# The behavior of essential dimension under specialization II

Zinovy Reichstein  Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2
Canada<EMAIL_ADDRESS>and  Federico Scavia  Department of Mathematics
University of California
Los Angeles, CA 90095-1555
USA<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a linear algebraic group over a field. We show that, under mild
assumptions, in a family of primitive generically free $G$-varieties over a
base variety $B$ the essential dimension of the geometric fibers may drop on a
countable union of Zariski closed subsets of $B$ and stays constant away from
this countable union. We give several applications of this result.

###### Key words and phrases:

Essential dimension, algebraic group, group action, torsor, complete
intersection.

###### 2020 Mathematics Subject Classification:

14L30, 20G15, 13A18

Zinovy Reichstein was partially supported by National Sciences and Engineering
Research Council of Canada Discovery grant 253424-2017.

Federico Scavia was partially supported by a graduate fellowship from the
University of British Columbia.

##  1\. Introduction

Let $X$ be a complex algebraic variety (that is, a separated reduced
$\mathbb{C}$-scheme of finite type) equipped with a faithful action of a
finite group $G$. We will refer to $X$ as a $G$-variety. Assume that the
$G$-action on $X$ is primitive, that is, $G$ transitively permutes the
irreducible components of $X$. In this paper we will be interested in the
essential dimension $\operatorname{ed}_{\mathbb{C}}(X;G)$ and how it behaves
in families. Essential dimension is an integer-valued birational invariant of
the $G$-variety $X$; its definition can be found in Section 2. When the group
$G$ is clear from the context, we will simply write
$\operatorname{ed}_{\mathbb{C}}(X)$ for $\operatorname{ed}_{\mathbb{C}}(X;G)$.

To date the study of essential dimension has been primarily concerned with
understanding versal $G$-varietes (once again, see Section 2 for the
definition). A complete versal $G$-variety $X$ has the following special
property: $X$ has an $A$-fixed rational point for every abelian subgroup
$A\subset G$; see [Mer09, Corollary 3.21]. At the other extreme are complete
$G$-varieties $X$, where the action of $G$ is free, i.e., no non-trivial
element has a fixed point. Existing methods for proving lower bounds on
$\operatorname{ed}_{\mathbb{C}}(X)$ usually fail here; we are aware of only a
small number of interesting examples of finite group actions, where
$\operatorname{ed}_{\mathbb{C}}(X)$ has been computed in this setting.

One such family concerns the action of $G=(\mathbb{Z}/p\mathbb{Z})^{n}$ on the
product of elliptic curves $X=E_{1}\times\ldots\times E_{n}$ over
$\mathbb{C}$. Here $p$ is a prime; the generator of the $i$th copy of
$\mathbb{Z}/p\mathbb{Z}$ acts on $E_{i}$ via translation by a point $x_{i}\in
E_{i}(\mathbb{C})$ of order $p$, and trivially on $E_{j}$ for $j\neq i$. J.-L.
Colliot-Thélène and O. Gabber [CT02, Appendice] showed that for a very general
choice of the elliptic curves $E_{i}$, a certain degree $n$ cohomological
invariant of $G$ does not vanish on $\mathbb{C}(X)^{G}$. This implies that
$\operatorname{ed}_{\mathbb{C}}(X)=n=\operatorname{ed}_{\mathbb{C}}(G)$.
Additional examples can be found in recent work of B. Farb, M. Kisin and J.
Wolfson [FKW21a, FKW21b] and N. Fakhruddin and R. Saini [FS21]. Some arise as
congruence covers of Shimura varieties, others from actions of subgroups of
$X[p]\simeq(\mathbb{Z}/p\mathbb{Z})^{2n}$ on a complex abelian variety $X$
(not necessarily a product of elliptic curves).

In the present paper we will work in a more general setting, where $G$ is a
linear algebraic group (not necessarily finite) over an algebraically closed
field $k$. Essential dimension and versality make sense in this more general
setting, provided that we require our $G$-actions to be generically free and
not just faithful; see Section 2.

If $\dim(G)>0$, by Borel’s Fixed Point Theorem $G$ cannot act freely on a
complete variety. Nevertheless, the notion of a free action of a finite group
on a projective variety can be generalized to the case of an arbitrary linear
algebraic group $G$ as follows: a generically free primitive $G$-variety $X$
is said to be _strongly unramified_ if $X$ is birationally $G$-equivariantly
isomorphic to the total space $X^{\prime}$ of a $G$-torsor $X^{\prime}\to P$
over some smooth projective irreducible $k$-variety $P$.

Our first main result is the following.

######  Theorem 1.1.

Let $G$ be a linear algebraic group over an algebraically closed field $k$ of
good characteristic (see Definition 2.1) and of infinite transcendence degree
over its prime field, and let $X$ be a generically free primitive $G$-variety.
Then there exist a strongly unramified $G$-variety $Y$ such that
$\dim(Y)=\dim(X)$ and $\operatorname{ed}_{k}(Y)=\operatorname{ed}_{k}(X)$.

Applying 1.1 to a versal $G$-variety $X$, we obtain a strongly unramified
$G$-variety $Y$ of maximal essential dimension, i.e. such that
$\operatorname{ed}_{k}(Y)=\operatorname{ed}_{k}(G)$. When $G$ is finite, $Y$
is itself smooth and projective. Thus by starting with an incompressible
$G$-variety $X$, we obtain examples analogous to those of Colliot-
Thélène–Gabber, Farb–Kisin–Wolfson and Fakhruddin–Saini for an arbitrary
finite group $G$. Note however, that Farb-Kisin-Wolfson and Fakhruddin-Saini
produce examples over $k=\overline{\mathbb{Q}}$, whereas Theorem 1.1 requires
$k$ to be of infinite transcendence degree over the prime field.

Our proof of 1.1 will rely on Theorems 1.2 and 1.4 below, which are of
independent interest.

######  Theorem 1.2.

Let $G$ be a linear algebraic group over a field $k$ of good characteristic.
Let $B$ be a noetherian $k$-scheme, $f\colon\mathcal{X}\to B$ be a flat
separated $G$-equivariant morphism of finite type such that $G$ acts trivially
on $B$ and the geometric fibers of $f$ are generically free and primitive
$G$-varieties (in particular, reduced). Then for any fixed integer $n\geqslant
0$ the subset of $b\in B$ such that
$\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}};G_{k(\overline{b})})\leqslant
n$ for every (equivalently, some) geometric point $\overline{b}$ above $b$ is
a countable union of closed subsets of $B$.

Furthermore, assume that $k$ is algebraically closed of infinite transcendence
degree over its prime field. (In particular, these conditions are satisfied if
$k$ is algebraically closed and uncountable.) Let $m\geqslant 0$ be the
maximum of
$\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}};G_{k(\overline{b})})$,
where $\overline{b}$ ranges over all geometric points of $B$. Then the set of
those $b\in B(k)$ such that $\operatorname{ed}_{k}(\mathcal{X}_{b};G)=m$ is
Zariski dense in $B$.

Informally Theorem 1.2 can be restated as follows: in a family of
$G$-varieties $\mathcal{X}\to B$, the essential dimension of the geometric
fibers drops on a countable union of Zariski closed subsets of $B$, and stays
constant away from this countable union. Several remarks concerning Theorem
1.2 are in order.

######  Remarks 1.3.

  1. (1)

The assumption that the $G$-action on every geometric fiber $\mathcal{X}_{b}$
of $f$ is generically free and primitive ensures that
$\operatorname{ed}_{k(b)}(\mathcal{X}_{b})$ is well defined.

  2. (2)

The countable union in the statement of 1.2(a) cannot be replaced by a finite
union in general; see Example 5.1.

  3. (3)

The assumption that $f$ is flat is necessary; see Example 5.2. On the other
hand, this assumption is rather mild. For example, when $\mathcal{X}$ and $B$
are smooth $k$-varieties, by “miracle flatness”, $f$ is flat if and only if
all its fibers have the same dimension; see [Mat89, Theorem 23.1]. In the
applications, one is usually interested in showing that the maximal value of
$\operatorname{ed}_{k}(\mathcal{X}_{b})$ is attained at a very general point
$b\in B(k)$. This can be done under a weaker flatness assumption on $f$; see
8.1.

  4. (4)

If $k$ is not algebraically closed, then the $k$-points $b\in B(k)$ such that
$\operatorname{ed}_{k}(\mathcal{X}_{b})\leqslant n$ do not necessarily lie on
a countable union of closed subvarieties of $B$; see Example 5.3. In other
words, 1.2 may fail if we consider fibers of arbitrary closed points instead
of just geometric fibers.

Our proof of 1.2 proceeds as follows. First we choose a subfield $k_{0}\subset
k$ finitely generated over $\mathbb{Q}$, such that
$G=G_{0}\times_{\operatorname{Spec}(k_{0})}\operatorname{Spec}(k)$,
$f=f_{0}\times_{\operatorname{Spec}(k_{0})}\operatorname{Spec}(k)$, and the
assumptions of 1.2 hold for $k_{0}$, $G_{0}$ and $f_{0}:\mathcal{X}_{0}\to
B_{0}$. Then using arguments inspired by Gabber’s appendix [CT02] we reduce
1.2 to the Specialization Property (3.1) and the Rigidity Property (Lemma
4.2).

Note that the Rigidity Property may fail if $k$ is not algebraically closed.
This is the reason why in 1.2 we only consider the geometric fibers; see
Remark 1.3(4).

######  Theorem 1.4.

Let $k$ be an infinite field, $G$ be a finite group, and let $X_{0}$ be an
equidimensional generically free $G$-variety of dimension $e\geqslant 1$ (not
necessarily primitive). Then there exist a smooth irreducible $k$-variety $B$,
a smooth irreducible $G$-variety $\mathcal{X}$ and a smooth $G$-equivariant
morphism $f\colon\mathcal{X}\to B$ of constant relative dimension $e$ defined
over $k$ such that:

  1. (i)

$G$ acts trivially on $B$ and freely on $\mathcal{X}$,

  2. (ii)

there exist $b_{0}\in B(k)$ such that $X_{0}$ is $G$-equivariantly
birationally isomorphic to a union of irreducible components of
$\mathcal{X}_{b_{0}}$,

  3. (iii)

there exists a dense open subscheme $U\subset B$ such that for every $b\in U$
the fiber $\mathcal{X}_{b}$ is smooth, projective and geometrically
irreducible.

In particular, for any a geometric point $b$ of $U$, the $G$-action on the
fiber $\mathcal{X}_{b}$ is strongly unramified.

