# Sparsity of integral points on moduli spaces of varieties

Jordan S. Ellenberg, Brian Lawrence, and Akshay Venkatesh

###### Abstract.

Let $X$ be a quasi-projective variety over a number field, admitting (after
passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period
mapping has zero-dimensional fibers. Then the integral points of $X$ are
sparse: the number of such points of height $\leq B$ grows slower than any
positive power of $B$.

For example, homogeneous integral polynomials in a fixed number of variables
and degree, with discriminant divisible only by a fixed set of primes, are
sparse when considered up to integral linear substitutions.

##  1\. Introduction

Let $K\subset\mathbb{C}$ be a finite field extension of the rational numbers,
and $S$ a finite set of primes of $K$. We will consider $S$-integral points on
quasi-projective $K$-varieties $X^{\circ}\subset\mathbb{P}^{m}$.

More precisely: in this situation we will write $X$ for the Zariski closure of
$X^{\circ}$ in $\mathbb{P}^{m}$, $\mathcal{L}=\mathcal{O}(1)$ the associated
hyperplane bundle, and $Z$ for $X\backslash X^{\circ}$. After choosing a good
integral model of $X$ and $Z$ (see §1.2 for details) we obtain a notion of
“$S$-integral point of $X^{\circ}$,” and the projective embedding allows us to
refer to the “height” of such a point. (By “height” we always mean the
_multiplicative_ height.) We will denote by $X_{\mathbb{C}}$ the complex
variety obtained by base extension via the fixed embedding
$K\hookrightarrow\mathbb{C}$, by $X(\mathbb{C})$ its complex points, and by
$X^{{\text{an}}}$ its complex analytification.

######  Theorem 1.1.

Let $X^{\circ}\subset\mathbb{P}^{m}$ be a quasi-projective variety over $K$
such that $X^{\circ}_{\mathbb{C}}$ admits a geometric variation of Hodge
structure, whose associated period map (see [11] for definitions) is locally
finite-to-one, i.e. has zero-dimensional fibers. Then integral points on
$X^{\circ}$ are sparse, in the sense that

(1) $\\#\\{x\text{ an $S$-integral point of $X^{\circ}$, of (multiplicative)
height at most $B$}\\}=O_{\epsilon}(B^{\epsilon}).$

By “geometric variation of Hodge structure” we mean a direct summand of a VHS
arising from a smooth projective family over $X^{\circ}_{\mathbb{C}}$; in
particular the Hodge structures arising here are pure. In the situation of the
Theorem, the “period map” is understood to be the complex-analytic map
$\widetilde{X^{\circ{\text{an}}}}\rightarrow D$ classifying the VHS, where
$\widetilde{X^{\circ{\text{an}}}}$ is the universal cover of
$X^{\circ{\text{an}}}$ and $D$ is the period domain associated to the varying
Hodge structures.

We note that the condition we are imposing on $X^{\circ}$ depends only on the
complex algebraic variety $X^{\circ}_{\mathbb{C}}$ and not on its rational
form over $K$.

The notation $O_{\epsilon}(B^{\epsilon})$ is that of analytic number theory:
for each $\epsilon>0$ there exists $c_{\epsilon}$ such that the left hand side
is at most $c_{\epsilon}B^{\epsilon}$.

Theorem 1.1 will be deduced from the following more general theorem.

######  Theorem 1.2.

Let $\pi:\mathfrak{X}\rightarrow X^{\circ}$ be a projective smooth morphism of
$K$-varieties, and (for some $i\geq 0$) let $\Phi$ be the period map
associated to the variation of Hodge structure
$\mathsf{V}=R^{i}\pi_{*}\mathbb{Q}$ on $(X^{\circ})^{{\text{an}}}$.

Then the $S$-integral points of $X^{\circ}$ with height at most $B$ are
covered by $O_{\epsilon}(B^{\epsilon})$ geometrically irreducible
$K$-varieties, each lying in a single fiber of $\Phi$.

We have abused language in the statement: for an irreducible analytic
subvariety $V\subset X^{\circ{\text{an}}}$ we shall say that $V$ “lies in a
fiber of $\Phi$” if some component (hence every component) of the preimage of
$V$ in the universal cover lies in such a fiber. We will use similar language
elsewhere in the paper – if a property of the period map is local, we will
phrase it on $X^{\circ}$ rather than the universal cover.

The bound (1) applies to many natural moduli spaces of varieties. For
instance, one may take $X$ to be the projective space parametrizing
hypersurfaces of a given degree and dimension, and $Z$ the locus of singular
hypersurfaces, and obtain the following corollary, which we will derive from
Theorem 1.1 in §3.

######  Corollary 1.3.

Fix $n\geq 2$ and $d\geq 3$ and a finite set $S$ of rational primes.

Then the $S$-integral homogeneous degree $d$ polynomials $P=\sum a_{i_{1}\dots
i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$ in $n$ variables with $S$-integer
discriminant and $\max_{i,v\in S}|a_{i}|_{v}\leq B$ lie in at most
$O_{S,d,n,\epsilon}(B^{\epsilon})$ orbits of the group of integral linear
substitutions $\operatorname{GL}_{n}(\mathbb{Z}[\frac{1}{S}])$.

Similarly the set of such integral polynomials (i.e. with $\mathbb{Z}$
coefficients), with discriminant exactly equal to a nonzero integer
$N\in\mathbb{Z}$, and $\max|a_{i}|\leq B$, lie in at most
$O_{N,d,n,\epsilon}(B^{\epsilon})$ orbits of the group of unimodular integral
linear substitutions $\operatorname{SL}_{n}(\mathbb{Z})$.

Recall that the discriminant of a homogeneous polynomial is a certain
homogeneous polynomial in the coefficients of $P$ which vanishes precisely
when the hypersurface defined by $P=0$ is singular; see [25] for more
discussion of its significance.

In the paper [21] of the second- and third- named author, it is proved by a
much more complicated argument that for “large enough” $n,d$ the set of
polynomials with $S$-integer discriminant is not Zariski dense in the ambient
affine hypersurface (which of course neither implies nor is implied by what we
prove here). It is expected ([16, Conj. 1.4]) that this set is finite as part
of a “Shafarevich-type” statement about hypersurfaces with good reduction; in
fact, if one assumes the Lang-Vojta conjecture, then results on the
hyperbolicity of period domains imply finiteness of $S$-integral points in the
general context of Theorem 1.2; see [30] and [16, Thm. 1.5].111 The theorem in
loc. cit. is stated only for complete intersections, but the proof applies in
the generality of our Theorem 1.2. We note that the Shafarevich assertion can
hold even in cases where there is no Torelli theorem; for instance, it holds
for the case $(n,d)=(4,3)$ of cubic surfaces, by a result of Scholl [27]. In
addition to cubic surfaces, the Shafarevich assertion is also known for cubic
and quartic threefolds [16].

###  1.1. Discussion of the proof

The reduction of Theorem 1.1 to Theorem 1.2 is relatively formal, and so we
will focus on the latter Theorem here.

The idea of the proof goes back to ideas initiated in the work of Bombieri-
Pila [3], Heath-Brown [15] (and, independently and in a different context,
Coppersmith [7]): One can show that an excess of rational (or integral) points
on any $X$ as above must lie on the zero-loci of some auxiliary functions,
i.e., must be covered by a certain collection of divisors. Then we iterate the
process, replacing $X$ by these divisors, and covering the points by
codimension-$2$ subvarieties, etc.; since we have almost no control over what
these subvarieties look like, it is crucial to have, as in Bombieri-Pila and
Heath-Brown, that the basic bounds are uniform in the ambient variety. See
Remark 2.7 for a quick review of this method.

Unfortunately, this process is quite lossy when executed in dimensions larger
than $1$. The problem is the usual one in studying rational points on higher
dimensional varieties: Even if a variety has high degree, it may contain
subvarieties of much lower degree, such as hyperplanes. For this reason, there
are very few situations where one can achieve $B^{\epsilon}$-type bounds (see
e.g. [23] for an example in a different context).

The key point of our strategy here is to use the fact that the bounds of [4]
improve under étale covers, which are plentiful under the conditions on
$\mathfrak{X}\rightarrow X$ we have imposed; these covers can be used to raise
the degree of intermediate subvarieties. To deploy this in a way suitable for
an iterative argument, we use the global invariant cycle theorem of Hodge
theory to construct a cover $\widetilde{X}\rightarrow X$ which remains
nontrivial when restricted to any subvariety of $X$. In some sense, this
strategy generalizes the approach of [10], which used étale covers to slightly
improve on Heath-Brown’s bounds in case $X$ is a non-rational curve (i.e. a
curve with interesting étale covers); in the case $\dim X=1$, of course, the
issue of restriction of covers to subvarieties becomes trivial.

In the end, the core part of the proof is quite short – the main idea is the
induction on dimension laid out in §4.2, with the key induction step being
provided by Lemma 4.2. The reader may want to skip directly to these, and
refer back as necessary. One of the reasons for the length of the text is the
technicalities involved in making various results uniform over all
subvarieties of $X$.

One way of thinking about the strategy executed here is in terms of “profinite
repulsion” between low-height integral points. For simplicity of exposition we
take $K=\mathbb{Q}$ for this paragraph. For any prime $p$, the embedding of
$X(\mathbb{Z}[1/S])$ into $X(\mathbb{Q}_{p})$ induces a topology on the former
set, and the methods of Bombieri-Pila and Heath-Brown rely on an argument that
low-height points tend to repel each other in this topology. In the cases
considered in this paper, there is an alternate profinite topology on
$X^{\circ}(\mathbb{Z}[1/S])$ afforded by the map

$X^{\circ}(\mathbb{Z}[1/S])\to
H^{1}(G_{\mathbb{Q}},\pi_{1}(X^{\circ}_{\overline{\mathbb{Q}}}))$

and our approach can be thought of as exploiting the fact that in this
topology, too, the low-height $S$-integral points repel each other. The two
profinite repulsion phenomena working in tandem is what allows us to get a
better upper bound on the number of low-height points. To be more precise: to
say that two points $P_{1}$ and $P_{2}$ of $X^{\circ}(\mathbb{Z}[1/S])$ are
close together in the alternate profinite topology is to say that there is
some high-degree cover $\widetilde{X}^{\circ}\to X^{\circ}$, with
$\widetilde{X}^{\circ}$ irreducible, such that $P_{1}$ and $P_{2}$ both lie in
the image of $\widetilde{X}^{\circ}(\mathbb{Q})$. Then the method of Heath-
Brown is used to control the low-height points of
$\widetilde{X}^{\circ}(\mathbb{Q})$, which is aided by the fact that this new
variety has very high degree.

