frac{B}{\left\lvert
a_{1}c_{0}c_{1}\right\rvert}}\mathop{}\\!\mathrm{d}c_{0}\ll\frac{B^{3/4}}{\left\lvert
a_{1}\right\rvert^{3/4}\left\lvert b_{1}\right\rvert^{1/2}\left\lvert
c_{1}\right\rvert^{1/2}}$ $\displaystyle=\frac{B}{\left\lvert
a_{1}b_{1}c_{1}\right\rvert}\left(\frac{B}{\left\lvert
a_{1}b_{1}^{2}c_{1}^{2}\right\rvert}\right)^{-1/4}$

for $V_{1}$. Since $\left\lvert a_{1}b_{1}^{2}c_{1}^{2}\right\rvert\leq B$,
Proposition 4.3 in [Der09] yields the desired result, for which are only left
to check that the constant is indeed $\prod_{p}c_{p}$ with

$\displaystyle c_{p}$
$\displaystyle=\frac{1}{p^{2}}\left(1-\frac{1}{p}\right)\left(1-\frac{1}{p}\right)^{2}+2\frac{1}{p}\left(1-\frac{1}{p}\right)^{2}\left(1-\frac{1}{p}\right)+\left(1-\frac{1}{p}\right)^{2}$
$\displaystyle=\left(1-\frac{1}{p^{2}}\right)\left(1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}\right).\qed$

###### Proof of Theorem 1.0.1.

We are only left to provide an asymptotic expansion of $V(B)$. The error we
introduce when removing the condition $\left\lvert a_{1}\right\rvert\geq 1$ in
the integral, while keeping the condition $\max\\{\left\lvert
b_{0}^{2}\right\rvert,\left\lvert b_{1}^{2}\right\rvert\\}\max\\{\left\lvert
c_{0}^{2}\right\rvert,\left\lvert c_{1}^{2}\right\rvert\\}\leq B$ implied by
the others, is at most

$2\int_{\begin{subarray}{c}\left\lvert c_{0}\right\rvert,\left\lvert
c_{1}\right\rvert\geq 1,\\\ \max\\{\left\lvert
b_{0}^{2}\right\rvert,\left\lvert b_{1}^{2}\right\rvert\\}\max\\{\left\lvert
c_{0}^{2}\right\rvert,\left\lvert c_{1}^{2}\right\rvert\\}\leq
B\end{subarray}}\mathop{}\\!\mathrm{d}b_{0}\mathop{}\\!\mathrm{d}b_{1}\mathop{}\\!\mathrm{d}c_{0}\mathop{}\\!\mathrm{d}c_{1}\
\ll\ \int_{\left\lvert c_{0}\right\rvert,\left\lvert c_{1}\right\rvert\geq
1}\frac{B}{\max\\{\left\lvert c_{0}^{2}\right\rvert,\left\lvert
c_{1}^{2}\right\rvert\\}}\mathop{}\\!\mathrm{d}c_{0}\mathop{}\\!\mathrm{d}c_{1}\
\ll\ B\log B\text{.}$

Using the symmetry in the integral, we get

$V(B)=\int_{\begin{subarray}{c}\left\lvert b_{0}\right\rvert,\left\lvert
b_{1}\right\rvert,\left\lvert c_{0}\right\rvert,\left\lvert
c_{1}\right\rvert\geq 1\\\ \left\lvert b_{0}\right\rvert\leq\left\lvert
b_{1}\right\rvert,\left\lvert c_{0}\right\rvert\leq\left\lvert
c_{1}\right\rvert,\\\ \left\lvert b_{1}^{2}c_{1}^{2}\right\rvert\leq
B\end{subarray}}\frac{B}{\left\lvert b_{1}^{2}\right\rvert\left\lvert
c_{1}^{2}\right\rvert}\mathop{}\\!\mathrm{d}b_{0}\mathop{}\\!\mathrm{d}b_{1}\mathop{}\\!\mathrm{d}c_{0}\mathop{}\\!\mathrm{d}c_{1}+O(B\log
B)\text{.}$