Our proof of 1.4 can be found in Section 7. It was motivated by J.-P. Serre’s
construction of a smooth projective $n$-dimensional complete intersection with
a free $G$-action, for an arbitrary finite group $G$ and an arbitrary positive
integer $n$; see [Ser58, Proposition 15]. 1.1 is then deduced from Theorems
1.2 and 1.4 in Section 8.

This paper is a sequel to [RS21]. The main result of [RS21] is used in the
proof of 3.1 (the specialization property of essential dimension). Other than
that, this paper can be read independently of [RS21].

##  2\. Notation and preliminaries

### Group actions and essential dimension

Let $k$ be a field, $\overline{k}$ be an algebraic closure of $k$, $G$ be a
linear algebraic group over $k$, and $X$ be a $G$-variety, i.e., a separated
reduced $k$-scheme of finite type endowed with a $G$-action over $k$. We will
say that the $G$-variety $X$ is primitive if $X\neq\emptyset$,
$G(\overline{k})$ transitively permutes the irreducible components of
$X_{\overline{k}}:=X\times_{k}\overline{k}$, and generically free if there
exists a dense open subscheme $U\subset X$ such that for every $u\in U$ the
scheme-theoretic stabilizer $G_{u}$ of $u$ is trivial.

By a $G$-compression of $X$ we will mean a dominant $G$-equivariant rational
map $X\dashrightarrow Y$, where the $G$-action on $Y$ is again generically
free and primitive. The essential dimension of $X$, denoted by
$\operatorname{ed}_{k}(X;G)$, or $\operatorname{ed}_{k}(X)$ if $G$ is clear
from the context, is defined as the minimal value of $\dim(Y)$, where the
minimum is taken over all $G$-compressions $X\dashrightarrow Y$. The essential
dimension $\operatorname{ed}_{k}(G)$ of the group $G$ is defined as the
supremum of $\operatorname{ed}_{k}(X)$, where $X$ ranges over all faithful
primitive $G$-varieties.

A $G$-variety $X$ is called weakly versal if every generically free primitive
$G$-variety $T$ admits a $G$-equivariant rational map $T\dashrightarrow U$. We
will say that $X$ is versal if every dense open $G$-invariant subvariety
$U\subset X$ is weakly versal.

### Good characteristic

######  Definition 2.1.

Let $G$ be a linear algebraic group defined over a field $k$. We will say that
$G$ is in good characteristic if

  * •

either $\operatorname{char}(k)=0$, or

  * •

$\operatorname{char}(k)=p>0$, $G^{\circ}$ is smooth reductive and there exists
a finite subgroup $S\subset G(\overline{k})$ of order prime to $p$ such that
the induced map $H^{1}(K,S)\to H^{1}(K,G)$ is surjective for every field
extension $K/\overline{k}$, or

  * •

$G$ is a finite discrete group, and if $\operatorname{char}(k)=p>0$ then the
only normal $p$-subgroup of $G$ is the trivial subgroup (that is, $G$ is
weakly tame in the sense of [BRV18]).

Here are two large families of examples in prime characteristic.

######  Example 2.2.

Suppose $G$ is a smooth group over a field $k$ of characteristic $p>0$. Assume
that the connected component $G^{0}$ of $G$ is reductive. Let $T$ be a maximal
torus in $G^{0}$, $r=\dim(T)\geqslant 0$, and $W=N_{G}(T)/T$ be the Weyl
group. If

(a) $G$ is split and defined over $\operatorname{Spec}(\mathbb{Z})$ and $p$
does not divide $2^{r}|W|$, or

(b) $G$ is connected and $p$ does not divide $|W|$,

then $G$ is in good characteristic. For a proof of (a), see [RS21, Proposition
5.1]. For a proof of (b), see [CGR06, Theorem 1.1(c)] and [CGR08, Remark 4.1].

The following example shows that conditions (a) and (b) above can sometimes be
relaxed.

######  Example 2.3.

The split orthogonal group $\operatorname{O}_{n}$, special orthogonal group
$\operatorname{SO}_{n}$ and the spin group $\operatorname{Spin}_{n}$ over a
field $k$ are in good characteristic as long as $\operatorname{char}(k)\neq
2$. Indeed, let $S$ be the group of diagonal $n\times n$ matrices of the form
$\operatorname{diag}(\epsilon_{1},\ldots,\epsilon_{n})$, where each
$\epsilon_{i}=\pm 1$, $S_{0}=S\cap\operatorname{SL}_{n}$, and $\tilde{S}$ be
the preimage of $S_{0}$ under the natural map
$\operatorname{Spin}_{n}\to\operatorname{SO}_{n}$. Then
$|S|=|\tilde{S}|=2^{n}$, $|S_{0}|=2^{n-1}$, and if $\operatorname{char}(k)\neq
2$, then the natural maps

$H^{1}(K,S)\to H^{1}(K,\operatorname{O}_{n}),\;\;H^{1}(K,S_{0})\to
H^{1}(K,\operatorname{SO}_{n}),\;\;\text{and}\;\;H^{1}(K,\tilde{S})\to
H^{1}(K,\operatorname{Spin}_{n})$

are all surjective. The surjectivity of the first two maps follows from the
fact that every quadratic form over a field of characteristic $\neq 2$ can be
diagonalized. The surjectivity of the third map is proved in [BRV07, Lemma
13.2].

##  3\. Specialization Property

The purpose of this section is to prove the following specialization property
of essential dimension.

######  Proposition 3.1.

Let $k$ be a field, $R$ be a discrete valuation ring containing $k$ and with
residue field $k$, and $l$ be the fraction field of $R$. Let $G$ be a linear
algebraic group over a field $k$ of good characteristic. Let $X$ be a flat
separated $R$-scheme of finite type endowed with a $G$-action over $R$, whose
fibers are generically free and primitive $G$-varieties. Then
$\operatorname{ed}_{\overline{l}}(X_{\overline{l}})\geqslant\operatorname{ed}_{\overline{k}}(X_{\overline{k}})$.

Our proof will be based on reduction to the case where $X$ is a $G$-torsor
over $\operatorname{Spec}(R)$. In the latter case the inequality
$\operatorname{ed}_{\overline{l}}(X_{\overline{l}})\geqslant\operatorname{ed}_{\overline{k}}(X_{\overline{k}})$
of 3.1 is established in [RS21, Theorem 6.4].

###### Proof of 3.1.

Let $k$ be the residue field of $R$, and $l$ be the fraction field of $R$. By
assumption, $X_{k}$ (resp. $X_{l}$) is a primitive generically free
$G_{k}$-variety (resp. $G_{l}$-variety). We fix algebraic closures
$\overline{k}$ and $\overline{l}$ of $k$ and $l$, respectively. Our proof will
be in several steps.

######  Claim 3.2.

There exists an integer $d\geqslant 0$ such that the irreducible components of
$X_{\overline{k}}$ and of $X_{\overline{l}}$ are all of dimension $d$.

For any finite field extensions $k^{\prime}\supset k$ and $l^{\prime}\supset
l$, there exists a discrete valuation ring $R^{\prime}\supset R$, finite and
free over $R$, such that the residue field of $R^{\prime}$ contains
$k^{\prime}$ and the fraction field of $R^{\prime}$ contains $l^{\prime}$; see
[Ser79, I.4, Proposition 9 and Remark] and [Ser79, I.6, Proposition 15]. Thus,
extending $R$ if necessary, we may assume that the irreducible components of
$X_{k}$ (resp. $X_{l}$) are geometrically irreducible and transitively
permuted by $G(k)$ (resp. $G(l)$).

After this reduction, the problem becomes to find $d\geqslant 0$ such that the
irreducible components of $X_{k}$ and of $X_{l}$ are all of dimension $d$.
Since $G$ acts transitively on the irreducible components of the fibers, it
suffices to exhibit one irreducible component of $X_{k}$ and one irreducible
component of $X_{l}$ of the same dimension.

Since $X$ is $R$-flat, by [Liu02, Lemma 4.3.7] every irreducible component of
$X$ dominates $\operatorname{Spec}R$. In other words, the open subscheme
$X_{l}\subset X$ is dense. Therefore each irreducible component of $X$ is the
closure of an irreducible component of $X_{l}$. Thus, since
$X_{k}\neq\emptyset$, there exists an irreducible component $X^{\prime}\subset
X$ such that $X^{\prime}_{k}$ contains some irreducible component of $X_{k}$
and such that $X^{\prime}_{l}$ is an irreducible component of $X_{l}$.

The composition $X^{\prime}\hookrightarrow X\to\operatorname{Spec}R$ is
surjective, hence [SP, 0B2J] implies that every irreducible component of
$X^{\prime}_{k}$ has dimension $\operatorname{dim}(X^{\prime}_{l})$. Since
$X^{\prime}_{k}$ contains an irreducible component of $X_{k}$, this completes
the proof of 3.2.

######  Claim 3.3.

There exists a $G$-invariant $R$-fiberwise dense open subscheme $U\subset X$
such that $G$ acts freely on $U$.

Since $G_{l}$ acts generically freely on $X_{l}$, there exists a closed
nowhere dense $G_{l}$-invariant subscheme $Z\subset X_{l}$ such that $G_{l}$
acts freely on $X_{l}\setminus Z$. Let $W\subset Z$ be an irreducible
component, and let $\overline{W}$ be the closure of $W$ in $X$. By [SP, Tag
0B2J], either $(\overline{W})_{k}$ is empty, or

$\dim((\overline{W})_{k})=\dim(W)\leqslant\dim(X_{k})-1.$

It now follows from 3.2 that $\overline{W}$ does not contain any irreducible
component of $X_{k}$. Therefore, the closure $\overline{Z}$ of $Z$ does not
contain any irreducible component of $X_{k}$.

Since $G_{k}$ acts generically freely on $X_{k}$, there exists a closed
nowhere dense $G_{k}$-invariant subscheme $Z^{\prime}\subset X_{k}$ such that
$G_{k}$ acts freely on $X_{k}\setminus Z^{\prime}$. It follows that
$U:=X\setminus(\overline{Z}\cup Z^{\prime})$ is a fiber-wise dense
$G$-invariant open subscheme of $X$, such that $G$ acts freely on $U_{l}$ and
$U_{k}$. To prove 3.3, it remains to show that $G$ acts freely on $U$, i.e.,
that the stabilizer $U$-group scheme

$\mathcal{G}:=U\times_{(U\times_{R}U)}(G\times_{R}U)$

is trivial. Here the fibered product is taken over the diagonal morphism $U\to
U\times_{R}U$ and the action morphism $G\times_{R}U\to U\times_{R}U$. Since
$G$ acts freely on $U_{l}$ and $U_{k}$, the $U_{l}$-group scheme
$\mathcal{G}_{l}$ and the $U_{k}$-group scheme $\mathcal{G}_{k}$ are both
trivial. Hence so is $\mathcal{G}$, as desired. This proves 3.3.