In the course of writing this paper we learned of the paper [5] of Brunebarbe,
which uses the same basic idea of increasing the positivity of line bundles by
passing to étale covers, in the context of a VHS with finite period map, in
order to prove uniform algebro-geometric assertions about covers of varieties
carrying such a VHS. Indeed Brunebarbe gives a more analytic (and therefore
effective) proof of a result ([5, Theorem 1.13]) of the type of Lemma 4.1,
whereas our argument uses general finiteness statements of algebraic geometry.
As in our paper, a crucial input to [5] is the invariant cycle theorem of
Hodge theory.

Finally, let us mention some potential refinements of the result.

  * •

The reader will observe that the geometricity of the variation of Hodge
structure is, in fact, barely used at all in our proof. It permits us to
phrase the argument in quite an explicit way, but it should be possible to
entirely avoid it.

  * •

Indeed, the whole apparatus of variations of Hodge structures is used simply
to provide examples of varieties $X^{\circ}$ with the property that the image
of $\pi_{1}(W^{\circ{\text{an}}})\to\pi_{1}(X^{\circ{\text{an}}})$ has
infinite order for every subvariety $W^{\circ}$ of $X^{\circ}$. We note that
this is essentially the same as Kollár’s condition of large fundamental group
from [20]. For an $X^{\circ}$ which possessed this property for some reason
unrelated to Hodge structures, the arguments here should work just as well to
control its rational points. For example, abelian varieties do not admit
variations of Hodge structure but they do have large fundamental group, so the
arguments of this paper should in principle imply sparseness of rational
points; but for abelian varieties, we know this already, by the Mordell-Weil
theorem.

  * •

One might ask what happens for varieties $X^{\circ}$ carrying a variation of
mixed Hodge structures; it seems plausible that a theorem like the one proved
here will still hold.

  * •

It would be interesting to refine our statement $O_{\epsilon}(B^{\epsilon})$
to a more precise upper bound. To do this would, at the very least, require
that the inexplicit dependence on degree in Broberg’s bounds be made explicit.
Questions in this vein have been the topic of much recent progress; see, e.g.,
the recent result of Castryck, Cluckers, Dittman, and Nguyen [6, Theorem 2],
which gives bounds with explicit polynomial dependence on degree. The
quantitative approach of [5] will also likely be needed in order to get
effective lower bounds for the degrees of covers that arise in the argument.

###  1.2. Notation and integral models

As above, $K$ is a number field embedded in $\mathbb{C}$; let $\overline{K}$
be the algebraic closure of $K$ in $\mathbb{C}$ (so $\overline{K}$ is just
another name for $\overline{\mathbb{Q}}$).

By a variety $V$ (over $K$), or $K$-variety for short, we mean, as usual, an
integral, separated, finite-type scheme over $K$. In this situation, $V$ will
always denote the scheme over $K$, and $V_{\mathbb{C}}$ the base extension of
$V$ to a complex scheme via $K\hookrightarrow\mathbb{C}$. (Note that a
$K$-variety is not required to be geometrically irreducible; thus,
$V_{\mathbb{C}}$ may have multiple components.) We will sometimes use notation
such as $Y_{\mathbb{C}}$ even when $Y$ is not a base-change from $K$, simply
to emphasize that $Y_{\mathbb{C}}$ is a complex variety. We will write
$V^{{\text{an}}}$ for the (complex) analytification of $V_{\mathbb{C}}$.

As above let $S$ be a finite set of primes of $K$. Let $\mathfrak{o}_{S}$ be
the ring of $S$-integers of $K$, i.e. the set of those elements of $K$ that
are integral outside $S$.

######  Definition 1.4.

Let $X^{\circ}\subset\mathbb{P}^{m}$ be a locally closed subvariety, and $X$
its Zariski closure. Let $Z$ be the complement $X\backslash X^{\circ}$, a
Zariski-closed subset of $\mathbb{P}^{m}$.

We will call a _good integral model_ for $(X,Z)$ a choice of a projective flat
$\mathfrak{o}_{S}$-scheme $X_{S}\subset\mathbb{P}^{N}_{S}$ extending $X$ on
the generic fiber. Given a good integral model, we define $Z_{S}$ to be the
Zariski closure of $Z$ inside $X_{S}$, and take $X^{\circ}_{S}=X_{S}-Z_{S}$.

If we are given a projective smooth morphism $f:\mathfrak{X}\rightarrow
X^{\circ}$ and we have chosen a good integral model for $(X,Z)$, a good
integral model for $f$ will be a projective smooth morphism
$f_{S}:\mathfrak{X}_{S}\rightarrow X^{\circ}_{S}$.

Good integral models exist after possibly increasing $S$, by standard
spreading arguments (a nice reference here is Theorem 3.2.1 and Appendix C,
Table 1 of [24]).

A $K$-point of $X$ will be said to be $S$-integral if it extends to a morphism
$\mathrm{Spec}(\mathfrak{o}_{S})\rightarrow X_{S}-Z_{S}$. Explicitly, assuming
$\mathfrak{o}_{S}$ to have class number one for simplicity, any $K$-point,
represented by a point with relatively prime homogeneous coordinates
$\mathbf{x}=[x_{0}:\dots:x_{N}]\in\mathfrak{o}_{S}^{N+1}$, is _integral_ at a
prime $p\notin S$ if there exists a homogenous function
$f\in\mathfrak{o}_{S}[x_{0},\dots,x_{N}]$ in the ideal of $Z_{S}$ such that
$f(\mathbf{x})$ is nonzero mod $p$; it is $S$-integral if it is integral at
all primes $p\notin S$.

If we choose two different choices of good integral model as above, there
exists a finite set $T$ such that any $S$-integral point for the first model
is $S\coprod T$-integral for the second model, so the choice of good integral
model is irrelevant to the statement of the theorem.

Finally, we fix some other notation to be used: For any field $F$ and any
$F$-variety $Y$ mapping to $X$, we denote by $Y^{\circ}$ the preimage of
$X^{\circ}$ inside $Y$. If the map $f:Y\rightarrow X$ is finite, the “degree”
of $Y$ will be the degree of the pullback $f^{*}\mathcal{L}$ of the hyperplane
bundle $\mathcal{L}$, characterized by the asymptotic formula

(2) $\dim_{F}\Gamma(Y,f^{*}\mathcal{L}^{\otimes
k})\sim\frac{\mathrm{deg}Y}{(\dim Y)!}k^{\dim Y}$

for large $k$.

###  1.3. Acknowledgements

We are grateful to Michael Kemeny for helpful discussions about the quasi-
finiteness of period maps, and to Brian Conrad for help and corrections to §2.
The first author was supported by NSF grant DMS-2001200, the second by NSF
grant DMS-2101985, and the third by NSF grant DMS-1931087.

##  2\. Some results used in the proof

We collect results we will use in the proof. These are a few lemmas from
algebraic geometry, and a crucial bound on rational points due to Broberg.

We emphasize again that, for $F$ a field, a variety over $F$ is assumed
reduced and irreducible, but not geometrically irreducible.

######  Lemma 2.1.

Suppose that $\pi:\mathfrak{X}\rightarrow X$ is a projective smooth morphism
of algebraic varieties over a subfield $\kappa\subset\mathbb{C}$, with $X$
smooth, and let $\mathcal{T}$ be the tangent bundle of $X$. Let $\Phi$ be the
period map associated to the variation of Hodge structure
$\mathsf{V}=R^{i}\pi_{*}\mathbb{Q}$ on $(X)^{{\text{an}}}$, for some $i>0$.

Then there is a morphism

(3) $g\colon\mathcal{H}_{1}\otimes\mathcal{T}\rightarrow\mathcal{H}_{2}$

of vector bundles over $X$ (everything defined over $\kappa$) such that the
derivative of the period map along a tangent vector $t$ at $x$ vanishes if and
only if $g_{x}(-,t)$ is the zero map from $\mathcal{H}_{1}$ to
$\mathcal{H}_{2}$.

###### Proof.

This is a consequence of standard facts in Hodge theory, in particular the
algebraicity of the Gauss–Manin connection (see [18] and [19, §1]).

The $i$th relative algebraic de Rham cohomology of $\mathfrak{X}/X$ gives a
variation of Hodge structure $H_{\text{dR}}$ on $X$. As a variation of Hodge
structure, $H_{\text{dR}}$ is a vector bundle, equipped with a flat connection
and a filtration by algebraic subbundles $F^{p}H_{\text{dR}}$. These data
determine the period map $\widetilde{X^{\circ{\text{an}}}}\rightarrow D$,
where $D$ is a flag variety classifying filtrations of a fixed vector space of
dimension $\operatorname{rank}H_{\text{dR}}$ by subspaces of dimensions
$\operatorname{rank}F^{p}H_{\text{dR}}$.

For each step $F^{p}H_{\text{dR}}$ of the filtration, the connection on
$H_{\text{dR}}$ defines an $\mathcal{O}_{X}$-linear map

$g_{p}\colon F^{p}H_{\text{dR}}\otimes\mathcal{T}\rightarrow
H_{\text{dR}}/F^{p}H_{\text{dR}},$

which describes how the subspace $F^{p}H_{\text{dR},x}$ varies in $x\in X$.
Combining the maps $g_{p}$ over all $p$, we obtain

$g\colon\bigoplus_{p}F^{p}H_{\text{dR}}\otimes\mathcal{T}\rightarrow\bigoplus_{p}H_{\text{dR}}/F^{p}H_{\text{dR}}.$

This map $g$ determines the differential of the period map. More precisely, at
every point, the tangent space to $D$ is identified with a subspace of

$\operatorname{Hom}\left(\bigoplus_{p}F^{p}H_{\text{dR}},\bigoplus_{p}H_{\text{dR}}/F^{p}H_{\text{dR}}\right),$

and with this identification, $g$ gives the differential of the period map.

Finally, we note that $g$ is defined over $\kappa$ by [18] and [19, §1].

∎

######  Lemma 2.2.

Let $F$ be a field of characteristic zero.