Removing $\left\lvert b_{0}\right\rvert\geq 1$ introduces an error of at most

$2\int_{\begin{subarray}{c}\left\lvert b_{1}\right\rvert,\left\lvert
c_{1}\right\rvert\geq 1\\\ \left\lvert c_{0}\right\rvert\leq\left\lvert
c_{1}\right\rvert\leq\sqrt{B}/\left\lvert
b_{1}\right\rvert\end{subarray}}\frac{B}{\left\lvert
b_{1}^{2}\right\rvert\left\lvert
c_{1}^{2}\right\rvert}\mathop{}\\!\mathrm{d}b_{1}\mathop{}\\!\mathrm{d}c_{0}\mathop{}\\!\mathrm{d}c_{1}\ll\int_{1\leq\left\lvert
c_{1}\right\rvert\leq B}\frac{B}{\left\lvert
c_{1}\right\rvert}\mathop{}\\!\mathrm{d}c_{1}\ll B\log B\text{,}$

as, analogously, does removing $\left\lvert c_{0}\right\rvert\geq 1$. We thus
arrive at

$\displaystyle V(B)$ $\displaystyle=4\int_{\begin{subarray}{c}\left\lvert
b_{1}\right\rvert,\left\lvert c_{1}\right\rvert\geq 1,\\\ \left\lvert
b_{1}\right\rvert\leq\sqrt{B}/\left\lvert
c_{1}\right\rvert\end{subarray}}\frac{B}{\left\lvert
b_{1}\right\rvert\left\lvert
c_{1}\right\rvert}\mathop{}\\!\mathrm{d}b_{1}\mathop{}\\!\mathrm{d}c_{1}+O(B\log
B)$ $\displaystyle=4\int_{1\leq\left\lvert
c_{1}\right\rvert\leq\sqrt{B}}\frac{B\log B}{\left\lvert
c_{1}\right\rvert}+O(B\log B)=4B(\log B)^{2}+O(B\log B)\text{.}\qed$

###  3.3. Interpretation of the result

As $E_{1}$ and $E_{2}$ intersect and this intersection has real points, while
$M$ meets neither of the exceptional divisors, the analytic Clemens complex of
$D$ consists of a $1$-simplex $A=\\{E_{1},E_{2}\\}$ and an isolated vertex
$\\{M\\}$ (which we will also simply denote by $M$). Integral points tend to
accumulate around the boundary divisor; their number is dominated by those
points lying near the intersection of a maximal number of boundary components.
It is for this reason that the dimension of the Clemens complex is part of the
exponent in the main theorem of [CLT10b].

$M$$E_{1}$$E_{2}$$A$ Figure 4. The Clemens complex of $D$.

For the toric variety $X$, this does not hold. There is an obstruction to the
existence of points near the intersection $E_{1}\cap E_{2}$ (and even to the
existence of integral points near $E_{1}\cup E_{2}$): Let us consider the
rational function $f=a_{1}xy/a_{0}$ (in fact, a character of $T$) on $X$. It
is a nonconstant regular function on $U_{A}=X\setminus H$, so there is an
obstruction in the sense of Corollary 2.3.2.

Concretely, this means the following: The function $f$ is a regular in a
neighborhood of $E_{1}\cap E_{2}$, vanishing on $E_{1}\cap E_{2}$. If a point
$p$ is near $E_{1}\cap E_{2}$, $\left\lvert f(p)\right\rvert$ should thus be
small. However, since $f$ is a regular function on $\mathfrak{U}$, its value
is an integer at any integral point—and thus $\left\lvert f(p)\right\rvert\geq
1$ except for points on the subvariety $\\{f=0\\}$. This means that the only
integral points that are close to $E_{1}\cap E_{2}$ can be points on this
subvariety (which we excluded in our counting problem). For this reason we
cannot expect a contribution of the maximal face $A$ of the Clemens complex to
our asymptotic formula. Since $f$ is even regular on neigborhoods of both
$E_{1}$ and $E_{2}$, there can in fact be no integral points near either of
those divisors, and we cannot expect a contribution of those two nonmaximal
faces (a general phenomenon by Remark 2.3.4). The existence of this function
also has an effect on the Picard group. That $f$ vanishes on $E_{1}$, $E_{2}$,
and $M^{\prime}=V(a_{1})$, and that it has a pole on $M$ means that we have
$[E_{1}]+[E_{2}]+[M^{\prime}]=[M]$ in $\operatorname{Pic}(X)$, and thus
$[E_{1}]+[E_{2}]+[M^{\prime}]=0$ in $\operatorname{Pic}(X\setminus M)$. All
three classes are nontrivial, hence the effective cone of $X\setminus M$
contains a plane. It is thus not strictly convex, and its characteristic
function is identically $0$.