######  Claim 3.4.

For the purpose of proving 3.1, we may assume that $X$ is the total space of a
$G$-torsor $X\to Y$, where $Y$ is a separated $R$-scheme of finite type.

After replacing $X$ by the open $R$-fiberwise dense subscheme $U$ constructed
in 3.3, we may assume that $G$ acts freely on $U$. Let
$X/G\to\operatorname{Spec}R$ denote the fppf-quotient of
$X\to\operatorname{Spec}R$ by the $G$-action. By a theorem of M. Artin [Ana73,
Théorème 3.1.1], $X/G$ is represented by an algebraic space of finite type $Y$
over $R$. (Equivalently, the quotient stack $[X/G]$ has trivial stabilizers,
hence it is represented by an algebraic space.) Since $G$ is smooth, the
projection $X\to X/G$ is an étale torsor.

We claim that $X$ is flat over $R$. If $Y$ were a scheme, this would follow
directly from [SP, Tag 02JZ], since $X$ is flat over both $R$ (this is one of
the assumptions of 3.1) and $Y$ (because $X\to Y$ is a $G$-torsor). If general
we use the fact that $R$-flatness is an étale local property: to prove that a
representable morphism of algebraic spaces is flat, it suffices to do so
locally after an étale base change. This way we reduce the claim to the case,
where $Y$ is a scheme and [SP, Tag 02JZ] applies.

Let $Y^{\prime}\to Y$ be surjective étale morphism and
$X^{\prime}:=X\times_{Y}Y^{\prime}$. For every $y\in Y$ and every geometric
point $\overline{y}$ of $Y$ lying above $y$, $\overline{y}$ factors through
some $y^{\prime}\in Y^{\prime}$ and we have a $G_{k(y^{\prime})}$-equivariant
isomorphism $(X_{y})_{k(y^{\prime})}\simeq(X^{\prime})_{y^{\prime}}$. It
follows that passage to an étale cover of $Y$ does not alter the essential
dimension of the geometric fibers of $X\to Y$. Therefore, we may assume that
$Y$ is a scheme, that is, $X\to Y$ is an étale $G$-torsor in the category of
schemes. This proves 3.4.

######  Claim 3.5.

For the purpose of proving 3.1, we may assume that:

  * •

$X$ is the total space of a $G$-torsor $X\to\operatorname{Spec}A$, where $A$
is a discrete valuation ring containing $R$ and the inclusion $R\subset A$ is
a local homomorphism, and

  * •

$\operatorname{ed}_{l}(X_{l})=\operatorname{ed}_{\overline{l}}(X_{\overline{l}})$.

By 3.4, we may assume that $X$ is the total space of a $G$-torsor $X\to Y$,
where $Y$ is a separated $R$-scheme of finite type. Since $G$ acts primitively
on $X_{k}$ and $X_{l}$, the varieties $Y_{k}$ and $Y_{l}$ are geometrically
irreducible. By [SP, Lemma 054F], there exists a morphism
$\operatorname{Spec}A\to Y$, where $A$ is a discrete valuation ring whose
generic point $\eta$ maps to the generic point $y$ of $Y$ so that the induced
inclusion $k(y)\subset k(\eta)$ is an equality, and whose closed point $s$
maps to the generic point $y^{\prime}\in Y_{k}$. We have

$\operatorname{trdeg}_{k}k(y^{\prime})=\dim(Y_{k})=\dim(Y_{l})-1=\operatorname{trdeg}_{k}k(y)-1=\operatorname{trdeg}_{k}k(\eta)-1=\operatorname{trdeg}_{k}k(s),$

hence the field extension $k(s)/k(y^{\prime})$ is algebraic. We thus obtain
the following Cartesian diagram.

$\textstyle{X_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spec}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spec}(R),}$

where $X_{A}=X\times_{\operatorname{Spec}(R)}\operatorname{Spec}(A)$. By
construction, the morphism $\operatorname{Spec}A\to\operatorname{Spec}R$ sends
the closed point of $\operatorname{Spec}A$ to the closed point of
$\operatorname{Spec}R$, and so it is local. As in 3.4, replacing the
$G$-torsor $X\to Y$ by the $G$-torsor $X_{A}\to\operatorname{Spec}A$ does not
alter the essential dimension of the geometric fibers over the generic and the
closed points of $R$. This proves the first assertion of 3.5.

Since every $G_{\overline{l}}$-equivariant compression of $X_{\overline{l}}$
over $\overline{l}$ is defined over some finite extension of $l$, there is a
finite subextension $l\subset l^{\prime}\subset\overline{l}$ such that
$\operatorname{ed}_{l^{\prime}}(X_{l^{\prime}})=\operatorname{ed}_{\overline{l}}(X_{\overline{l}})$.
Let $R^{\prime}\supset R$ be a discrete valuation ring with fraction field
$l^{\prime}$, and let $k^{\prime}\supset k$ be the residue field of
$R^{\prime}$. The $G_{R}$-torsor $X$ over $R$ lifts to a
$G_{R^{\prime}}$-torsor on $X_{R^{\prime}}$, which is $R^{\prime}$-fiberwise
generically free and primitive. Since
$\operatorname{ed}_{k^{\prime}}(X_{k^{\prime}})\geqslant\operatorname{ed}_{\overline{k}}(X_{\overline{k}})$,
we are allowed to replace $R$ by $R^{\prime}$. This completes the proof of
3.5.

We are now ready to complete the proof of 3.1. We may place ourselves in the
setting of 3.5. Since $k$ is algebraically closed and $G$ is in good
characteristic, the assumptions of [RS21, Theorem 6.4] are satisfied, hence
$\operatorname{ed}_{l}(X_{l})\geqslant\operatorname{ed}_{k}(X_{k})$. Therefore

$\operatorname{ed}_{\overline{l}}(X_{\overline{l}})=\operatorname{ed}_{l}(X_{l})\geqslant\operatorname{ed}_{k}(X_{k})\geqslant\operatorname{ed}_{\overline{k}}(X_{\overline{k}}).\qed$

##  4\. Proof of Theorem 1.2

######  Lemma 4.1.

Let $k/k_{0}$ be a field extension of infinite transcendence degree such that
$k$ is algebraically closed. Let $B_{0}$ be an irreducible $k_{0}$-variety and
$B:=B_{0}\times_{k_{0}}k$. Then the set of $k$-rational points of $B$ mapping
to the generic point of $B_{0}$ is dense in $B$.

###### Proof.

Let $U$ be a non-empty open $k$-subscheme of $B$. It suffices to prove the
following

Claim: $U$ has a $k$-point which maps to the generic point of $B_{0}$.

To prove this claim, note that the open embedding $U\hookrightarrow B$ is
defined over some intermediate subfield $k_{0}\subset k_{1}\subset k$ such
that the extension $k_{1}/k_{0}$ is finitely generated. In other words,
$U\hookrightarrow B$ is obtained by base change from an affine open embedding
$U_{1}\hookrightarrow B_{0}\times_{k_{0}}k_{1}$ defined over $k_{1}$. In
particular, the morphism $U_{1}\to B_{0}$ is dominant. Let
$\eta_{1}:\operatorname{Spec}K_{1}\to U_{1}$ be the generic point of $U_{1}$.

Now consider a subfield $k_{1}\subset L_{1}\subset K_{1}$ such that
$L_{1}/k_{1}$ is purely transcendental of finite transcendence degree and
$K_{1}/L_{1}$ is finite. Since $k/k_{1}$ has infinite transcendence degree,
there exists a field embedding $\iota:L_{1}\hookrightarrow k$ compatible with
the $k_{1}$-algebra structures of $L_{1}$ and $k$. Since $k$ is algebraically
closed and $K_{1}/L_{1}$ is finite, we may extend $\iota$ to a field embedding
$K_{1}\hookrightarrow k$, again compatible with the $k_{1}$-algebra structures
of $K_{1}$ and $k$. This gives rise to a scheme morphism

$u_{1}:\operatorname{Spec}k\to\operatorname{Spec}K_{1}\xrightarrow{\eta_{1}}U_{1}.$

Since $U=U_{1}\times_{k_{1}}k$, $u_{1}$ uniquely lifts to a $k$-point $u$ of
$U$ mapping to the generic point of $U_{1}$. Since the morphism $U_{1}\to
B_{0}$ is dominant, the $k$-point $u$ maps to the generic point of $B_{0}$.
This completes the proof of the Claim and thus of Lemma 4.1. ∎

We will make use of the following “rigidity property” of essential dimension.
For a proof, see [RS21, Lemma 2.2].

######  Lemma 4.2.

Let $k$ be an algebraically closed field, $G$ be a $k$-group, and $X$ be a
generically free primitive $G$-variety defined over $k$. Then
$\operatorname{ed}_{k}(X)=\operatorname{ed}_{l}(X_{l})$ for any field
extension $l/k$. ∎

Not let $f\colon\mathcal{X}\to B$ be as in 1.2. For every integer $n$, we set

$\Phi_{f}(n):=\\{b\in
B\,|\,\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}})\leqslant
n\text{ for some geometric point $\overline{b}$ with image $b$}\\}.$

######  Lemma 4.3.

(a) A point $b\in B$ belongs to $\Phi_{f}(n)$ if and only
$\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}})\leqslant n$
for every geometric point $\overline{b}$ with image $b$.

(b) Let $\pi:B^{\prime}\to B$ be a morphism of schemes, and let
$f^{\prime}:X\times_{B}B^{\prime}\to B^{\prime}$ be the base of change of $f$
along $\pi$. Then $\Phi_{f^{\prime}}(n)=\pi^{-1}(\Phi_{f}(n))$.

###### Proof.

(a) Let $\overline{b}_{1}$ and $\overline{b}_{2}$ be two geometric points of
$B$ with image $b$. By [Liu02, Exercise 3.1.10(b)], the ring
$A:=k(\overline{b}_{1})\otimes_{k(b)}k(\overline{b}_{2})$ is not zero. If
$\mathfrak{m}$ is a maximal ideal of $A$, the quotient $A/\mathfrak{m}$ is a
field containing $k(\overline{b}_{1})$ and $k(\overline{b}_{2})$. By
considering an algebraic closure of $A/\mathfrak{m}$, we are thus reduced to
the case when there is a field homomorphism
$k(\overline{b}_{1})\hookrightarrow k(\overline{b}_{2})$. We may thus assume
that $k(\overline{b}_{1})\subset k(\overline{b}_{2})$. In this case, (a)
follows from Lemma 4.2.