Suppose that $Y$ is a proper $F$-variety equipped with an ample line bundle
$\mathcal{L}$, and let $g:\widetilde{Y}\rightarrow Y$ be finite, with
$\dim(\widetilde{Y})=\dim(Y)$. Writing $\deg g$ for the degree of $g$ at the
generic point, we have

(4) $\deg_{g^{*}\mathcal{L}}(\widetilde{Y})=(\deg g)\deg_{\mathcal{L}}(Y).$

We will apply this only when $\widetilde{Y}$ is a variety, but the argument
does not require that, taking (2) as the definition of degree.

###### Proof.

By the projection formula we have

$\Gamma(\widetilde{Y},g^{*}\mathcal{L}^{\otimes
k})=\Gamma(Y,g_{*}\mathcal{O}\otimes\mathcal{L}^{\otimes k}).$

Now $g_{*}\mathcal{O}$ is isomorphic to $\mathcal{O}^{\oplus(\mathrm{deg}g)}$
away from a set of positive codimension on $Y$. Therefore
$\Gamma(\widetilde{Y},g^{*}\mathcal{L}^{\otimes k})$ coincides with
$(\mathrm{deg}\ g)\frac{\mathrm{deg}_{\mathcal{L}}Y}{(\dim Y)!}k^{\dim Y}$ for
large $k$, up to terms $O(k^{\dim Y-1})$. We conclude using
$\dim\widetilde{Y}=\dim Y$. ∎

The next Lemma is [28, Exposé I, Corollaire 10.8].

######  Lemma 2.3.

Suppose that $X$ is an irreducible normal noetherian scheme and
$f:Y\rightarrow X$ a finite étale cover. Then the irreducible components of
$Y$ are disjoint.

The following result asserts, essentially, “boundedness” of the set of
irreducible varieties in a fixed projective space and bounded degree. Results
of this type are also stated and used in work of Salberger [26, Lemma 1.4,
Thm. 3.2] in a similar context; in the interest of self-containedness we give
a proof of precisely what we use.

######  Lemma 2.4.

Let $F$ be a field of characteristic zero.

  * (a)

Suppose that $V\subset\mathbb{P}^{m}_{F}$ is a closed subvariety (irreducible,
reduced closed subscheme) of degree $d$. Then there are bounds, depending only
on $m,d$ (not on $F$) for each coefficient of the Hilbert polynomial of $V$,
and in particular the homogeneous ideal $I(V)$ of $V$ is generated in degree
$O_{m,d}(1)$.

  * (b)

Suppose given integers $m,n,d,R$. Then there exist bounds $D$ and $N$ with the
following property: Suppose $V_{1},\ldots,V_{r}$ (with $r\leq R$) is a
collection of closed subvarieties of $\mathbb{P}^{m}_{F}$, with each $V_{i}$
of dimension $\leq n$ and degree $\leq d$. Let $Z$ be the intersection

(5) $Z=\bigcap_{i=1}^{r}V_{i}.$

Then the number of irreducible components of $Z$ is at most $N$, and the
degree of each such component (endowed with the reduced scheme structure) is
at most $D$. Furthermore, the bounds $D$ and $N$ are independent of the field
$F$.

  * (c)

Suppose that $V\subset\mathbb{P}^{m}_{F}$ is a closed variety of dimension $n$
and degree $d$ which is not geometrically irreducible. Then there exist
finitely many subvarieties $V_{1},\ldots,V_{N}\subseteq V$, defined over $F$,
such that:

    * –

Each $V_{i}$ is irreducible of dimension $\leq n-1$,

    * –

$V(F)=\bigcup V_{i}(F)$, and

    * –

the number $N$ of $V_{i}$’s, and degree of each $V_{i}$ can be bounded in
terms of $n,m,d$ (but independently of the field $F$ and the variety $V$).

  * (d)

Suppose that $V\subset\mathbb{P}^{m}_{F}$ is a closed variety of dimension $n$
and degree $d$. Then the set of points in $V(F)$ which are singular on $V$ can
again be covered by varieties $V_{1},\dots,V_{N}$ with the same properties as
(c).

######  Example 2.5.

As an example of “bad” examples for part (a): take a $d$-dimensional variety,
and adjoin to it a large set of disjoint points; this modification does not
affect the degree, and shows the need for irreducibility or at least
equidimensionality. Similarly, consideration of embedded points shows that
“reduced” is also important.

As an example of the situation in part (c), consider the plane curve defined
by $x^{2}+y^{2}=0$. This is irreducible over $\mathbb{Q}$ but not
geometrically irreducible; it only has one rational point $(0,0)$, which is
contained in a zero-dimensional, geometrically irreducible subvariety. More
generally, suppose $F$ is a number field, and consider “the affine line over
$F$, with the origin reduced to a $\mathbb{Q}$-point” – that is,
$\operatorname{Spec}(\mathbb{Q}+TF[T])$. If $F\neq\mathbb{Q}$, this scheme is
irreducible, but geometrically reducible, and its only $\mathbb{Q}$-point is
the origin. (Taking $F=\mathbb{Q}[i]$ recovers the original example.)

###### Proof.

The first assertion of (a) follows from [1, Exposé XIII, Corollary 6.11(a)].
That assertion applies as formulated to “special positive cycles” over an
algebraically closed field; in our situation
$V_{\overline{F}}\subset\mathbb{P}^{m}_{\overline{F}}$ is reduced
equidimensional and its decomposition into irreducible components give the
closed subschemes appearing in loc. cit. Définition 6.9.

The consequence on bounded generation of $I(V)$ follows, because such a bound
on generation of the defining ideal is valid in any finite type subscheme of
the Hilbert scheme.

For (b), we may as well consider one fixed $r$. Let $\mathcal{P}$ be the
finite set of polynomials arising from (a), and let $\operatorname{Hilb}$ be
the Hilbert scheme parametrizing closed subschemes of $\mathbb{P}^{m}$ with
Hilbert polynomial in $\mathcal{P}$. This is a finite-type $\mathbb{Q}$-scheme
by (a). Now tuples $(V_{1},\ldots,V_{r})$ are classified by suitable
$K$-points of $\operatorname{Hilb}^{r}$, which is again of finite type; and
the result now follows from standard results on families over a finite-type
base.

Specifically, we have the universal schemes
$\mathcal{V}_{1},\ldots,\mathcal{V}_{r}$ over $\operatorname{Hilb}^{r}$; let
$\mathcal{Z}\subseteq\mathbb{P}^{m}\times\operatorname{Hilb}^{r}$ be their
fiber product over $\mathbb{P}^{m}$, so fiberwise $\mathcal{Z}$ gives the
intersection of the $\mathcal{V}$s (although with a possibly non-reduced
schematic structure). We will work by Noetherian induction. Let $\eta$ be the
generic point of a closed irreducible subscheme
$H\subseteq\operatorname{Hilb}^{r}$. We will show that there exists a
relatively open subset $U\subseteq H$ (that is, open in $H$, but not
necessarily in $\operatorname{Hilb}^{r}$) such that the number, dimensions,
and degrees of the irreducible components of fibers $\mathcal{Z}_{h}$, for
$h\in U$, are bounded. Here, as in the statement, “degree” is taken with
reference to the reduced scheme structure.

The number of irreducible components of any geometric fiber of $\mathcal{Z}$
is bounded by [13, 9.7.9]. This bounds the number of geometric components, and
so also the number of irreducible components, of any $Z$ as in (5).

Now we turn to the degree. For each $j$ with $0\leq j\leq n$, let $Z_{j}$ be
the closure in $\mathcal{Z}$ of the union of $j$-dimensional components of
$\mathcal{Z}_{\eta}$ (thought of, for now, merely as a closed subset of
$\mathcal{Z}$ in the Zariski topology; we will revisit the issue of scheme
structure shortly.) In particular, the fiber over $\eta$ of each $Z_{j}$ is
equidimensional of dimension $j$. By [13, 9.5.1, 9.5.5], we can restrict to an
open $U\subseteq H$, on which:

  * (i)

For each $s\in U$, the fiber $(Z_{j})_{s}$ is equidimensional of dimension
$j$,

  * (ii)

For each $s\in U$, the fiber $\mathcal{Z}_{s}$ is set-theoretically covered by
the various $(Z_{j})_{s}$, and

  * (iii)

For each $s\in U$, and for all $j^{\prime}<j$, the intersection
$(Z_{j})_{s}\cap(Z_{j^{\prime}})_{s}$ has all components of dimension strictly
less than $j^{\prime}$.

Take $s\in U$ and let $K$ be any irreducible component of $\mathcal{Z}_{s}$,
say of dimension $q$; by (ii) it is contained in some irreducible component of
some $(Z_{j})_{s}$. Since $K$ is maximal among irreducible subsets, we must
have equality here, i.e. $K$ coincides with an irreducible component of this
$(Z_{j})_{s}$, and since $(Z_{j})_{s}$ is equidimensional of dimension $j$ we
must have $j=q$.

Now let us endow $Z_{j}$ (so far merely a closed set) with its reduced scheme
structure. But now by generic flatness ([14, 6.9.1]), we further shrink $U$ so
that each $Z_{j}$ is flat over $U$. Then for each $j$, the degree of each
$(Z_{j})_{s}$ is independent of $s$. Now, the scheme structure on
$(Z_{j})_{s}$ need not be reduced, but nonetheless there is an inequality of
degrees $\mathrm{deg}\ (Z_{j})_{s}\geq\mathrm{deg}\
(Z_{j})_{s}^{\mathrm{red}}$ (where $(Z_{j})_{s}^{\mathrm{red}}$ denotes the
fiber taken with the reduced scheme structure) and the degree of
$(Z_{q})_{s}^{\mathrm{red}}$ bounds from above the degree of $K$ taken with
its reduced scheme structure.

Now we turn to (c). Consider the geometrically irreducible components
$W_{1},\dots,W_{h}$ inside the base change $V_{\overline{F}}$ of $V$ to an
algebraic closure. Note that $h$ is bounded by the degree of $V$, which we
have assumed bounded, and similarly the degree of each $W_{i}$ is bounded by
the degree of $V$. The intersection $\cap_{i=1}^{h}W_{i}$ (with its reduced
structure) is a Galois-stable closed subscheme of $V_{\overline{F}}$ and
thereby descends to a reduced $F$-subscheme $W\subset V$. Since $V$ is not
geometrically irreducible, we know that
$\operatorname{dim}W\leq\operatorname{dim}V-1$. Also (b), applied with $F$
replaced by $\overline{F}$, implies that $W_{\bar{F}}$ has a bounded number of
irreducible components, each of bounded degree. The same is then true for the
$F$-scheme $W$. We claim, further, that $W(F)=V(F)$. To see this, note that
any $F$-point of $V$ must be Galois-invariant; since Galois permutes the
geometric components of $V$ transitively, the associated element of
$V(\overline{F})$ belongs to all $W_{i}$. (See [29, Lemma 0G69] for a similar
argument.)