Since a value of the characteristic function is a factor of the leading
constant in op. cit., this means that, for this variety, the leading constant
is zero, contrary to their claim in Lemma 3.11.4. In particular, this variety
is an example for the obstruction in 2.3, and, more precisely, the situation
considered in Theorem 2.4.1 (i). The exponent of $\log B$ in Proposition 1.0.1
is one less than the one given by Chambert-Loir and Tschinkel. We can however
interpret our asymptotic formula analogously to the formula given by Chambert-
Loir and Tschinkel: There is no obstruction at the only remaining maximal face
$M$ of the Clemens complex. Substituting this face for the maximal dimensional
face $A$ of the Clemens complex, we get the correct asymptotic formula.
Summarizing, the situation is as follows:

######  Proposition 3.3.1.

  1. (i)

The cone $\operatorname{Eff}_{A}=\overline{\operatorname{Eff}}_{X\setminus
M}\subset\operatorname{Pic}(X\setminus M)_{\mathbb{R}}$, associated with the
unique maximal-dimensional face $A$ of the Clemens complex, is not strictly
convex.

  2. (ii)

The cone $\operatorname{Eff}_{M}$, associated with the unique other maximal
face $M$, is strictly convex. The constant associated with this face is
$\alpha_{M}=1/8$, and the exponent associated with it is
$b_{M}=b^{\prime}_{M}=3$.

###### Proof.

The Picard group $\operatorname{Pic}(U;A)=\operatorname{Pic}(X\setminus M)$ is
the quotient

$\operatorname{Pic}(X)/[M]\cong\mathbb{Z}^{5}/\langle(1,0,0,0,0)\rangle\cong\mathbb{Z}^{4}.$

The effective cone is generated by the classes of the torus-invariant prime
divisors

$\displaystyle(0,0,-1,-1),\ (1,0,0,0),\ (1,0,-1,0),\ (0,1,0,0),$
$\displaystyle(0,1,0,-1),\ (0,0,1,0),\ \text{and }(0,0,0,1),$

and thus contains the plane $\\{(0,0,x,y)\mid x,y\in\mathbb{R}\\}$; in
particular, it is not strictly convex.

The Picard group $\operatorname{Pic}(U;H)=\operatorname{Pic}(U_{M})$ for
$U_{M}=X\setminus(E_{1}\cup E_{2})$ is the quotient

$\operatorname{Pic}(X)/\langle[E_{1}],[E_{2}]\rangle\cong\mathbb{Z}^{5}/\left\langle(0,0,0,1,0),(0,0,0,0,1)\right\rangle\cong\mathbb{Z}^{3}\text{.}$

Its rank is $b_{M}^{\prime}=3$, so it coincides with

$b_{M}=\operatorname{rk}\operatorname{Pic}(U)-\operatorname{rk}E(U)+\\#\\{M\\}=2-0+1.$

The effective cone $\operatorname{Eff}_{M}=\operatorname{Eff}_{U_{M}}$ is
smooth and generated by

$(1,0,0),\ (0,1,0),\ \text{and }(0,0,1).$

The image of the log anticanonical class in this quotient is $(1,2,2)$. The
characteristic function of $\operatorname{Eff}_{M}$ thus evaluates to $1/4$,
and

$\alpha_{M}=\frac{1}{(b_{M}^{\prime}-1)!}\frac{1}{4}=\frac{1}{8}.\qed$

######  Remark 3.3.2.

This exemplifies a gap in the proof of [CLT10b, Lem. 3.11.4] of which the
authors were already aware and because of which they no longer believed in the
correctness of the final result of their preprint: they do not check that the
characteristic function
$\mathscr{X}_{\Lambda^{\prime}_{A}}(\pi(\tilde{\lambda}))$ (in the notation of
op. cit., equal to
$\mathcal{X}_{\operatorname{Eff}_{A}}(\pi(\omega_{X}(D)^{\vee}))=(b^{\prime}_{A}-1)!\alpha_{A}$
in our notation) is nonzero, and this example demonstrates that it can be
zero, making the leading constant 0. Note that the formula in op. cit. is
still correct if interpreted as $N(B)=0\cdot B(\log B)^{3}+O(B(\log B)^{2})$,
that is, as an upper bound. Here, we prove an asymptotic formula, of the form
$N(B)\sim cB(\log B)^{2}$.