(b) Let $b^{\prime}\in B^{\prime}$ and $b\in B$ be such that
$\pi(b^{\prime})=b$. Let $\overline{b}^{\prime}$ be a geometric point of $B$
with image $b^{\prime}$, so that $\overline{b}:=\pi\circ\overline{b}^{\prime}$
is geometric point of $B$ with image $b$. Then there is a natural isomorphism
$\mathcal{X}_{\overline{b}^{\prime}}\simeq\mathcal{X}_{\overline{b}}\times_{k(\overline{b})}k(\overline{b}^{\prime})$
of $G_{\overline{b}^{\prime}}$-varieties, and

$\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}})=\operatorname{ed}_{k(\overline{b}^{\prime})}(\mathcal{X}_{\overline{b}}\times_{k(\overline{b})}k(\overline{b}^{\prime}))=\operatorname{ed}_{k(\overline{b}^{\prime})}(\mathcal{X}_{\overline{b}^{\prime}})$

by Lemma 4.2(c). In particular, $b\in\Phi_{f}(n)$ if and only if
$b^{\prime}\in\Phi_{f^{\prime}}(n)$, as desired. ∎

###### Proof of 1.2.

We must show that $\Phi_{f}(n)\subset B$ is a union of countably many closed
subsets of $B$. By noetherian approximation (see [GD64, IV, §8.10] or [TT90,
Appendix C]), the $G$-action on $\mathcal{X}$ descends to a subfield $k_{0}$
of $k$ which is finitely generated over its prime field. In other words, there
exist

  * •

a field $k_{0}\subset k$ finitely generated over its prime field,

  * •

a smooth group scheme $G_{0}$ of finite type over $k_{0}$,

  * •

$k_{0}$-schemes of finite type $B_{0}$ and $\mathcal{X}_{0}$,

  * •

a $G_{0}$-action on $\mathcal{X}_{0}$ over $k_{0}$,

  * •

a flat separated $G_{0}$-invariant morphism $f_{0}\colon\mathcal{X}_{0}\to
B_{0}$, and

  * •

a Cartesian diagram

(4.4)
${\mathcal{X}}$${\mathcal{X}_{0}}$${B}$${B_{0},}$$\scriptstyle{f}$$\scriptstyle{f_{0}}$$\scriptstyle{\pi}$

such that $G=G_{0}\times_{k_{0}}k$, and the base change of the $G_{0}$-action
on $\mathcal{X}_{0}/B_{0}$ along $\pi$ is isomorphic to the $G$-action on
$\mathcal{X}/B$.

By Lemma 4.3(b), we have $\Phi_{f}(n)=\pi^{-1}(\Phi_{f_{0}}(n))$. Thus, since
$\pi$ is continuous, it suffices to prove that $\Phi_{f_{0}}(n)$ is a
countable union of closed subsets of $B_{0}$. In other words, we may assume
that $k$ is finitely generated over its prime field and that $B$ is of finite
type over $k$. In this case, $B$ is countable, hence $\Phi_{f}(n)$ is
countable. It remains to show that $\Phi_{f}(n)$ is a union of closed subsets
of $B$. By elementary topology it suffices to show that $\Phi_{f}(n)$ is
closed under specialization; see [SP, Tag 0EES]. In other words, if
$b^{\prime}\in B$ is a specialization of $b\in\Phi_{f}(n)$, i.e.,
$b^{\prime}\in\overline{\\{b\\}}$, then we want to show that
$b^{\prime}\in\Phi_{f}(n)$.

By [GD61, Proposition 7.1.4], there exist a discrete valuation ring $R$ with
closed point $s$ and generic point $\eta$, and a morphism
$\operatorname{Spec}R\to B$ sending $s$ to $b^{\prime}$ and $\eta$ to $b$.
Pre-composing with the completion map
$\operatorname{Spec}\hat{R}\to\operatorname{Spec}R$, we may assume that $R$ is
complete. Since $B$ is a $k$-scheme, the residue fields of
$b,b^{\prime},s,\eta$ all have the same characteristic as $k$. Thus $R$ is
complete and equicharacteristic and hence, by Cohen’s Structure Theorem we
have an isomorphism $R\simeq k(s)[[t]]$. In particular, the residue field
$k(s)$ is contained in $R$. By 3.1(b), letting $\overline{\eta}$ and
$\overline{s}$ be geometric points of $\operatorname{Spec}R$ lying above
$\eta$ and $s$, respectively, we deduce that

$\operatorname{ed}_{k(\overline{\eta})}\big{(}X_{k(\overline{\eta})}\big{)}\geqslant\operatorname{ed}_{k(\overline{s})}\big{(}X_{k(\overline{s})}\big{)}.$

Now Lemma 4.3(a) tells us that

$n\geqslant\operatorname{ed}_{{k(\overline{b})}}\big{(}X_{k(\overline{b})}\big{)}\geqslant\operatorname{ed}_{k(\overline{b}^{\prime})}\big{(}X_{k(\overline{b}^{\prime})}\big{)},$

where $\overline{b}$ and $\overline{b}^{\prime}$ are geometric points of $B$
lying above $b$ and $b^{\prime}$, respectively. This shows that $\Phi_{f}(n)$
is closed under specialization.

Assume now that $k$ is algebraically closed and of infinite transcendence
degree over its prime field, and let $m$ be the maximum of
$\operatorname{ed}_{k(\overline{b})}(\mathcal{X}_{\overline{b}};G_{k(\overline{b})})$,
where $\overline{b}$ ranges over all geometric points of $B$. Consider the
diagram (4.4). Since $\Phi_{f_{0}}(m-1)$ is a union of closed subsets of
$B_{0}$ and it does not equal $B_{0}$, it does not contain the generic point
of $B_{0}$. By Lemma 4.3(b), we have
$\Phi_{f}(m-1)=\pi^{-1}(\Phi_{f_{0}}(m-1))$, hence for every $k$-point $b$ of
$B$ mapping to the generic point of $B_{0}$ we have
$\operatorname{ed}_{k}(\mathcal{X}_{b})=m$. By Lemma 4.1, the set of such
$k$-points is Zariski dense in $B$. ∎

######  Remark 4.5.

To put 1.2 in perspective, we will conclude this section by recalling an
analogous result for cohomological invariants from [CT02, Appendix]. For an
overview of the theory of cohomological invariants, see [GMS03].

Let $k$ be a field, $G$ be a linear algebraic $k$-group, and
$f\colon\mathcal{X}\to B$ be a morphism as in 1.2. Let $i$ be a non-negative
integer, $C$ be a finite $\operatorname{Gal}(k_{s}/k)$-module of order prime
to the characteristic of $k$, and $F\in\operatorname{Inv}^{i}(G,C)$ be a
cohomological invariant over $k$ with values in the Galois cohomology ring
$H^{i}(-,C)$. Passing to a dense open subscheme of $B$ if necessary, we may
assume that $F(k(X))$ comes from a cohomology class $\alpha\in
H^{i}_{\textrm{\'{e}t}}(X,C)$. In this case, by the compatibility of the
specialization map in étale and Galois cohomology [GMS03, Page 15, Footnote],
this implies that $\alpha_{\overline{s}}=F(k(X_{\overline{s}}))$ (up to sign)
for every geometric point $\overline{s}$ of $B$. From [CT02, Proposition A7],
we deduce the following:

$B_{0}:=\\{s\in B:F\big{(}k(X_{\overline{s}})\big{)}=0\text{ for some
geometric point $\overline{s}$ above $s$}\\}$

is a countable union of closed subsets of $B$. Note that by the Rigidity
Property for étale cohomology [Mil80, Corollary VI.2.6], one may replace
“some” by “every” in the definition of $B_{0}$, as in Lemma 4.3(a).

##  5\. Counterexamples

######  Example 5.1.

The following example shows that in 1.2 we may not replace “countable union”
by “finite union”. In this example $l$ will denote an odd prime. We will
assume that the base field $k=\mathbb{C}$ is the field of complex numbers and
will write $\operatorname{ed}$ in place of $\operatorname{ed}_{\mathbb{C}}$.

Let $A$ be a complex abelian variety. Any choice of $v_{1},\dots,v_{r}\in
A[\ell]$ gives rise to a $(\mathbb{Z}/\ell\mathbb{Z})^{r}$-action on $A$ via
$(n_{1},\ldots,n_{r})\colon a\mapsto a+n_{1}v_{1}+\ldots+n_{r}v_{r}$. This
action is free if and only if $v_{1},v_{2},\ldots,v_{r}$ are linearly
independent over $\mathbb{Z}/\ell\mathbb{Z}$. When we view $A$ as a
$(\mathbb{Z}/\ell\mathbb{Z})^{r}$-variety via this action, we will denote it
by $(A;v_{1},\dots,v_{r})$. We will focus on the case, where $r=2$ and
$A=E\times E$ is the direct product of two copies of a complex elliptic curve
$E$. More specifically we will investigate how $\operatorname{ed}(E\times
E;v_{1},v_{2})$ depends on the choice of $E$, $v_{1}$ and $v_{2}$.

Recall that the endomorphism ring of an elliptic curve over $\mathbb{C}$ is
either $\mathbb{Z}$ or an order in an imaginary quadratic field extension of
$\mathbb{Q}$, and that all such rings arise as endomorphism rings of a complex
elliptic curve. By the Chinese Remainder Theorem, there exist infinitely many
negative integers $d\equiv 2,3\pmod{4}$ such that $d$ is not a square modulo
$\ell$.

(i) Let $E$ be an elliptic curve over $\mathbb{C}$ such that
$\operatorname{End}(E)\simeq\mathbb{Z}[\sqrt{d}]$; see [Sil09, p. 426]. We
claim that $\operatorname{ed}(E\times E;(q,0),(0,q))=1$ for any $q\in
E(\mathbb{C})[\ell]\setminus\\{0\\}$.