The argument for (d) is similar to that for (b). Again we can parameterize all
such $V$ by a suitable finite type Hilbert scheme $\mathrm{Hilb}$ and, writing
$\mathcal{H}\rightarrow\mathrm{Hilb}$ for the universal subscheme, the smooth
locus of $\pi$ coincides with the set of points of $\mathcal{H}$ that are
smooth in their fiber over $\mathrm{Hilb}$ (see e.g. [12, 17.5.1]). Let
$\mathcal{Z}$ be the complement of this smooth locus, endowed with the reduced
structure. Then proceed as in (b). ∎

Finally, the following theorem of Broberg [4] builds on fundamental ideas of
Heath-Brown [15] and Bombieri-Pila [3]:

######  Theorem 2.6 (Broberg, 2004).

Let $V\subset\mathbb{P}^{M}_{K}$ be an irreducible closed subvariety of
dimension $n$ and degree $d$. Then the points of $V(K)$ whose naive height is
at most $H$ are contained in a set of $K$-rational divisors of cardinality
$\ll_{\epsilon,M}H^{\frac{n+1+\epsilon}{d^{1/n}}}$, and each of which has
degree $O_{\epsilon,M}(1)$.

We note that, as stated in [4], Broberg requires a bound on the generation of
the ideal of $V$. This bound is however automatic from Lemma 2.4 part (a).
Also “divisor” in the statement means “effective Cartier divisor.”

######  Remark 2.7.

Because Theorem 2.6 is so crucial, particularly its dependence on degree, we
briefly outline where it comes from, taking $K=\mathbb{Q}$ to simplify
notation; this is not needed in the remainder of the paper.

One chooses a large integer $k$ and embeds $V\hookrightarrow\mathbb{P}^{e-1}$
via a basis of sections of $\Gamma(V,\mathcal{O}(1)^{\otimes k})$. In fact, we
can and do choose from $\Gamma(\mathbb{P}^{M},\mathcal{O}(1)^{\otimes k})$ a
set of monomials $f_{1},\ldots,f_{e}$ of degree $k$ in the $M+1$-variables
which freely span $\Gamma(V,\mathcal{O}(1)^{\otimes k})$.

Choose a “good” prime $p$ and examine a collection of $e$ points $P_{i}\in
V(\mathbb{Q})$ which all reduce to the same point modulo $p$ of
$\mathbb{P}^{M}$, and whose height is at most $H$. Expressing each $P_{i}$ in
coprime integer coordinates, we can speak of the evaluation
$f_{i}(P_{j})\in\mathbb{\mathbb{Z}}$.

Consider

$\Delta:=\det\left[f_{i}(P_{j})\right]_{1\leq i,j\leq e}\in\mathbb{Z}$

which measures the volume of the $e$-simplex in $\mathbb{Z}^{e}$ spanned by
the $P_{i}$ and the origin. On the one hand, $\Delta$ is bounded by a constant
multiple of $H^{ke}$.

On the other hand, $\Delta$ is highly divisible by $p$, because the values of
$f_{j}$ modulo power of $p$ are highly constrained, and therefore there are
many relations (mod $p^{k}$) between rows of $\Delta$. To see this more
formally, fix $r\geq 1$ and let denote by $V(\mathbb{Z}_{p})_{0}$ the subset
of $V(\mathbb{Z}_{p})$ consisting of points with a given reduction modulo $p$.
Set

$M_{r}:=\\{\mbox{functions:
$V(\mathbb{Z}_{p})_{0}\rightarrow\mathbb{Z}/p^{r}$}\\}.$

Each $f_{j}$ gives an element of $M_{r}$, by evaluation and reduction modulo
$p^{r}$. For $r=1$, all these functions (for varying $j$) lie in a
$\mathbb{Z}/p$-module of rank one: the constant functions. For $r=2$, these
functions depend only on the “constant term” and “derivative” of $f_{j}$, and
thereby lie in a $\mathbb{Z}/p^{2}$-submodule of $M_{2}$ of rank $n+1$. For
$r=3$ we get $\mathbb{Z}/p^{3}$-submodule of $M_{3}$ of rank $(n+1)(n+2)/2$,
where the number comes from counting possible Taylor expansions of $f_{j}$ up
to degree two. Each such statement gives linear constraints on the rows of
$f_{i}(P_{j})$, and therefore leads to divisibility for $\Delta$.

Computing with this we find

$v_{p}(\Delta)\gtrsim ke\cdot\frac{d^{1/n}}{1+1/n},$

where $d$ arises on the right-hand side eventually through the asymptotic
behaviour of $e=\dim\Gamma(V,\mathcal{O}(k))$, cf. (2). Choosing $p$ so that
$p^{v_{p}(\Delta)}$ is larger than the size bound $H^{ke}$ then forces
$\Delta=0$; so the points $P_{i}$ lie on a hyperplane of $\mathbb{P}^{e-1}$,
i.e. all the points of $V$ with a fixed mod $p$ reduction lie on a divisor. So
we produce $\approx p^{n}$ divisors covering the points of height $\leq H$.
The argument above is so flexible – in particular, using freedom to choose $p$
– that one can achieve bounds that are uniform in $V$.

Notice the crucial point: as the degree of $\mathcal{O}(1)$ on $V$ increases,
$\mathcal{O}(1)^{\otimes k}$ has more sections for a fixed $k$, giving
stronger divisibility for $\Delta$ and thus stronger bounds.

##  3\. Reduction to Theorem 1.2

We describe how Corollary 1.3 is reduced to Theorem 1.2.

### Deduction of Corollary 1.3 from Theorem 1.1

We will focus on the first statement of the Corollary, with $S$-integral
points, and remark at the end of the proof on the only modification needed to
handle the statement about fixed discriminant.

For $n=2$ (binary forms of degree three and above) one in fact knows
finiteness (Birch–Merriman [2]). The same is true for the case $n=4,d=3$ of
cubic surfaces (Scholl [27]).

We may now restrict to the remaining cases $n\geq 3,d\geq 3$ and
$(n,d)\neq(4,3)$. We will apply Theorem 1.2 taking $X=\mathbb{P}^{M}$ the
projective space parameterizing polynomials of degree $d$ in $n$ variables up
to scaling, with $Z$ the zero-locus of the discriminant, and taking the
geometric variation of Hodge structure to arise from the middle cohomology of
the universal family of hypersurfaces over $X^{\circ}$. Here the infinitesimal
Torelli theorem is known, see [11, Thm. 9.8(b)]; that is, every fiber of the
period map is locally contained in an orbit of $\mathrm{PGL}_{n}(\mathbb{C})$;
so, noting that the Weil height of $[a_{0}:\dots:a_{M}]$ is given by
$\prod_{v}\max_{i}|a_{i,v}|$ and is thereby bounded by a power of $B$ in the
situation of the Corollary, Theorem 1.2 shows that the integral points in
question are covered by $O_{\epsilon}(B^{\epsilon})$ orbits of
$\operatorname{PGL}_{n}(\mathbb{C})$ or equivalently
$\operatorname{GL}_{n}(\mathbb{C})$.

We must replace $\operatorname{GL}_{n}(\mathbb{C})$ by
$\operatorname{GL}_{n}(\mathbb{Z}[S^{-1}])$. This is not difficult but one
must take care because a hypersurface could have automorphisms in
characteristic $p$ that do not lift to characteristic zero.

Fix $P_{0}$ as in the Corollary. We will show that the number of
$\operatorname{GL}_{n}(\mathbb{Z}[S^{-1}])$-orbits on $S$-integral polynomials
$P\in\operatorname{GL}_{n}(\mathbb{C})P_{0}$ with $S$-integral discriminant is
bounded in terms of $n,d,S$. Let $h$ be the degree of the discriminant
polynomial. For any $S$-integral $P\in\operatorname{GL}_{n}(\mathbb{C})P_{0}$
with $S$-integral discriminant, there exists a rescaling of $P$ by an $S$-unit
whose discriminant has $p$-valuation between $0$ and $h$. It suffices, then,
to show that for any integer $N=\prod_{p\in S}p^{a_{p}}$ (with $0\leq
a_{p}<h$) the set of $S$-integral polynomials in
$\operatorname{GL}_{n}(\mathbb{C})P_{0}$ with discriminant $N$ lie in a union
of $O_{n,d,S}(1)$ orbits of $\operatorname{SL}_{n}(\mathbb{Z}[S^{-1}])$.

Now, write $Y$ for the affine hypersurface defined by $\mathrm{disc}(P)=N$,
which we can regard as an affine scheme over $\mathbb{Q}$ (and even over
$\mathbb{Z}$). It is equipped with an action of the $\mathbb{Q}$-algebraic
group $G=\mathrm{SL}_{n}$. We must show that the intersection of
$Y(\mathbb{Z}[S^{-1}])$ with any $G(\mathbb{C})$-orbit is covered by
$O_{n,d,S}(1)$ orbits of $G(\mathbb{Z}[S^{-1}])$. We will need:

> Claim 1: The action morphism $G\times Y\rightarrow Y\times Y$ is a finite
> morphism of algebraic varieties over $\mathbb{Q}$.

This follows essentially from the theorem of Matsumura and Monsky [22] that
each stabilizer $G_{y}$ for $y\in Y(\mathbb{C})$ is finite. The deduction can
be carried out using results of geometric invariant theory to show that the
stack of smooth hypersurfaces is separated; see [17]. We give a self-contained
argument using similar ideas.

###### Proof.

(of Claim 1.) It is sufficient to prove this over $\mathbb{C}$, and, since the
morphism is quasi-finite, it is sufficient to prove that it is proper.

Using the singular value decomposition one reduces to checking that the action
of the diagonal subgroup $T=\\{(t_{1},\dots,t_{n})\text{ with
}t_{i}\in\mathbb{R}_{+}\\}$ on $Y(\mathbb{C})$ is proper, which can be checked
for the analytic topology [28, XII, Prop 3.2]. In other words, one must show
that for any compact regions $\Omega_{1}$ and $\Omega_{2}$ in $Y(\mathbb{C})$,
the set $\\{g\in T:g\Omega_{1}\cap\Omega_{2}\mbox{ nonempty}\\}$ is bounded.
It suffices to show that, if $P=\sum a_{i_{i}\dots i_{n}}x_{1}^{i_{1}}\dots
x^{i_{n}}$ and $Q=\sum b_{i_{1}\dotsi_{n}}x_{1}^{i_{1}}\dots x^{i_{n}}$ and
$(t_{1},\ldots,t_{n})\cdot P=Q$ then the absolute values of the $t_{j}$ can be
bounded in terms of the coefficients of $P$ and $Q$.