To finish the proof of Theorem 3.0.1, we are only left to compute the relevant
Tamagawa volumes. To this end, consider the chart

$\displaystyle X\setminus V(a_{1}b_{1}c_{1}xy)\quad$
$\displaystyle\to\quad\qquad\mathbb{A}^{3},$
$\displaystyle(a_{0}:a_{1}:b_{0}:b_{1}:c_{0}:c_{1}:x:y)$
$\displaystyle\mapsto\left(\frac{a_{0}}{a_{1}xy},\frac{b_{0}}{b_{1}x},\frac{c_{0}}{c_{1}y}\right)$

and its inverse $\mathbb{A}^{3}\to X$

$(a,b,c)\mapsto(a:1:b:1:c:1:1:1).$

######  Lemma 3.3.3.

Under this chart, the integral points $\mathfrak{U}(\mathbb{Z}_{p})$
correspond to

$\\{(a,b,c)\in\mathbb{Z}_{p}^{3}\mid\text{either }\left\lvert
a\right\rvert=1,\text{ or }\left\lvert a\right\rvert>1\text{ and }\left\lvert
b\right\rvert,\left\lvert c\right\rvert\leq 1\\}\text{.}$

###### Proof.

Analogously to (18), $\mathbb{Z}_{p}$-integral points are the images of
$(1,a_{1},b_{0}:,b_{1},c_{0},c_{1},1,1)\in Y(\mathbb{Q}_{p})$ for
$a_{1},\dots,c_{1}\in\mathbb{Z}_{p}$ such that (16) holds. In particular,
$|a|=|1/a_{1}|\geq 1$. If $|a|>1$, then $|a_{1}|<1$; the coprimality
conditions then imply $b_{1}c_{1}\in\mathbb{Z}_{p}^{\times}$, and thus
$|b|=|b_{0}|,|c|=|c_{0}|\leq 1$.

On the other hand, let $(a,b,c)$ be a point in the above set. If $|a|=1$, let
$a_{1}$ and $a_{0}=a^{-1}$. If $|b|\leq 1$, let $b_{0}=b$ and $b_{1}=1$, else,
let $b_{0}=1$ and $b_{1}=b^{-1}$, and set $c_{0}$, $c_{1}$ analogously.
Finally, if $|a|>1$, let $a_{0}=a$, $b_{0}=b$, $c_{0}=c$, and the remaining
coordinates be $1$. In each case,
$((a_{0}:a_{1}),(b_{0}:b_{1}),(c_{0}:c_{1}))$ is an integral point that maps
to $(a,b,c)$. ∎

######  Lemma 3.3.4.

We have

$\tau_{M,\infty}(M(\mathbb{R}))=16\quad\text{and}\quad\tau_{U,p}(\mathfrak{U}(\mathbb{Z}_{p}))=1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}$

for all primes $p$.

###### Proof.

In order to get a metric on the canonical bundle inducing the Tamagawa
measures, consider the isomorphism from $\omega_{X}$ to the bundle
$\mathcal{L}_{\omega_{X}}$ whose sections are elements of degree $\omega_{X}$
in the Cox ring that maps
$\mathop{}\\!\mathrm{d}a\wedge\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c$
to $a_{1}^{-2}b_{1}^{-2}c_{1}^{-2}x^{-1}y^{-1}$; then pull back the metric
along this isomorphism. For the archimedean volume, we want to integrate

$\left\lVert
1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{M}(E_{1}+E_{2})}=\left\lVert
a^{-1}1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}a\wedge\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{X}(D)}$

over $M(\mathbb{R})$ (regarding $a^{-1}$ as an element in
$\Gamma(U,\mathcal{O}(-M))\subset\Gamma(U,\mathcal{K}_{\mathbb{A}^{3}})$).
Outside $M$, we have $a^{-1}=a^{-1}1_{M}$, where the first factor is a section
in $\Gamma(\mathbb{A}^{3}\setminus M,\mathcal{O}_{\mathbb{A}^{3}})$, and thus

$\left\lVert
1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{M}(E_{1}+E_{2})}=\lim_{a\to
0}\left(\left\lvert a^{-1}\right\rvert\left\lVert
1_{M}1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}a\wedge\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{X}(D)}\right).$

The norm $\left\lVert
1_{M}1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}a\wedge\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{X}(D)}$
is