It is obvious from the definition that $\operatorname{ed}(E\times
E:(q,0),(0,q))\geqslant 1$, so we only need to show that
$\operatorname{ed}(E\times E;(q,0),(0,q))\leqslant 1$. Let
$\phi\in\operatorname{End}(E)$ be such that $\phi^{2}\colon E\to E$ is
multiplication by $d$. Since $\phi$ is an endomorphism, it restricts to a
group homomorphism $E(\mathbb{C})[\ell]\to E(\mathbb{C})[\ell]$. Fixing a
$(\mathbb{Z}/\ell\mathbb{Z})$-basis of
$E(\mathbb{C})[\ell]\simeq(\mathbb{Z}/\ell\mathbb{Z})^{2}$, $\phi$ corresponds
to a matrix $A\in\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$. The matrix
$A$ does not have any eigenvalues in $\mathbb{Z}/\ell\mathbb{Z}$. Indeed, if
$Av=\lambda v$ for some non-zero $v\in(\mathbb{Z}/\ell\mathbb{Z})^{2}$ and
$\lambda\in\mathbb{Z}/\ell\mathbb{Z}$, then $dv=A^{2}v=\lambda^{2}v$, hence
$d=\lambda^{2}$ in $\mathbb{Z}/\ell\mathbb{Z}$, which is impossible as $d$ is
not a square modulo $\ell$. It follows that $q$ and $\phi(q)$ are linearly
independent, and so form a basis of $E(\mathbb{C})[\ell]$. Now
$\phi\colon(E;q)\to(E;\phi(q))$ is a $\mathbb{Z}/\ell\mathbb{Z}$-equivariant
morphism and the composition

$(E\times E;(q,0),(0,q))\xrightarrow{(\operatorname{id},\phi)}(E\times
E;(q,0),(0,\phi(q)))\xrightarrow{+}(E;(q,\phi(q)))$

is a $(\mathbb{Z}/\ell\mathbb{Z})^{2}$-compression. Thus
$\operatorname{ed}(E\times E;(q,0),(0,q))\leqslant 1$, as desired.

(ii) Let $E$ be an elliptic curve such that
$\operatorname{End}(E)=\mathbb{Z}$, and let $q\in
E(\mathbb{C})[\ell]\setminus\\{0\\}$. We claim that $\operatorname{ed}(E\times
E;(q,0),(0,q))=2$.

Indeed, assume the contrary. Then there exists a dominant
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$-equivariant rational map

$f\colon E\times E\dashrightarrow C,$

where $E\times E$ stands for the $(\mathbb{Z}/\ell\mathbb{Z})^{2}$-variety
$(E\times E;(q,0),(0,q))$ and $C$ is some curve on which
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$ acts faithfully. We may assume that $C$ is
smooth and projective. Since $\ell$ is odd, $(\mathbb{Z}/\ell\mathbb{Z})^{2}$
cannot act faithfully on $\mathbb{P}^{1}$. Thus $C$ is not isomorphic to
$\mathbb{P}^{1}$. For all but finitely many $v$, $f$ restricts to a well-
defined surjective morphism $E\simeq E\times\\{v\\}\to C$. We deduce from
Hurwitz’s formula that $C$ has genus $1$. After suitably choosing an origin
for $C$, $f$ becomes an everywhere defined homomorphism of abelian varieties.
The restrictions of $f$ to $E\times\\{0\\}$ and $\\{0\\}\times E$ give
isogenies $f_{1},f_{2}\colon E\to C$ such that th element $(1,0)$ of
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$ acts on $C$ via translation by $f_{1}(q)$,
and the element $(0,1)$ acts on $C$ via translations by $f_{2}(q)$. Since the
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$-action on $C$ is faithful, we conclude that
$f_{1}(q)$ and $f_{2}(q)$ form a basis of $C[\ell]$. On the other hand, recall
from [Sil09, Lemma 4.2(b)] that $\operatorname{Hom}(E,C)$ is torsion-free
$\mathbb{Z}$-module. Since

$\operatorname{Hom}(E,C)\otimes_{\mathbb{Z}}\mathbb{Q}\simeq\operatorname{Hom}(E,E)\otimes_{\mathbb{Z}}\mathbb{Q}\simeq\mathbb{Q},$

we conclude that $\operatorname{Hom}(E,C)=\mathbb{Z}$. This implies that there
exists homomorphism $h\colon E\to C$ such that $f_{1}$ and $f_{2}$ are
multiples of $h$. In particular, $f_{1}(q)$ and $f_{2}(q)$ are linearly
dependent, a contradiction. We conclude that $C$ does not exist, and thus
$\operatorname{ed}(E\times E;(q,0),(0,q))=2$, as claimed.

For every prime $\ell$, there exists a complex curve $B$ and a family of
elliptic curves $\mathcal{E}\to B$, together with a nowhere zero
$\ell$-torsion section, such that every pair $(E;q)$ where $E$ is a complex
elliptic curve and $q\in E(\mathbb{C})[\ell]\setminus\\{0\\}$ arises as a
fiber of $\mathcal{E}\to B$; see [CT02, Proposition A4]. The group
$\mathbb{Z}/\ell\mathbb{Z}$ acts freely on $\mathcal{E}$ over $B$ by
translations by the $\ell$-torsion section, and so
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$ acts freely on the self-product
$\Phi:\mathcal{E}\times_{B}\mathcal{E}\to B$ by translation. The fibers of
$\Phi$ are triples $(E\times E;(q,0),(0,q))$, where $q\in
E(\mathbb{C})[\ell]\setminus\\{0\\}$. There are infinitely many fibers as in
(i), and the fibers as in (ii) form a very general subset. Thus, the above
discussion shows that the fiber of $\Phi$ over a very general point of the
curve $B$ has essential dimension $2$, while countably many fibers have
essential dimension $1$.

######  Example 5.2.

The following example shows that the flatness assumption in 1.2 is necessary.

Let $n$ be a positive integer, and let $k$ be an algebraically closed field of
characteristic not dividing $n$. Consider the affine plane
$\mathbb{A}^{2}_{k}=\operatorname{Spec}k[x,y]$ with coordinates $x,y$, let
$X\subset\mathbb{A}^{2}$ be defined by the equation $x(y^{n}-1)=0$, let
$B=\mathbb{A}^{1}_{k}=\operatorname{Spec}k[x]$, and let $f$ be the projection
induced by the inclusion $k[x]\subset k[x,y]$. The group
$\mu_{n}=\mathbb{Z}/n\mathbb{Z}$ acts on $\mathbb{A}^{2}_{k}$ by
$\zeta\cdot(x,y)\mapsto(x,\zeta y)$. Then $X$ is $\mu_{n}$-invariant, $f$ is
$\mu_{n}$-equivariant, and the $\mu_{n}$-action on the fibers of $f$ is
generically free and primitive. We have $\operatorname{ed}_{k}(X_{a})=0$ for
every $a\in k^{\times}$, but
$\operatorname{ed}_{k}(X_{0})=\operatorname{ed}_{k}(\mu_{n})=1$.

######  Example 5.3.

The following example shows that 1.2 fails if $k$ is not algebraically closed.
More precisely, in this case the $k$-points $s\in B(k)$, where
$\operatorname{ed}_{k}(X_{s})\leqslant n$ do not necessarily lie on a
countable union of closed subvarieties of $B$.

Indeed, let $k=\mathbb{R}$ be the field of real numbers and $G$ be the
orthogonal group $\operatorname{O}_{2}$ defined over $\mathbb{R}$. Consider
the action of $G=\operatorname{O}_{2}$ on $X=\operatorname{GL}_{2}$ via
multiplication on the right. Note that $X$ is the total space of a $G$-torsor
$\tau\colon X\to Y$, where $Y=\operatorname{GL}_{2}/O_{2}$ is naturally
identified with the space of symmetric $2\times 2$ matrices via $\tau\colon
A\mapsto AA^{T}$.

Now consider the morphism

$f\colon X=\operatorname{GL}_{2}\longrightarrow
B=\mathbb{A}^{1}\setminus\\{0\\},$

sending a matrix $A$ to $\det(A)^{2}$. This morphism factors through $\tau$ as
follows:

$f\colon
X\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}Y\stackrel{{\scriptstyle\det}}{{\longrightarrow}}B=\mathbb{A}^{1}\setminus\\{0\\}.$

Denote that fibers of $X$ and $Y$ over $s\in B$ by $X_{s}$ and $Y_{s}$,
respectively. Then $X_{s}$ is a $G$-torsor over $Y_{s}$.

Claim: View a non-zero real number $s$ as an $\mathbb{R}$-point of $B$. Then

(5.4) $\operatorname{ed}(X_{s})=\begin{cases}\text{$0$, if $s<0$, and}\\\
\text{$1$, if $s>0$.}\end{cases}$

Note that $Y_{s}$ is the variety of symmetric matrices
$B=\begin{pmatrix}a&b\\\ b&c\end{pmatrix}$ such that $\det(B)=s$. Thus $Y_{s}$
is a rational surface over $\mathbb{R}$ whose function field can be identified
with $\mathbb{R}(a,b)$. Passing to the generic point of $Y_{s}$, we see that
$\operatorname{ed}_{\mathbb{R}}(X_{s})=\operatorname{ed}_{\mathbb{R}}(\tau_{s})$,
where $\tau_{s}\in H^{1}(\mathbb{R}(a,b),\operatorname{O}_{2})$ is the
$\operatorname{O}_{2}$-torsor over $\mathbb{R}(a,b)$ obtained by pulling back
$\tau$ to the generic point of $Y_{s}$. Examining the long exact cohomology
sequence associated to the exact sequence
$1\to\operatorname{O}_{2}\to\operatorname{GL}_{2}$ of algebraic groups and
remembering that $H^{1}(\mathbb{R}(a,b),\operatorname{GL}_{2})=1$ by Hilbert’s
Theorem 90, we see that $H^{1}(\mathbb{R}(a,b),\operatorname{O}_{2})$ is in a
natural bijective correspondence with the set of 2-dimensional non-singular
quadratic forms over $\mathbb{R}(a,b)$, up to equivalence, and the quadratic
form $q_{s}$ corresponding to $\tau_{s}$ is the form whose Gram matrix is
$\begin{pmatrix}a&b\\\ b&c\end{pmatrix}$, where $c=\dfrac{s+b^{2}}{a}$. Note
that, by definition,
$\operatorname{ed}_{\mathbb{R}}(\tau_{s})=\operatorname{ed}_{\mathbb{R}}(q_{s})$
and the discriminant of $q_{s}$ is $s$.