Write $\Sigma$ for the set of $I=(i_{1},\ldots,i_{n})$ such that $a_{I}\neq
0$. For each $t=(t_{1},\ldots,t_{n})$, write $t^{I}$ for
$\prod_{j}t_{j}^{i_{j}}=\exp(\sum i_{j}\log t_{j})$. Then the maximal absolute
value of a coefficient of $(t_{1},\ldots,t_{n})\cdot P$ is
$\max_{I\in\Sigma}|a_{I}|t^{I}$, so the condition that
$(t_{1},\ldots,t_{n})\cdot P=Q$ provides upper bounds on $|a_{I}|t^{I}$ for
all $I\in\Sigma$.

An upper bound on $|a_{I}|t^{I}$ restricts $(\log t_{1},\ldots\log t_{n})$ to
a half-space; we are done once we show that the intersection of these half-
spaces over all $I\in\Sigma$ is a compact region in $T$. This is the case
exactly when the region

(6) $\\{t\in T:\sum i_{j}\log t_{j}>0\mbox{ for all $I\in\Sigma$}\\}$

is empty. Suppose otherwise; then there exists $t$ in this region of the form
$t_{0}^{m_{1}},\ldots,t_{0}^{m_{n}}$ for some (whence any)
$t_{0}\in\mathbb{R}_{+}$ and with $m_{j}\in\mathbb{Z}$. By assumption on $t$,
the limit in the analytic topology $P_{0}:=\lim_{t_{0}\rightarrow
0}(t_{0}^{m_{1}},\ldots,t_{0}^{m_{n}})\cdot P$ exists; explicitly,
$P_{0}=\sum_{I\in\Sigma_{0}}a_{I}t^{I}$ where $\Sigma_{0}$ is that subset of
$\Sigma$ consisting of those $I$ with $\sum i_{j}m_{j}=0$. Then
$\mathrm{disc}(P_{0})=\lim_{t_{0}\to 0}\mathrm{disc}(t\cdot
P)=\mathrm{disc}(P)$ is nonzero, so $P_{0}$ cuts out a smooth affine
hypersurface. But the identity $\sum m_{j}X_{j}\frac{d}{dX_{j}}P_{0}=0$ means
that the $n$ conditions $\frac{dP}{dX_{j}}=0$ cutting out the nonsmooth locus
are dependent, contradicting smoothness of $P_{0}=0$ (cf. proof in [22]). Thus
region (6) is empty, so the action of $T$ is proper, so the action of $G$ is
proper as well. This concludes the proof of Claim 1. ∎

Take $y_{1},y_{2}\in Y(\mathbb{Z}[S^{-1}])$. We will now prove

> Claim 2 : The action of
> $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the stabilizers
> $G_{y_{i}}(\overline{\mathbb{Q}})$ and also on the set $G_{12}:=\\{g\in
> G(\overline{\mathbb{Q}}):gy_{1}=y_{2}\\}$ is unramified outside a set of
> primes $\mathcal{P}$ depending only on $n,d,S$.

###### Proof.

(of Claim 2). It is enough to prove the claim for $G_{12}$; take $y_{1}=y_{2}$
to get the claim about stabilizers.

From finiteness of the action map we see that the matrix entries of $g^{\pm
1}$ for $g\in G_{12}$ satisfy monic polynomials whose coefficients are
rational polynomials in the coordinates of $y_{i}$. Take
$A_{1}(y_{1},y_{2}),\ldots,A_{M}(y_{1},y_{2})$ to be the finite collection of
all coefficients arising in this way.

Now, for any extension of the $p$-valuation on $\mathbb{Q}$ to
$\overline{\mathbb{Q}}$, and for every matrix entry $g^{\pm 1}_{ij}$ of
$g^{\pm 1}$ we have

(7) $|g^{\pm 1}_{ij}|_{p}\leq\max_{k}|A_{k}(y_{1},y_{2})|_{p}.$

Take $P$, larger than any prime in $S$, such that the coefficients of each
$A_{i}$ are $p$-integral for all $p>P$; it now follows from (7) that, for all
$p>P$, any element $g$ which lies in $G_{12}$ for any $y_{1},y_{2}\in
Y(\mathbb{Z}[S^{-1}])$ has for entries roots of monic polynomial with
$p$-integral coefficients; in other words, it is $p$-integral. Thus, for all
such $p$ there is an induced map

$G_{12}\rightarrow\operatorname{SL}_{n}(\overline{\mathbb{Z}}_{p})\rightarrow\operatorname{SL}_{n}(\overline{\mathbb{F}}_{p})$

which is necessarily injective if we also require $p$ to be larger than the
order of $G_{12}$, since any torsion element of the kernel of
$\operatorname{SL}_{n}(\overline{\mathbb{Z}}_{p})\rightarrow\operatorname{SL}_{n}(\overline{\mathbb{F}}_{p})$
must have $p$-power order. For general constructibility reasons (similar to
Lemma 2.4) the order of $G_{12}$ is bounded above; choose $P$ to be larger
than this bound.

Now for any $\sigma$ belonging to the inertia group at $p>P$, and any $g\in
G_{12}$, $g^{\sigma}$ and $g$ have the same image in
$\operatorname{SL}_{n}(\overline{\mathbb{F}_{p}})$, so must coincide. This is
precisely to say that the Galois action on $G_{12}$ is unramified at $p$, so
we have proved Claim 2 with $\mathcal{P}$ the set of primes less than $P$. ∎

Now, with $y\in Y(\mathbb{Z}[S^{-1}])$, there is an injection

$\mbox{$G(\mathbb{Q})$-orbits on $Gy\cap
Y(\mathbb{Q})$}\hookrightarrow\mbox{$G_{y}$-torsors over $\mathbb{Q}$}$

sending $y^{\prime}\in Y(\mathbb{Q})$ to the right torsor
$\\{g:gy=y^{\prime}\\}$. The Claim means that, under this map, elements of
$Y(\mathbb{Z}[S^{-1}])$ are sent to unramified-away-from-$\mathcal{P}$ torsors
for the unramified-away-from-$\mathcal{P}$-group $G_{y}$, whose order is
moreover bounded. Hermite-Minkowski gives an upper bound on the size of
unramified Galois $H^{1}$ in this setting, and we conclude that the
$S$-integral points lying in $Gy\cap Y(\mathbb{Q})$ lie in a collection of
$G(\mathbb{Q})$-orbits whose cardinality is bounded in terms of $d,n,S$.
Finally, we pass to $G(\mathbb{Z}[S^{-1}])$-orbits using (7). This proves the
first statement of the Corollary.

In the last sentence of the argument we used (7) only at $p\notin S$; but
using it also for $p\in S$ allows one to conclude similarly that

(8) $\\{y\in Y(\mathbb{Z}):\mathrm{disc}(y)=N,\mbox{ht.}(y)\leq B\\}$

is covered by $O_{\epsilon}(B^{\epsilon})$ orbits of
$\operatorname{SL}_{n}(\mathbb{Z})$. Here $\mathrm{ht}(y)$ just refers to the
largest coefficient of $y$, rather than a Weil height. This is the second
statement of the Corollary.

### Reduction of Theorem 1.1 to Theorem 1.2

Suppose that $X^{\circ}$ is quasi-projective and $X^{\circ}_{\mathbb{C}}$
admits a geometric variation of Hodge structure, i.e. there is a morphism

$\pi_{\mathbb{C}}:\mathfrak{X}_{\mathbb{C}}\rightarrow X_{\mathbb{C}}^{\circ}$

such that the variation of Hodge structure in the theorem statement is a
direct summand of $R^{i}\pi_{\mathbb{C}*}\mathbb{C}$, for some $i\geq 0$.
(That such $\mathfrak{X}$ exists is understood to be the content of the word
“geometric.”)

Theorem 1.1 assumes that the period morphism associated to $\mathsf{V}$ is
locally finite-to-one. In this case, the period morphism associated to
$R^{i}\pi_{\mathbb{C}*}\mathbb{C}$ has the same property. The issue to be
handled is that $\mathfrak{X}_{\mathbb{C}}$ is defined only over $\mathbb{C}$
whereas Theorem 1.2 requires a $K$-morphism as input.

To verify sparsity it will suffice, by descending Noetherian induction and
extending the base field, to produce a proper Zariski-closed subset $E\subset
X_{\overline{K}}$ with the property that integral points on $X^{\circ}$ that
do not lie inside $E$ are sparse. (Warning: this is not the same as studying
“integral points on the quasi-projective variety $X^{\circ}\backslash E$.”)

By a standard spreading out technique we may extend
$\pi_{\mathbb{C}}:\mathfrak{X}_{\mathbb{C}}\rightarrow X_{\mathbb{C}}$ over
the spectrum of a subring $R\supset K$ of $\mathbb{C}$, finitely generated
over $K$:

$\pi:\mathfrak{X}_{R}\rightarrow X_{R}=X\times_{\mathrm{Spec}\ K}S,$

where $S$ is the spectrum of $R$. This recovers $\pi_{\mathbb{C}}$ upon taking
the pullback via the map $s:\operatorname{Spec}\mathbb{C}\rightarrow S$
associated to $R\rightarrow\mathbb{C}$.

The image of $s$ is the generic point $\eta_{S}$ of $S$. Since $S$ is the
spectrum of the finitely generated integral domain $R$,
$S\rightarrow\operatorname{Spec}K$ is smooth at $\eta_{S}$. Then, deleting the
nonsmooth locus, we may suppose that $S$ is a smooth $K$-variety.

Consider the morphism of vector bundles supplied by Lemma 2.1, which, applied
to the morphism $\mathfrak{X}_{R}\rightarrow X\times S$, gives a morphism of
locally free sheaves

(9) $g:T_{X\times S}\otimes\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$

over $X\times S$. Let $T_{X}\subset T_{X\times S}$ be the sub-bundle defined
by the pullback of the tangent bundle of $T_{X}$. Since $\pi_{\mathbb{C}}$ has
finite-to-one period map, the specialization $g_{x,s}|_{T_{X}}$ is injective
for each $x\in X(\mathbb{C})$; the same is then true for a Zariski-open
neighbourhood $U$ of $X\times\\{\eta_{S}\\}$ inside $X\times S$.