$\frac{\max\\{\left\lvert a_{0}\right\rvert,\left\lvert
a_{1}xy\right\rvert\\}}{\left\lvert a_{0}xy\right\rvert\max\\{\left\lvert
a_{1}^{2}b_{0}^{2}c_{0}^{2}\right\rvert,\left\lvert
a_{1}^{2}b_{1}^{2}c_{0}^{2}x^{2}\right\rvert,\left\lvert
a_{1}^{2}b_{0}^{2}c_{1}^{2}y^{2}\right\rvert,\left\lvert
a_{1}^{2}b_{1}^{2}c_{1}^{2}x^{2}y^{2}\right\rvert,\left\lvert
a_{0}^{2}b_{1}^{2}c_{1}^{2}\right\rvert\\}}$

at a point $(a_{0}\mathbin{\mathchar 58\relax}\nobreak a_{1}\mathbin{\mathchar
58\relax}\nobreak b_{0}\mathbin{\mathchar 58\relax}\nobreak
b_{1}\mathbin{\mathchar 58\relax}\nobreak c_{0}\mathbin{\mathchar
58\relax}\nobreak c_{1}\mathbin{\mathchar 58\relax}\nobreak
x\mathbin{\mathchar 58\relax}\nobreak y)\in X(\mathbb{Q})$ given in Cox
coordinates. Evaluating in the image of a point $(a,b,c)$ yields

$\left\lVert\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert=\lim_{a\to
0}\frac{\left\lvert a\right\rvert\max\\{1,\left\lvert
a\right\rvert\\}}{\left\lvert a\right\rvert\max\\{\left\lvert
b^{2}c^{2}\right\rvert,\left\lvert c^{2}\right\rvert,\left\lvert
b^{2}\right\rvert,1,\left\lvert
a^{2}\right\rvert\\}}=\frac{1}{\max\\{1,\left\lvert
b\right\rvert^{2}\\}\max\\{1,\left\lvert c\right\rvert^{2}\\}}$

Integrating this results in the archimedean Tamagawa volume

$\tau_{(M,E_{1}+E_{2}),\infty}(M(\mathbb{R}))=\int_{\mathbb{R}^{2}}\frac{1}{\max\\{1,\left\lvert
b\right\rvert^{2}\\}\max\\{1,\left\lvert
c\right\rvert^{2}\\}}\mathop{}\\!\mathrm{d}b\mathop{}\\!\mathrm{d}c=16,$

which has to be renormalized with the factor $c_{\mathbb{R}}=2$.

In order to determine the Tamagawa volumes at the nonarchimedean places, we
integrate

$\left\lVert
1_{M}1_{E_{1}}1_{E_{2}}\mathop{}\\!\mathrm{d}a\wedge\mathop{}\\!\mathrm{d}b\wedge\mathop{}\\!\mathrm{d}c\right\rVert_{\omega_{X}(D)}$

over $\mathfrak{U}(\mathbb{Z}_{p})$. Using the same description as above, this
results in

$\left(1-\frac{1}{p}\right)\int_{b,c\in\mathbb{Q}_{p}}\frac{1}{\max\\{1,\left\lvert
b\right\rvert^{2}\\}\max\\{1,\left\lvert
c\right\rvert^{2}\\}}\mathop{}\\!\mathrm{d}b\mathop{}\\!\mathrm{d}c+\int_{\begin{subarray}{c}\left\lvert
a\right\rvert>1\\\ \left\lvert b\right\rvert,\left\lvert c\right\rvert\leq
1\end{subarray}}\frac{1}{\left\lvert
a\right\rvert^{2}}\mathop{}\\!\mathrm{d}a\mathop{}\\!\mathrm{d}b\mathop{}\\!\mathrm{d}c.$

The first integral is

$\left(\int_{b\in\mathbb{Q}_{p}}\frac{1}{\max\\{1,\left\lvert
b\right\rvert^{2}\\}}\mathop{}\\!\mathrm{d}b\right)^{2}=\left(1+\int_{\left\lvert
b\right\rvert>1}\frac{1}{\left\lvert
b\right\rvert^{2}}\mathop{}\\!\mathrm{d}b\right)^{2}=\left(1+\frac{1}{p}\right)^{2}\text{,}$

and the second is

$\int_{\left\lvert a\right\rvert>1}\frac{1}{\left\lvert
a\right\rvert^{2}}\mathop{}\\!\mathrm{d}a=\frac{1}{p}\text{,}$

so, in total, we get

$\tau_{U,p}(\mathfrak{U}(\mathbb{Z}_{p}))=\left(1-\frac{1}{p}\right)\left(1+\frac{1}{p}\right)^{2}+\frac{1}{p}=1+\frac{2}{p}-\frac{1}{p^{2}}-\frac{1}{p^{3}}\text{.}\qed$

###### Proof of Theorem 3.0.1.

Comparing $\alpha_{\boldsymbol{A}}$ as computed in Proposition 3.3.1 and the
descriptions of the Tamagawa volumes in Lemma 3.3.4 to the asymptotic formula
in Theorem 1.0.1 finishes the proof. ∎

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