Since $q$ assumes the value $a$ and has discriminant $s$, $q_{s}$ is
isomorphic to $\left\langle{a,\,as}\right\rangle$, Here
$\left\langle{a,as}\right\rangle$ denote the $2$-dimensional quadratic form
$q_{s}(z,w)=az^{2}+asw^{2}$ over $\mathbb{R}(a,b)$. If $s<0$, then $q$ is
isotropic over $\mathbb{R}(a,b)$. Hence, $q$ is hyperbolic over
$\mathbb{R}(a,b)$, i.e., $q_{s}$ is isomorphic to
$\left\langle{1,-1}\right\rangle$; see [Lam05, Theorem I.3.2]. In particular,
$q_{s}$ descends to $\mathbb{R}$ and hence,
$\operatorname{ed}_{\mathbb{R}}(q_{s})=0$.

On the other hand, suppose that $s>0$. Then $s$ is a complete square in
$\mathbb{R}(a,b)$, so $q\simeq\left\langle{a,\,a}\right\rangle$. Clearly
$q_{s}$ descends to $\mathbb{R}(a)\subset K$, so
$\operatorname{ed}_{\mathbb{R}}(q_{s})\leqslant 1$. In order to complete the
proof of (5.4), it remains to show that
$\operatorname{ed}_{\mathbb{R}}(q_{s})\neq 0$. We argue by contradiction.
Assume $\operatorname{ed}_{\mathbb{R}}(q_{s})=0$, i.e., $q_{s}$ descends to
some intermediate extension $\mathbb{R}\subset K\subset\mathbb{R}(a,b)$, where
$\operatorname{trdeg}_{\mathbb{R}}(K)=0$. In other words, $K$ is algebraic
over $\mathbb{R}$. Since $\mathbb{R}$ is algebraically closed in
$\mathbb{R}(a,b)$, this is only possible if $K=\mathbb{R}$, i.e., $q$ descends
to a $2$-dimensional form $q_{0}$ defined over $\mathbb{R}$. Since $s>0$, $q$
is anisotropic over $\mathbb{R}(a,b)$, and hence, so is $q_{0}$. Let
$\nu_{a}\colon K^{\times}\to\mathbb{Z}$ the valuation associated to the
variable $a$. It is now easy to see that for any
$(0,0)\neq(f,g)\in(K^{\times})^{2}$, $\nu_{a}(q_{0}(f,g))$ is even, where as
$\nu_{a}(q(f,g))$ is odd. This tells us that $q$ and $q_{0}$ have no values in
common, contradicting our assumption that $q$ descends to $q_{0}$. Thus
completes the proof of (5.4). ∎

##  6\. Transversal intersections in projective space

This section contains several preliminary results which will be used in the
proof of 1.4. The common theme is transversal intersections of projective
varieties with linear subspaces in projective space. Note that there are no
algebraic groups or group actions here; they will come into play in the next
section.

Recall that a commutative ring with identity is said to be regular if it is
noetherian and all its localizations at prime ideals are regular local rings.

######  Lemma 6.1.

Let $A$ be a regular semi-local noetherian ring and
$\mathfrak{m}_{1},\mathfrak{m}_{2},\dots,\mathfrak{m}_{r}$ be the maximal
ideals of $A$. For each $1\leqslant i\leqslant r$, let
$P_{i}\subset\mathfrak{m}_{i}$ be a prime ideal such that
$P_{i}\not\subset\mathfrak{m}_{j}$ for any $j\neq i$ and such that each local
ring $A/P_{i}$ is regular. Assume that the prime ideals $P_{1},\ldots,P_{r}$
have the same height,
$\operatorname{ht}(P_{1})=\dots=\operatorname{ht}(P_{r})=c$. Then there exist
$h_{1},h_{2},\dots,h_{c}\in A$ such that
$P_{i}A_{\mathfrak{m}_{i}}=(h_{1},\dots,h_{c})A_{\mathfrak{m}_{i}}$ and
$h_{1},\dots,h_{c}$ form a regular sequence in $A_{m_{i}}$ for each $i$.

###### Proof.

For any $a\neq b$, the ideal $P_{a}+P_{b}$ is not contained in any
$\mathfrak{m}_{i}$. Hence $P_{a}+P_{b}=A$. By the Chinese Remainder Theorem
the natural ring homomorphism $A/(P_{1}^{2}\cdots
P_{r}^{2})\to(A/P_{1}^{2})\times\dots\times(A/P_{r}^{2})$ is an isomorphism;
see [SP, Tag 00DT] or [Eis95, Exercise 2.6]. In particular, the natural
projection $A\to(A/P_{1}^{2})\times\dots\times(A/P_{r}^{2})$ is surjective.
Since $A/P_{i}$ is regular for every $1\leqslant i\leqslant r$, there exist
$h_{i,1},\dots,h_{i,c}\in P_{i}$ whose images
$\overline{f}_{i,1},\dots,\overline{f}_{i,c}$ in the
$A/\mathfrak{m}_{i}$-vector space $\mathfrak{m}_{i}/\mathfrak{m}_{i}^{2}$ form
a basis of the subspace $P_{i}/\mathfrak{m}_{i}^{2}$, and such that
$h_{i,j}-1\in P_{s}^{2}$ for $s\neq i$. In particular
$h_{i,1},\dots,h_{i,c}\in P_{i}\setminus(\cup_{s\neq i}P_{s})$. By Nakayama’s
Lemma $P_{i}A_{\mathfrak{m}_{i}}=(h_{i,1},\dots,h_{i,c})A_{\mathfrak{m}_{i}}$.

Set $h_{j}:=\prod_{i=1}^{n}h_{i,j}$ for each $j=1,\ldots,r$. Then
$h_{j}-h_{i,j}\in P_{i}^{2}$ for all $i$, hence $h_{j}\in\cap_{i=1}^{r}P_{i}$
for all $j$. Moreover, the images of $h_{1},\ldots,h_{c}$ in
$\mathfrak{m}_{i}/\mathfrak{m}_{i}^{2}$ form a basis of
$P_{i}/\mathfrak{m}_{i}^{2}$ for all $i$. By [SP, Tag 00SC],
${h_{1}},\ldots,{h_{c}}$ form a regular sequence in
$\mathfrak{m}_{i}A_{\mathfrak{m}_{i}}$ for each $i$. Since
$\operatorname{ht}(P_{i})=c$, the $h_{1},\ldots,h_{c}$ generate
$P_{i}A_{\mathfrak{m}_{i}}$ for each $i=1,\ldots,r$. ∎

Let $k$ be a field. For every $0\leqslant d\leqslant n$, we denote by
$\operatorname{Gr}(n,n-d)$ the Grassmannian of codimension $d$ hyperplanes of
$\mathbb{P}^{n}_{k}$. If $W\subset\mathbb{P}^{n}_{k}$ is a $k$-subspace of
codimension $d$, we will denote by $[W]\in\operatorname{Gr}(n,n-d)(k)$ the
$k$-point representing $W$ in the Grassmannian. We will say that $[W]$
intersects $Z$ transversely at a smooth point $z\in Z$ if $z\in W$ and the
tangent space $T_{z}(Z)$ intersects $W$ transversely. Equivalently, $[W]$
intersects $Z$ transversely at a smooth point $z\in Z$ if $W$ can be cut out
by linear forms
$h_{1},\ldots,h_{d}\in\Gamma(\mathbb{P}^{n}_{k},\mathcal{O}(1))$ such that
$h_{1}(z)=\ldots,h_{d}(z)=0$ and $\displaystyle h_{1}/h,\ldots,h_{d}/h$ form a
regular sequence in the local ring $\mathcal{O}_{Z,z}$ for some (and thus any)
$h\in\Gamma(\mathbb{P}^{n}_{k},\mathcal{O}(1))$ with $h(z)\neq 0$; see [Eis95,
Section 10.3].

######  Lemma 6.2.

Let $Z$ be a closed subscheme of $\mathbb{P}^{n}_{k}$, $0\leq c\leq n$ be an
integer,

$I_{Z,c}\subset\operatorname{Gr}(n,n-c)\times Z$

be the incidence correspondence parametrizing pairs $([W],v)$ such that $v\in
W$, and

$\displaystyle\phi\colon$ $\displaystyle
I_{Z,c}\;\to\;\operatorname{Gr}(n,n-c)$ $\displaystyle([V],v)\to V$

be the projection to the first component. Assume that $z$ is a smooth
$k$-point of $Z$ and a codimension $c$ linear subspace $W_{0}$ of
$\mathbb{P}^{n}$ intersects $Z$ transversely at $z$. Then $\phi$ is smooth at
$([W_{0}],z)$.

###### Proof.

We claim that $\phi$ is flat at $([W_{0}],z)$. Note that the fiber $W\cap Z$
of $\phi$ over $[W]$ is smooth at $z$, so the lemma follows from this claim by
[SP, Tag 01V8].

To prove the claim, we argue by induction on $c$. In the base case, $c=0$,
$\operatorname{Gr}(n,n-c)$ is a point, $\phi$ is the identity map, and the
claim is obvious.

For the induction step, assume that $c\geqslant 1$ and the claim holds when
$c$ is replaced by $c-1$, for every $n\geqslant c$ and every closed subscheme
$Z$ of $\mathbb{P}^{n}$. Choose linear forms
$h_{1},\ldots,h_{c}\in\Gamma(\mathbb{P}^{n}_{k},\mathcal{O}(1))$ such that
$h_{1},\ldots,h_{c}$ cut out $W_{0}$, and $\displaystyle
h_{1}/h,\ldots,h_{c}/h$ form a regular sequence in the local ring
$\mathcal{O}_{Z,z}$ for some $h\in\Gamma(\mathbb{P}^{n}_{k},\mathcal{O}(1))$
such that $h(z)\neq 0$.