Choose $s^{\prime}\in S(\overline{K})$ such that $X(\overline{K})\times
s^{\prime}$ meets $U(\overline{K})$. The fiber of $\mathfrak{X}_{R}\rightarrow
X_{R}$ over $s^{\prime}$ gives a proper smooth morphism of $\bar{K}$-varieties
$\mathfrak{X}_{s^{\prime}}\rightarrow X_{\bar{K}}$. The associated period map
has generically injective derivative; let $E\subset X_{\bar{K}}$ be the locus
where its derivative has a nontrivial kernel, i.e., where
$g_{x,s^{\prime}}|T_{X}$ fails to be injective. $E$ is a proper Zariski closed
subset of $X$ defined over some finite extension $K_{1}\supset K$, and Theorem
1.2 (applied after passage to $K_{1}$) implies that integral points on
$X^{\circ}$ that lie on the complement of $E$ are sparse. This establishes the
inductive step, and therefore concludes the reduction of Theorem 1.1 to
Theorem 1.2.

##  4\. Proof of Theorem 1.2

The remainder of the paper is devoted to the proof of Theorem 1.2. We use
notation as in the statement. We will fix throughout a good integral model for
both $(X,Z)$ and the morphism $\mathfrak{X}\rightarrow X^{\circ}$, as in
Definition 1.4.

As we have noted, the proof involves proving that integral points of $X$ lie
on various collections of subvarieties, whose dimension will be steadily
reduced until they are either points or fibers of the period map. The key
inductive statement used to reduce the dimension of the subvarieties is Lemma
4.2.

It may be helpful to note, in advance, that we will not need to keep track of
any integral structure on our subvariety. The notion of “integral point on a
subvariety” will simply mean a $K$-rational point of the subvariety that is
integral as a rational point on $X$.

###  4.1. Large fundamental group.

Enlarging $S$ if necessary, we fix a prime $p\in S$ for which the integral
cohomology of complex fibers of $\mathfrak{X}$ over $X^{\circ}$ is
$p$-torsion-free, say of rank $r$ over $\mathbb{Z}$. Fixing $x\in
X^{\circ}(\mathbb{C})$ we get a monodromy representation of the topological
fundamental group

(10) $\pi_{1}(X^{\circ,{\text{an}}},x)\longrightarrow
G_{n}:=\mathrm{Aut}(H^{i}(\mathfrak{X}_{x},\mathbb{Z}/p^{n}))\simeq\operatorname{GL}_{r}(\mathbb{Z}/p^{n}\mathbb{Z}).$

In this setting, the global invariant cycle theorem ([9, Corollaire 4.1.2 and
4.1.3.3]) implies the following statement: For any complex irreducible
subvariety $\iota:V\hookrightarrow X_{\mathbb{C}}$, not contained in
$Z_{\mathbb{C}}$ and with $V^{\circ}$ not contained in a fiber of the period
map, the image of the monodromy representation

(11) $\pi_{1}(V^{\circ,{\text{an}}})\rightarrow G_{n}$

(topological $\pi_{1}$, taken for an arbitrary choice of basepoint) has size
that grows without bound as $n\rightarrow\infty$. Indeed [9, Corollaire 4.1.2
and 4.1.3.3] applies, after passing from $V$ to the smooth part $V^{\prime}$
of its intersection with $X^{\circ}$, to show that the image of

$\pi_{1}({V^{\prime}}^{{\text{an}}})\longrightarrow\mathrm{Aut}\
H^{i}(\mathfrak{X}_{x},\mathbb{Q})$

is infinite. In particular, the image of the monodromy representation of
$\pi_{1}((V^{\circ})^{{\text{an}}})$ on $H^{i}(\mathfrak{X}_{x},\mathbb{Z})$
is infinite, so the size of the image of monodromy on
$H^{i}(\mathfrak{X}_{x},\mathbb{Z})/p^{n}$ grows without bound as
$n\rightarrow\infty$.

These results transpose, as usual, to the étale topology. Indeed,
$R^{i}\pi^{\mathrm{et}}_{*}(\mathbb{Z}/p^{n}\mathbb{Z})$ defines a locally
constant étale sheaf of $\mathbb{Z}/p^{n}$-modules on $X^{\circ}$, which, by
the local constancy of direct images for a smooth proper morphism ([8, Theorem
5.3.1]), extends to a locally constant étale sheaf on the good integral model
$X^{\circ}_{S}$ over $\mathfrak{o}_{S}$.

For $V$ as above, standard comparison theorems show that the homomorphism in
(11) factors as

$\pi_{1}(V^{\circ,{\text{an}}})\rightarrow\pi_{1}^{\mathrm{et}}(V^{\circ})\rightarrow
G_{n},$

so the image of étale $\pi_{1}$

(12) $\pi_{1}^{\mathrm{et}}(V^{\circ})\rightarrow G_{n}$

(again, with an arbitrary geometric basepoint in $(V^{\circ})$) has size that
grows without bound as $n\rightarrow\infty$.

Next, let $E\supset K$ be an arbitrary algebraically closed field, with base
change $X_{E}:=X\times_{K}E$; let $i:V\hookrightarrow X_{E}$ be a closed
$E$-subvariety not contained in $Z_{E}$ or a fiber of the period map. (Note
that we can make sense of the latter condition without reference to
$\mathbb{C}$ by using Lemma 2.1: by “$V$ is contained in a fiber of the period
map” we mean that the associated morphism of vector bundles is zero on the
smooth locus of $V$.)

We get by base change $\pi_{E}:\mathfrak{X}_{E}\rightarrow X_{E}$ and an étale
local system $R^{i}\pi^{\mathrm{et}}_{E*}(\mathbb{Z}/p^{n}\mathbb{Z})$ on
$X_{E}^{\circ}$; and the same conclusion as above holds, i.e. the monodromy
representation (12) for $V$ on
$R^{i}\pi^{\mathrm{et}}_{*}(\mathbb{Z}/p^{n}\mathbb{Z})$ has “large image” in
the sense specified above. We will use this only in the case when $E$ is the
algebraic closure of a finitely generated field; we may then choose a
$K$-embedding $\sigma:E\rightarrow\mathbb{C}$, and thus also
$V_{\mathbb{C}}\subset X_{\mathbb{C}}$ compatibly with $V\subset X_{E}$. The
local system $R^{i}\pi^{\mathrm{et}}_{E*}(\mathbb{Z}/p^{n}\mathbb{Z})$ on
$V_{E}$ pulls back to the similarly defined system on $V_{\mathbb{C}}$, so
“large monodromy” for $V$ follows from the same statement for
$V_{\mathbb{C}}$.

### Construction of a suitable cover of $X$

Our proof will involve an induction over higher and higher-codimension
subvarieties of $X$ about which we know almost nothing apart from their
degree. It is thus crucial to have at hand covers of $X$ whose monodromy is
uniformly bounded below on restriction to every subvariety of bounded
dimension and degree.

######  Lemma 4.1.

Fix $d,n,D\geq 1$, and let $H$ be any (locally closed, finite type) complex
subvariety of the Hilbert scheme of subschemes of $X_{\mathbb{C}}$ of degree
$\leq d$ and dimension $n$.

There are a finite group $G$ and a finite morphism $f:\widetilde{X}\to X$ of
$K$-varieties, equipped with an injection
$G\hookrightarrow\mathrm{Aut}(\widetilde{X}/X)$, such that:

  * (a)

$f|_{X^{\circ}}$ is finite étale Galois with deck group $G$, and, moreover,
extends to a finite étale cover of the good integral model $X^{\circ}_{S}$.

  * (b)

Let $U\subset X$ be any $n$-dimensional irreducible closed complex subvariety
of degree $\leq d$. Suppose that:

    * –

The point of the Hilbert scheme classifying $U$ lies in $H(\mathbb{C})$, and

    * –

$U$ is not contained in $Z$ and $U^{\circ{\text{an}}}$ is not contained in a
single fiber of the period map $\Phi$.

Let $Q$ be any irreducible component of $f^{-1}U$, endowed with the reduced
structure, and such that the induced finite map $f:Q\rightarrow U$ is dominant
(note that it is automatically étale over $U^{\circ}$). Then the degree of
$f:Q\rightarrow U$ at the generic point is $\geq D$, i.e., the induced map of
function fields has degree $\geq D$.

###### Proof.

Through the rest of this proof, $U$ will represent a single subvariety of $X$,
classified by a point of the Hilbert scheme, and we will use $\mathcal{U}$ for
the universal family.

We are going to find a cover $f:\widetilde{X}\to X$ and a proper Zariski-
closed subset $H_{1}\subseteq H$ such that the conclusion of (b) holds for any
$U=U_{h}$, satisfying the assumptions of (b), and with
$h\in(H-H_{1})(\mathbb{C})$. The result will follow by Noetherian induction.
In particular, removing the singular locus of $H$ at the start, we may suppose
that $H$ is smooth.

Take a geometric generic point $\eta\rightarrow H$. Let $U_{\eta}\subset
X_{\eta}$ be the corresponding generic subscheme. We may assume without loss
of generality that:

> Situation:
>
>   * (a)
>
> Every geometric fiber of $\mathcal{U}\rightarrow H$ is integral;
>
>   * (b)
>
> Every fiber of $\mathcal{U}\rightarrow H$ meets $X^{\circ}$;
>
>   * (c)
>
> On each fiber $\mathcal{U}_{h}$ for $h\in H(\mathbb{C})$ the period map is
> not locally constant.
>
>

For (a), note that the locus of points with geometrically integral fiber by
[13, 12.2.1(x)] is open on the base, so if there exists one $h\in
H(\mathbb{C})$ for which the fiber $U_{h}$ is integral, then (after shrinking
$H$ to a suitable nonempty open neighbourhood) we can suppose it is true for
all $h$. If there is no such $h$, then the conclusion of the theorem holds for
$H$ vacuously.

For (b), note that the set of $h$ for which $U_{h}$ meets $X^{\circ}$ is
constructible. To see this, first note that $U_{h}$ is reduced for every $h$.
Now note that $U_{h}$ meets $X^{\circ}$ if and only if $(U_{h}\cap
Z)_{s}\rightarrow U_{s}$ is not surjective, and apply [13, 9.6.1(i)]. Thus,
restricting to an open subset of $H$, we can assume that $U_{h}$ meets
$X^{\circ}$ either for no $h$ or for all $h$. In the former case, the
statement is vacuously true; so we can assume that (b) holds for all $h$.