Denote the zero locus of $h_{1}$ by $\mathbb{P}^{n-1}$, the intersection
$Z\cap\mathbb{P}^{n-1}$ by $Z^{\prime}$, the preimage of $Z^{\prime}$ under
$\phi$ by $I_{Z,c}^{\prime}$, and the restriction of $\phi$ to $I_{Z,c}$ by
$\phi^{\prime}$. By [Eis95, Corollary 6.9] it suffices to show that
$\phi^{\prime}\colon I_{Z,c}^{\prime}\to\operatorname{Gr}(n,n-c)^{\prime}$ is
flat at $([W_{0}],z)$. Here $\operatorname{Gr}(n,n-c)^{\prime}$ denotes the
hypersurface in $\operatorname{Gr}(n,n-c)$ consisting of $(n-c)$-dimensional
linear subspaces of $\mathbb{P}^{n}$ which are contained in
$\mathbb{P}^{n-1}$. We will view $Z^{\prime}$ as a closed subscheme of
$\mathbb{P}^{n-1}$. Since $h_{1}$ cuts $Z$ transversely at $z$, $z$ is a
smooth point of $Z^{\prime}$. Now observe that
$\operatorname{Gr}(n,n-c)^{\prime}$ is naturally isomorphic to
$\operatorname{Gr}(n-1,n-c)$ and $I_{Z,c}^{\prime}$ is naturally isomorphic to
$I_{Z^{\prime},c-1}$ over $Z^{\prime}$ so that the following diagram commutes

${I_{Z^{\prime},c-1}}$${I^{\prime}_{Z,c}}$${I_{Z,c}}$${\operatorname{Gr}(n-1,n-c)}$${\operatorname{Gr}(n,n-c)^{\prime}}$${\operatorname{Gr}(n,n-c)}$$\scriptstyle{\simeq}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\phi}$$\scriptstyle{\simeq}$

Since $\operatorname{Gr}(n-1,n-c)=\operatorname{Gr}(n-1,\,(n-1)-(c-1))$, we
can use the induction assumption to conclude that $\phi^{\prime}$ is flat at
$([W_{0}],z)$, as desired. This completes the induction step. ∎

######  Lemma 6.3.

Let $Z$ be an irreducible quasi-projective $k$-variety, $Y\subset Z$ be a
closed equidimensional subvariety, with irreducible components
$Y_{1},\dots,Y_{r}$. For every $1\leqslant i\leqslant r$, let $z_{i}\in Y$ be
a closed point such that $Y$ and $Z$ are smooth at $z_{i}$, and $c$ be the
codimension of $Y_{i}$ in $Z$. Then there exist an integer $n\geqslant 0$, a
closed embedding $Z\hookrightarrow\mathbb{P}^{n}_{k}$ and a codimension $c$
subspace $W_{0}\subset\mathbb{P}^{n}_{k}$ such that $W_{0}$ intersects $Z$
transversely at $z_{i}$, and locally around $z_{i}$ we have $Y=Y_{i}=Z\cap
W_{0}$ (scheme-theoretically) for each $i=1,\ldots r$.

###### Proof.

Since $Z$ is quasi-projective, there exists an affine open subset
$\operatorname{Spec}B\subset Z$ containing $z_{1},\dots,z_{r}$. Let
$\operatorname{Spec}A\subset Z$ be the semi-localization of $Z$ at
$\\{z_{1},\dots,z_{r}\\}$. By definition, the ring $A$ is obtained from $B$ by
localizing at the multiplicative subset consisting of elements which do not
belong to the ideals corresponding to the $z_{i}$. Then $A$ is a regular semi-
local ring. By Lemma 6.1, there exist $f_{1},\dots,f_{c}\in A$, such that
$f_{1},\dots,f_{c}$ form a regular sequence in $\mathcal{O}_{Z,z_{i}}$ and
generates the ideal of $Y_{i}$ in $\mathcal{O}_{Z,z_{i}}$, for each
$1\leqslant i\leqslant r$.

Since $Z$ is a quasi-projective, there exists a locally closed embedding
$\iota:Z\hookrightarrow\mathbb{P}^{n}_{k}$. For every $0\leqslant i\leqslant
c$, $f_{i}$ is the restriction of a rational function $P_{i}/Q_{i}$ on
$\mathbb{P}^{n}_{k}$, where $P_{i}$ and $Q_{i}$ are homogeneous polynomials
and $Q_{i}(z_{h})\neq 0$. (Here, by $Q_{i}(z_{h})$ we mean the image of
$Q_{i}$ in the residue field of $k(z_{h})$, which is a finite extension of
$k$.) We deduce that around every $z_{h}$ the variety $Y$ is the scheme-
theoretic intersection of $Z$ and the closed subscheme defined by
$P_{1},\dots,P_{c}$. Multiplying the $P_{i}$ by some homogeneous polynomials
not vanishing at the $z_{h}$, we may assume that the $P_{i}$ have the same
degree $D\geqslant 1$. Replacing $m$ by $mD$ (which amounts to composing
$\iota$ with a suitable Veronese embedding of $\mathbb{P}^{n}_{k}$), we may
assume that the $P_{i}$ have degree $1$, that is, that
$\\{P_{i}=0\\}\subset\mathbb{P}^{n}$ are hyperplanes. Now the embedding
$\iota\colon Z\hookrightarrow\mathbb{P}^{n}_{k}$ and the linear subspace
$W_{0}$ of $\mathbb{P}^{n}_{k}$ given by $\\{P_{1}=\dots=P_{c}=0\\}$ have the
properties claimed in the lemma. ∎

##  7\. Proof of Theorem 1.4

Let $X_{0}=X_{0}^{(1)}\cup X_{0}^{(2)}\cup\ldots$ be the irreducible
decomposition of $X_{0}$. Choose rational functions
$\alpha_{1},\ldots,\alpha_{d-1}\colon X_{0}\dashrightarrow\mathbb{A}^{1}_{k}$
such that the restriction of $\alpha_{1},\ldots,\alpha_{d-1}$ to $X_{0}^{(i)}$
generate the function field $k(X_{0}^{(i)})$ for every $i$. After adjoining
all $G$-translates of $\alpha_{1},\ldots,\alpha_{d-1}$ to this set, we may
assume that $G$ permutes $\alpha_{1},\ldots,\alpha_{d-1}$. Consider the
$G$-equivariant rational map $\alpha\colon X_{0}\dasharrow\mathbb{P}(V)$
taking $x\in X_{0}$ to $(1:\alpha_{1}(x):\ldots:\alpha_{d}(x))$ for a suitable
linear (permutation) representation of $G$ on $V=k^{d+1}$. Note that since the
$G$-action on $X_{0}$ is assumed to be faithful, the $G$-action on
$\mathbb{P}(V)$ is faithful as well. Moreover, by our construction, $\alpha$
induces a $G$-equivariant birational isomorphism between $X_{0}$ and
$\alpha(X_{0})$. After replacing $X_{0}$ with the closure of $\alpha(X_{0})$
in $\mathbb{P}(V)$, we may assume that $X_{0}$ is a closed subvariety of
$\mathbb{P}(V)$.

Let $\mathbb{P}(V)_{\rm non-free}$ be the non-free locus for the $G$-action on
$V$, i.e., the union of the fixed point loci $\mathbb{P}(V)^{g}$ as $g$ ranges
over the non-trivial elements of $G$. Since $G$ acts faithfully on
$\mathbb{P}(V)$, we have

(7.1) $\dim\,\mathbb{P}(V)>\dim\,\mathbb{P}(V)_{\rm non-free}.$

Let $V^{r}$ be the direct sum of $r$ copies of $V$ (as a $G$-representation).
We claim that

(7.2) the codimension of $\mathbb{P}(V^{r})_{\rm non-free}$ in
$\mathbb{P}(V^{r})$ is $\geqslant r$.

Indeed, for each $1\neq g\in G$, let $\operatorname{mult}(g,V)_{\lambda}$ be
the dimension of the $\lambda$-eigenspace of $g$ and
$\operatorname{mult}(g,V)$ be the maximal value of
$\operatorname{mult}(g,V)_{\lambda}$, where $\lambda$ ranges over
$\overline{k}$. Then $\dim\,\mathbb{P}(V)_{\rm non-free}$ is the maximal value
of $\operatorname{mult}(g,V)-1$ as $g$ ranges over the non-trivial elements of
$G$. Clearly $\operatorname{mult}(g,V^{r})=r\operatorname{mult}(g,V)$ for
every $g$. Thus

$\dim\,\mathbb{P}(V^{r})_{\rm non-free}=(\dim\,\mathbb{P}(V)_{\rm non-
free}+1)r-1,$

and the codimension of $\mathbb{P}(V^{r})_{\rm non-free}$ in
$\mathbb{P}(V^{r})$ is

$\displaystyle\dim\,\mathbb{P}(V^{r})-\dim\mathbb{P}(V^{r})_{\rm non-free}$
$\displaystyle=r(\dim\,\mathbb{P}(V)+1)-1-\big{(}(\dim\,\mathbb{P}(V)_{non-
free}+1)r-1\big{)}$
$\displaystyle=r(\dim\,\mathbb{P}(V)-\dim\,\mathbb{P}(V)_{\rm non-
free})\geqslant r,$

where the last inequality follows from (7.1). This completes the proof of
(7.2).

The $G$-equivariant linear embedding $V\hookrightarrow V^{r}$, given by
$v\mapsto(v,0,\dots,0)$, induces a $G$-equivariant closed embedding

$X_{0}\subset\mathbb{P}(V)\hookrightarrow\mathbb{P}(V^{r}).$

This allows us to view $X_{0}$ as a $G$-invariant subvariety of
$\mathbb{P}(V^{r})$. By [PV94, Theorem 4.14] there exists a geometric quotient
map $\pi\colon\mathbb{P}(V^{r})\to\mathbb{P}(V^{r})/G$. Explicitly, write

$\mathbb{P}(V^{r})=\operatorname{Proj}(k[t_{0},\dots,t_{dr}]),$

where each variable $t_{i}$ has degree $1$, and let
$A:=k[t_{0},\dots,t_{dr}]^{G}$. Then
$\mathbb{P}(V^{r})/G\simeq\operatorname{Proj}(A)$ and $\pi$ is induced by the
inclusion $A^{G}\hookrightarrow A$ of graded rings. Restricting $\pi$ to
$X_{0}\subset\mathbb{P}(V^{r})$, we obtain the geometric quotient map
$X_{0}\to X_{0}/G$. Note that
$\dim\mathbb{P}(V^{r})/G=\dim\mathbb{P}(V^{r})=dr-1$ and
$\dim\,X_{0}/G=\dim\,X_{0}=e$, so every irreducible component of $X_{0}/G$ is
of codimension $c=dr-1-e$ in $\mathbb{P}(V^{r})/G$.