For (c) let $\mathcal{U}^{\prime}$ be the smooth locus of the morphism
$\mathcal{U}\rightarrow H$, which, by flatness of the morphism, coincides with
the locus of points which are smooth points of their fibers ([12, 17.5.1]).
Note that:

  * •

$\mathcal{U}^{\prime}$ is itself smooth over $\mathrm{Spec}\ K$, since it is
smooth over $H$ and $H$ was assumed smooth.

  * •

$\mathcal{U}^{\prime}$ contains an open dense subset of every fiber, since
these fibers are all integral.

  * •

$\mathcal{U}^{\prime}$ is a $K$-variety: this follows from the previous
conditions. It is reduced by smoothness, and since
$\mathcal{U}^{\prime}\rightarrow H$ is flat ([29, Tag 01VF]), $H$ is
irreducible, and the fibers are irreducible, it readily follows ([29, Tag
004Z]) that $\mathcal{U}^{\prime}$ is itself irreducible.

The tangent bundle $T_{\mathcal{U}^{\prime}}$ has a sub-bundle
$T_{\mathcal{U}^{\prime}/H}$ made up of‘vertical” vector fields. Restricting
the morphism of Lemma 2.1 to this sub-bundle we get

$g:\mathcal{H}_{1}\otimes
T_{\mathcal{U}^{\prime}/H}\rightarrow\mathcal{H}_{2}$

Now, we may certainly assume there is some $h\in H(\mathbb{C})$ such that
$\mathcal{U}_{h}$ satisfies the conditions of (b) in the statement of the
Lemma, or else the Lemma once again holds vacuously. In particular, there
exists a point $u\in\mathcal{U}_{h}(\mathbb{C})$, smooth in the fiber
$\mathcal{U}_{h}$, such that $g_{u}$ is nonzero. It follows that $g_{u}$ is
nonzero on a nonempty Zariski-open subset of $\mathcal{U}^{\prime}$; the image
of this Zariski-open by the dominant morphism $\mathcal{U}^{\prime}\rightarrow
H$ contains an nonempty open subset of $H$, and we replace $H$ by this open to
obtain the second part of the Situation.

So we proceed assuming ourselves to be in the Situation above. We continue to
write $\mathcal{U}^{\prime}$ for the smooth locus of $\mathcal{U}/H$ and
$\mathcal{U}^{\prime\circ}$ for the preimage of $X^{\circ}$ in
$\mathcal{U}^{\prime}$. Recall that our assumptions guarantee that
$\mathcal{U}^{\prime\circ}$ is fiberwise dense in $\mathcal{U}^{\prime}$.

By §4.1 we can find an $m$ for which the image of the geometric monodromy
representation of $\pi_{1}(U_{\eta}^{{}^{\prime}\circ})$ in $G_{m}$ has size
at least $D$ (same notation as in §4.1). This choice of $m$ determines a
finite étale Galois cover of $X^{\circ}$ with Galois group $G_{m}$, which
extends to a finite étale Galois cover of the $\mathfrak{o}_{S}$-model
$X^{\circ}_{S}$.

Let $f:\widetilde{X}\rightarrow X$ be the normalization of $X$ in this $G_{m}$
cover; then $\widetilde{X}$ is a normal $K$-variety and the morphism $f$ is
finite (although not necessarily flat). The action of $G_{m}$ by deck
transformations above $X^{\circ}$ extends uniquely to a $G_{m}$-action on the
morphism $f$.

The morphism $f$ gives of course a morphism $f:\widetilde{X}_{\eta}\rightarrow
X_{\eta}$ after base-change from $\operatorname{Spec}\ K$ to $\eta$. The
restricted map

(13) $f_{\eta}^{-1}\mathcal{U}_{\eta}^{\prime\circ}\rightarrow
U_{\eta}^{\prime\circ}$

is finite étale and has degree $\geq D$ restricted to each geometric component
of the source by choice of $f$. Now this morphism is the geometric generic
fiber of a finite étale morphism of smooth $H$-schemes:

(14) $f^{-1}\mathcal{U}^{\prime\circ}\rightarrow\mathcal{U}^{\prime\circ}$

and we want to draw the same conclusion about degrees for the fibers of (14)
over a nonempty open subset of $H$. This will imply the desired conclusion,
for – with $Q$ as in the statement – the assumed dominance implies that $Q\cap
f^{-1}U^{\prime\circ}$ is an open nonempty subset of $Q$.

We now use [29, Lemma 055A] (see also [13, Prop. 9.7.8] and references
therein). which guarantees the existence of a morphism
$g:H^{\prime}\rightarrow H$ (which in fact factors as a finite étale
surjection followed by an open immersion) such that, after base change of
$f^{-1}\mathcal{U}^{\prime\circ}\rightarrow\mathcal{U}^{\prime\circ}$ by $g$ –
i.e. replacing $\mathcal{U}^{\prime\circ}$ by
$\mathcal{U}^{\prime\circ}\times_{H}H^{\prime}$ and similarly for
$f^{-1}\mathcal{U}^{\prime\circ}$ – the following assertions hold:

  * (a)

Each irreducible component of the generic fiber
$(f^{-1}\mathcal{U}^{\prime\circ})_{\eta^{\prime}}$ (with $\eta^{\prime}$ the
generic point of $H^{\prime}$ – not a geometric generic point here) is in fact
a geometrically irreducible component of that generic fiber.

  * (b)

Let $\overline{Z_{1}},\dots,\overline{Z_{r}}$ be the Zariski closures of these
generic irreducible components $Z_{1},\dots,Z_{r}$ inside
$f^{-1}\mathcal{U}^{\prime\circ}$. These $\overline{Z_{i}}$ give, upon
intersection with the fiber $(f^{-1}\mathcal{U}^{\prime\circ})_{h^{\prime}}$
above any $h^{\prime}\in H^{\prime}$, the decomposition of that fiber into
irreducible components, and indeed each of these irreducible components are
geometrically irreducible.

In the decomposition of (b)

$(f^{-1}\mathcal{U}^{\prime\circ})=\coprod\overline{Z_{\alpha}}$

the sets $Z_{\alpha}$ are disjoint. Indeed, upon restriction to each fiber,
this decomposition recovers the decomposition of
$f^{-1}\mathcal{U}^{\prime\circ}_{h^{\prime}}$; however, this is finite étale
over $\mathcal{U}^{\prime\circ}_{h^{\prime}}$ and Lemma 2.3 implies the
disjointness. In particular, the $\overline{Z_{\alpha}}$ are both closed and
open, and in particular inherit a scheme structure as open sets in
$f^{-1}\mathcal{U}^{\prime\circ}$. The restriction of the map $f$ to each
$\overline{Z_{\alpha}}$ is then a finite étale map
$f_{\alpha}:\overline{Z_{\alpha}}\rightarrow\mathcal{U}^{\prime\circ}$. The
degree of such a map is locally constant on the base, and here, by assumption,
that degree is $\geq D$ everywhere on the generic fiber
$\mathcal{U}^{\prime\circ}_{\eta^{\prime}}$. That generic fiber is dense
because it contains every generic point of of $\mathcal{U}^{\prime\circ}$,
and, consequently the degree of $f_{\alpha}$ is everywhere $\geq D$.
Restricting to a single fiber $\mathcal{U}^{\prime\circ}_{h^{\prime}}$ for
$h^{\prime}\in H^{\prime}$ gives the desired conclusion – that is, the bound
stated in (b) of the Theorem holds for all fibers $U_{h}$ for all $h$ in a
nonenmpty open subset of $H$, explicitly, the image of $H^{\prime}\rightarrow
H$.

Finally, we conclude by Noetherian induction. We have shown that, given $H$,
there is a Zariski-closed $H_{1}\subseteq H$ and an $m>0$, such that the image
of geometric monodromy in $G_{m}$ gives an étale cover for $X$, that satisfies
the required properties for any $h\in(H-H_{1})(\mathbb{C})$. There is no harm
in replacing $G_{m}$ by $G_{m^{\prime}}$, for $m^{\prime}\geq m$. Thus we can
apply Noetherian induction to find one $m$ that works for all $h\in
H(\mathbb{C})$. ∎

### Bounding rational points on a subvariety

The following result is the key inductive step. We note that all constants
appearing in this discussion are permitted to depend on the variety $X$, and
indeed on the integral model chosen in §1.2, without explicit mention.

######  Lemma 4.2.

Let $V$ be a geometrically irreducible closed subvariety of $X$ defined over
$K$, of dimension $n$ and degree $d$, such that $V_{\mathbb{C}}$ is not
contained in $Z$ and $V^{\circ{\text{an}}}$ is not contained in a fiber of
$\Phi$.

Then all integral points of $X^{\circ}$ of height $\leq B$ that lie on $V$ can
be covered by $O_{d,\epsilon}(B^{\epsilon})$ irreducible (but not necessarily
geometrically irreducible) subvarieties, all defined over $K$, with dimension
$\leq n-1$ and degree $O_{d,\epsilon}(1)$.

###### Proof.

Choose $D$ so that $\frac{n+1}{D^{1/n}}<\epsilon$. Let $\mathcal{P}$ be the
(finite, by Lemma 2.4) set of Hilbert polynomials that arise from irreducible
subvarieties of $X$ of dimension $n$ and degree $d$, and let $H_{\mathcal{P}}$
be the associated Hilbert scheme. Write $H^{red}_{\mathcal{P}}$ for the
reduced induced closed subscheme of $H_{\mathcal{P}}$.

For this choice of $n,d,D,H=H^{red}_{\mathcal{P}}$ take a finite group $G$ and
a finite morphism $f:\widetilde{X}\rightarrow X$ as provided by Lemma 4.1. We
note that $H_{\mathcal{P}}(\mathbb{C})=H^{red}_{\mathcal{P}}(\mathbb{C})$, so
the passage to the reduced subscheme structure is irrelevant for the
statements on complex subvarieties proved in Lemma 4.1.

For every $x\in X^{\circ}(K)$, the action of $G$ on the fiber of
$\widetilde{X}$ over $x$ defines a class in the Galois cohomology group
$H^{1}(\operatorname{Gal}(\overline{K}/K),G)$. Concretely, since the Galois
action on $G$ is trivial, $H^{1}(\operatorname{Gal}(\overline{K}/K),G)$
classifies homomorphisms $\rho:\operatorname{Gal}(\overline{K}/K)\rightarrow
G$ up to conjugacy. Choosing a point $\widetilde{x}$ above $x$, we define a
homomorphism $\rho$ by the rule

(15) $\sigma(\widetilde{x})=\rho(\sigma)\cdot\widetilde{x}$

for $\sigma$ in the Galois group. If $x$ is $S$-integral, this homomorphism
$\rho$ is in fact unramified outside $S$. Such a $\rho$ can also be used to
twist $\widetilde{X}\rightarrow X$, namely, one modifies the Galois action on
$\widetilde{X}$ through $\rho$; and then (15) means precisely that $x$ will
lift to a $K$-rational point on the twist of $\widetilde{X}$ indexed by
$\rho$. (See $\S$ 4.5 and Thm. 8.4.1 of [24] for further discussion.)

There are only finitely many homomorphisms
$\operatorname{Gal}(\overline{K}/K)\rightarrow G$, unramified outside $S$;
call them $\rho_{1},\rho_{2},\dots,\rho_{R}$. This list does not depend on
$B$. Each such $\rho_{j}$ can be used to twist $f$ to a map
$f_{j}:\widetilde{X}_{j}\rightarrow X$. Our previous discussion now shows that
any integral point of $X^{\circ}$ lifts along some $f_{j}$ to a point of
$\widetilde{X}_{j}(K)$.

For a sufficiently large integer $e$ the pullback
$(f_{j}^{*}\mathcal{L})^{\otimes e}$ is very ample and defines, after fixing a
basis of sections, a projective embedding
$\widetilde{X}_{j}\hookrightarrow\mathbb{P}^{M_{j}}$. Now the data of the
diagram of $K$-varieties and line bundles

(16) $(X,\mathcal{L}^{\otimes e})\stackrel{{\scriptstyle
f_{j}}}{{\longleftarrow}}(\widetilde{X}_{j},(f_{j}^{*}\mathcal{L})^{\otimes
e})\hookrightarrow(\mathbb{P}^{M_{j}},\mathcal{O}(1))$

depends on various choices, but these choices can (and will) be made once and
for all depending only on $d,\epsilon$. Then for $P\in\widetilde{X}_{j}(K)$ we
get

(17) $H_{\mathcal{L}}(f_{j}(P))^{e}\asymp
H_{f_{j}^{*}\mathcal{L}}(P)^{e}\asymp H_{\mathbb{P}^{M_{j}}}(P)$

where the symbol $\asymp$ means that the ratio is bounded above and below by
constants that may depend on $f_{j}$. Since there are only finitely many
$f_{j}$, and their coefficients are bounded in terms of $d$ and $\epsilon$
(and, as always, $X$ and $S$) but don’t depend on $B$, these constants depend
only on $d$ and $\epsilon$.

Therefore, we have shown that the integral points of $X^{\circ}$ with height
$\leq B$ belonging to $V$ all have the form $f_{j}(P)$, where $P$ is a
$K$-rational point of $f_{j}^{-1}(V)$ with $H_{\mathbb{P}^{M_{j}}}(P)\leq
c_{d,\epsilon}B^{e}$. It will suffice to prove the conclusion for those $P$
for which $f_{j}(P)$ is a smooth point of $V$, simply by including each
irreducible component of the singular locus of $V$ in the list of subvarieties
(see Lemma 2.4 part (d) for the necessary bounds).

Let $V^{\prime}\subset V$ be the (open) smooth locus. Consider those geometric
components $Q^{\circ}\subset(f_{j}^{-1}V^{\prime})^{\circ}$ that have a
$K$-rational point. Because $(f_{j}^{-1}V^{\prime})^{\circ}$ is a finite étale
cover of the geometrically irreducible smooth $K$-variety $V^{\prime\circ}$,
its geometric components are pairwise disjoint (Lemma 2.3) and permuted by the
Galois group; so any such $Q^{\circ}$ is defined over $K$ and the number of
such $Q^{\circ}$ is bounded in number by the size of the group $G$.

The Zariski closure $Q$ of any $Q^{\circ}$ is again geometrically irreducible
and defined over $K$; we understand it to be endowed with its reduced scheme
structure. The map $f_{j}:\widetilde{X}_{j}\rightarrow X$ induces a compatible
map $f_{j}:Q\rightarrow V$, which is dominant since, by construction of $Q$,
the image contains a nonempty open set of $V^{\prime\circ}$. Indeed,
$f_{j}:Q\rightarrow V$ is étale over $V^{\circ}$, with degree between $D$ and
$(\\#G)$; the lower bound comes from (b) of Lemma 4.1, using also the fact
that $f_{j}$ is a twist of $f$.

The degree of $V$ with respect to $\mathcal{L}^{\otimes e}$ is $de^{n}$, and
therefore, by Lemma 2.2 the degree of $Q$, considered as a closed subvariety
of $\mathbb{P}^{M_{i}}$ via (16), satisfies

$Dde^{n}\leq\mathrm{deg}Q\leq(\\#G)de^{n}.$

We apply Theorem 2.6 to each $Q$ that arises in the above fashion, i.e. to the
Zariski closure of any irreducible geometric component of
$(f_{j}^{-1}V^{\prime})^{\circ}$ that has a $K$-point. Theorem 2.6 and our
choice of $D$ implies that the set of rational points of $Q$ of height $\leq
cB^{e}$ are supported on a set of proper closed subvarieties of $Q$ of degree
$O_{d,\epsilon}(1)$ with cardinality $\ll_{d,\epsilon}B^{2\epsilon}$. These
subvarieties are defined over $K$ and need not be geometrically irreducible.

For any such $Q$ and any such proper subvariety $Y\subset Q$, the scheme-
theoretic image $f_{j}(Y)$ under the finite map $f_{j}$ is a proper subvariety
$f_{j}(Y)\subset V$, in particular, of dimension $\leq n-1$. Moreover, $f_{j}$
restricts to a finite map $Y\rightarrow f_{j}(Y)$. By Lemma 2.2 the
$\mathcal{L}$-degree of $f_{j}(Y)$ is no larger than the
$f_{j}^{*}\mathcal{L}$-degree of $Y$, in particular, $O_{d,\epsilon}(1)$.

The number of maps $f_{j}$ depends only on $d,\epsilon$, and the number of $Q$
arising is then at most the number of $f_{j}$ multiplied by the order of $G$,
which is again $O_{d,\epsilon}(1)$. Consequently, the number of $Y$ arising as
in the prior paragraph is $O_{d,\epsilon}(B^{2\epsilon})$, concluding the
proof (after the obvious scaling $\epsilon\leftarrow\epsilon/2$.) ∎

###  4.2. Conclusion of the proof of Theorem 1.2

###### Proof.

Fix $\epsilon>0$. We use descending induction via Lemma 4.2. The inductive
statement is the following:

> $(\star)_{n}$: For every $n$ with $0\leq n\leq\dim X$, there exists an
> integer $d_{n}$ with the following property: for all $B>0$, the $S$-integral
> points of $X^{\circ}$ are covered by a collection of
>
> $O_{\epsilon}(B^{(\dim X-n)\epsilon})$
>
> irreducible subvarieties of $X$, all defined over $K$, each of which is
> either
>
>   * –
>
> $\textrm{(a)}_{n}$: a subvariety of dimension $\leq n$ and degree $\leq
> d_{n}$, or
>
>   * –
>
> (b): a geometrically irreducible subvariety that is contained in a single
> fiber of the period map.
>
>

The base case is given by $n=\dim X$, in which case, of course, the single
subvariety $X\subseteq X$ suffices.

The implication $(\star)_{n}\implies(\star)_{n-1}$ follows from Lemma 4.2: Let
$\mathcal{V}_{n}$ be the collection of $n$-dimensional varieties in the
statement of $(\star)_{n}$. For each $V\in\mathcal{V}_{n}$, we will construct
a set of varieties covering all the integral points of $X^{\circ}$ lying on
$V$. We subdivide into cases:

  * •

$V$ is not geometrically irreducible. In this case, we take the set
$\\{V_{i}\\}$ of subvarieties given by part (c) of Lemma 2.4. These varieties
number at most $O_{n,d_{n}}(1)$ and they have dimension $\leq n-1$ and degree
$O_{n,d_{n}}(1)$.

  * •

$V$ is geometrically irreducible but $V^{\circ{\text{an}}}$ is contained in a
fiber of $\Phi$: then we take the singleton set $\\{V\\}$.

  * •

$V$ is contained in $Z$; in this case we can take the empty set $\emptyset$.

  * •

$V$ is geometrically irreducible and not contained in $Z$, and
$V^{\circ{\text{an}}}$ is not contained in a fiber of $\Phi$; then we may
apply Lemma 4.2 to show that integral points of height $\leq B$ on $V$ are
covered by $O_{d_{n},\epsilon}(B^{\epsilon})$ irreducible $K$-varieties of
dimension $\leq n-1$ and degree $O_{d_{n},n,\epsilon}(1)$.

We take $d_{n-1}$ to be the largest of the implicit constants $O_{n,d_{n}}(1)$
and $O_{n,d_{n},\epsilon}(1)$ appearing in the above proof. Then, to sum up,
by $(\star)_{n}$ we know that the $S$-integral points of $X^{\circ}$ of height
at most $B$ are covered by $O_{\epsilon}(B^{(\dim X-n)\epsilon})$ subvarieties
$V$ satisfying either $\textrm{(a)}_{n}$ or (b), and we know that for each of
those $V$, the subset of those points lying on $V$ is covered by
$O_{\epsilon}(B^{\epsilon})$ subvarieties satisfying either
$\textrm{(a)}_{n-1}$ or (b); together, these facts yield $(\star)_{n-1}$.

We emphasize that this is the point in the argument where the uniformity in
Broberg’s result is crucial. We have no control of the heights of the
varieties making up the collection $\mathcal{V}_{n}$, and indeed these heights
will grow with $B$; but since the implicit constants in Lemma 4.2 depend only
on $X$ and $\epsilon$, not on $V$, this lack of control does not present a
problem.

The case $n=0$ gives the Theorem. ∎

##  5\. Author affiliations

Jordan S. Ellenberg, University of Wisconsin

Brian Lawrence, University of California, Los Angeles<EMAIL_ADDRESS>
Akshay Venkatesh, Institute for Advanced Study

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