We can choose smooth closed points $x_{1},\dots,x_{r}$, one on each
irreducible component of $X_{0}$, such that the (scheme-theoretic) stabilizer
$G_{x_{i}}$ is trivial for every $i$. We now apply Lemma 6.3 to
$Z=\mathbb{P}(V^{r})/G$, $Y=X_{0}/G$, $z_{1},\ldots,z_{r}$, where
$z_{i}=\pi(x_{i})$ for each $i$. We deduce that there exist a closed embedding
$\mathbb{P}(V^{r})/G\hookrightarrow\mathbb{P}^{n}$ defined over $k$ and a
subspace $W_{0}\subset\mathbb{P}^{n}$ of codimension $c$ such that
$X_{0}/G=W_{0}\cap(\mathbb{P}(V^{r})/G)$ locally around $z_{i}$ for each $i$.
Consider the diagram

${T}$${\operatorname{Gr}(n,n-c)\times\mathbb{P}(V^{r})}$${I_{Z,c}}$${\operatorname{Gr}(n,n-c)\times
Z}$${\operatorname{Gr}(n,n-c)}$$\scriptstyle{\overline{f}}$$\scriptstyle{\pi\times\operatorname{id}}$$\scriptstyle{\phi}$$\scriptstyle{\operatorname{pr}_{2}}$

Here $I_{Z,c}$ is the incidence correspondence parametrizing pairs $([W],q)$,
where $W$ is a linear subspace of $\mathbb{P}^{n}_{k}$ of codimension $c$ and
$q\in W$, as in Lemma 6.2, and $T$ is the preimage of $I_{Z,e}$ in
$\mathbb{P}(V^{r})\times\operatorname{Gr}(n,n-c)$.

We claim that the map $\phi$ is smooth at $([W_{0}],z_{i})$ for each
$i=1,\ldots,r$. To prove this claim, note that flatness is local with respect
to the fpqc topology; see [SP, 02L2]. Since the morphism
$\operatorname{Spec}\overline{k}\to\operatorname{Spec}k$ is fpqc, we may base-
change from $k$ to $\overline{k}$. By Lemma 6.2, $\phi_{\overline{k}}$ is
smooth; hence, so is $\phi$. This proves the claim. (Alternatively, one could
replace the fpqc topology by the étale topology as follows. The points
$x_{i}\in X_{0}$ may be chosen so that the finite extensions $k(x_{i})/k$ are
separable. Since flatness is étale-local, to check flatness of $\phi$ at
$([W_{0}],z_{i})$ we may first base change to the Galois closure of
$k(z_{i})$, and then apply Lemma 6.2. This yields an alternative proof of the
claim.) By our choice of $x_{1},\ldots,x_{r}$, $\pi$ is smooth at each of
these points; hence, $\overline{f}=\phi\circ(\pi\times\text{id})\colon
T\to\operatorname{Gr}(n,n-c)$ is smooth at $([W],x_{i})$ for each $i$.

We will construct the family $f\colon\mathcal{X}\to B$ by restricting
$\overline{f}$ to a dense open subset $\mathcal{X}=T\setminus C$, where
$C=T_{\rm sing}\cup T_{\rm non-free}$. Here $T_{\rm sing}$ is the singular
locus of $f$ and $T_{\rm non-free}=\mathbb{P}(V^{r})_{\rm non-
free}\times\operatorname{Gr}(n,n-c)$ is the non-free locus for the $G$-action
in $T$. (Recall that $\mathbb{P}(V^{r})_{\rm non-free}$ was defined at the
beginning of this section.) The base $B$ of our family will be obtained by
removing from $\operatorname{Gr}(n,n-c)$ the locus of points
$b\in\operatorname{Gr}(n,n-c)$ such that the entire fiber $T_{b}$ lies in $C$.
In particular, $\overline{f}(\mathcal{X})\subset B$. Since $C$ is closed in
$T$, $B$ is open in $\operatorname{Gr}(n,n-c)$. Note also that since
$\overline{f}$ is a proper morphism, $\overline{f}(C)$ is closed in $B$.

Let $b_{0}:=[W_{0}]\in\operatorname{Gr}(n,n-c)(k)$. By our choice of
$x_{1},\ldots,x_{r}$, none of the points $([W_{0}],x_{i})$ lie in $C$. Hence
$b_{0}\in B(k)$ and the union of the irreducible components of
$\mathcal{X}_{b_{0}}$ passing through $x_{1},\ldots,x_{r}$ remains
birationally isomorphic to $X_{0}$. Moreover, by our construction
$\mathcal{X}$ and $B$ are irreducible, the $G$-action on $\mathcal{X}$ is
free, $f$ is smooth of constant relative dimension
$e=\dim(X_{0})=\dim(\mathcal{X}_{b_{0}})$. In particular, conditions (i) and
(ii) of Theorem 1.4 hold for $f$. It remains to check that for $r>e$,

(a) $\overline{f}(T_{\rm non-free})\neq\operatorname{Gr}(n,n-c)$,

(b) $\overline{f}(T_{\rm sing})\neq\operatorname{Gr}(n,n-c)$,

(c) there exists a dense open subset $U\subset\operatorname{Gr}(n,n-c)$ such
that the fibers of $f$ over $U$ are projective and irreducible.

Together, (a), (b) and (c) this will ensure that condition (iii) of Theorem
1.4 holds as well.

To prove (a), recall that by (7.2), the codimension of $P(V^{r})_{\rm non-
free}$ in $P(V^{r})$ is $\geqslant r>e$, i.e.,
$\dim\,\pi(\mathbb{P}(V^{r})_{\rm non-free})<rd-1-e=c$. Consequently, an
$(n-c)$-dimensional linear subspace $W$ in $\mathbb{P}^{n}$ in general
position will intersect $\pi(\mathbb{P}(V^{r})_{\rm non-
free})=\mathbb{P}(V)_{\rm non-free}/G$ trivially. Consequently,
$\overline{f}^{-1}(W)\cap T_{\rm non-free}=\emptyset$. This proves (a).

To prove (b), recall that the fiber of the morphism $\phi\colon
I_{Z,c}\to\operatorname{Gr}(n,n-c)$ over $[W]$ is $W\cap Z$. By Bertini’s
Theorem, there exists a dense open subset $U\subset\operatorname{Gr}(n,n-c)$
consisting of $(n-c)$-dimensional linear subspaces $W$ of $\mathbb{P}^{n}$
such that $W\cap Z$ is smooth and irreducible. By generic flatness, after
replacing $U$ by a smaller open subset, we may assume that
$\phi\colon\phi^{-1}(U)\to U$ is flat. Appealing to [SP, Tag 01V8], as we did
in Lemma 6.2, we see that since $\phi\colon\phi^{-1}(U)\to U$ is a flat map
with smooth fibers, it is smooth. Finally, after intersecting $U$ with the
complement of $\overline{f}(T_{\rm non-free})$ (which we know is a dense open
subset of $\operatorname{Gr}(n,n-c)$ by (a)), we may assume that
$\pi\times\operatorname{id}$ is smooth over $\phi^{-1}(U)$. Hence, the map
$\overline{f}\colon\overline{f}^{-1}(U)\to U$, being a composition of two
smooth maps,
$\pi\times\operatorname{id}\colon\overline{f}^{-1}(U)\to\phi^{-1}(U)$ and
$\phi\colon\phi^{-1}(U)\to U$, is smooth. This proves (b).

Finally, to prove (c), recall that by definition, for $b=[W]\in B$, the fiber
$\phi^{-1}(b)=W\cap Z=W\cap\mathbb{P}(V^{r})/G$ is a complete intersection in
$\mathbb{P}(V^{r})/G$. The fiber
$\mathcal{X}_{[W^{\prime}]}=f^{-1}(W^{\prime})$ is cut out by the same
homogeneous polynomials, now viewed as elements of $k[V^{r}]$ instead of
$k[V^{r}]^{G}$. Thus $\mathcal{X}_{b}$ is a smooth complete intersection in
$\mathbb{P}(V^{r})$. Since $\dim\,\mathcal{X}_{b}=e\geqslant 1$, by [Ser55, n.
78], $\mathcal{X}_{b}$ is connected, hence irreducible. This completes the
proof of (c) and thus of Theorem 1.4. ∎

######  Remark 7.3.

The family $f\colon\mathcal{X}\to B$ constructed in the proof of 1.4 has the
additional property that the fibers of $f$ over the dense open subset
$U\subset B$ (the open subset in part (iii) of the statement of the theorem)
are complete intersections in projective space $\mathbb{P}^{n}$. With a bit of
extra effort one can make sure that every fiber of $f$ over $U$ is of general
type. Since we will not need this assertion in this paper, we leave the proof
as an exercise for the interested reader.

##  8\. Proof of Theorem 1.1

Let $d:=\dim(X)$ and $e:=\operatorname{ed}_{k}(X)$. We must show that there
exists an irreducible smooth projective variety $Z$ and a $G$-torsor $Y\to Z$
such that $\dim(Y)=\dim(X)$ and
$\operatorname{ed}_{k}(Y)=\operatorname{ed}_{k}(X)$. If $e=0$, Theorem 1.1 is
obvious: we can take $Y=G\times\mathbb{P}^{d}$, where $G$ acts by translations
on the first factor and trivially on the second. Thus we may assume without
loss of generality that $e\geqslant 1$ and in particular, $d\geqslant 1$. In
this case, we use the following strategy: construct a family $\mathcal{X}\to
B$ as in Theorem 1.4 with $X_{0}=X$, then take $Y=\mathcal{X}_{b}$ where $b$
is a $k$-point of $B$ in very general position. 1.4 tells us that $\dim(Y)=d$
and the $G$-action on $Y$ is strongly unramified. We would like to appeal to
1.2 to conclude that $\operatorname{ed}_{k}(Y)=e$. One difficulty in
implementing this strategy is that

(i) 1.4 requires $G$ to be a finite group, whereas in 1.1, $G$ is an arbitrary
linear algebraic group over a field of good characteristic.

Even if assume that $G$ is a finite group, there is another problem:

(ii) 1.2 requires the fibers of $f\colon\mathcal{X}\to B$ to be primitive
$G$-varieties, whereas in 1.4 the fiber $\mathcal{X}_{b_{0}}$ over $b_{0}$ may
not be primitive; we only know that it contains the primitive variety
$X_{0}=X$ as a union of its irreducible components.

We will overcome (i) by using reduction of structure to a finite group, and
(ii) by using the following variant of 1.2.

######  Theorem 8.1.

Let $G$ be a linear algebraic group over an algebraically closed field $k$ of
good characteristic (see Definition 2.1), $f:\mathcal{X}\to B$ be a
$G$-equivariant morphism of $k$-varieties such that $B$ is irreducible, $G$
acts trivially on $B$, and that the generic fiber of $f$ is a primitive and
generically free $G_{k(B)}$-variety. Let $b_{0}\in B(k)$,
$x_{0}\in\mathcal{X}_{b_{0}}(k)$, and $X_{0}$ be a $G$-invariant reduced open
subscheme of $\mathcal{X}_{b_{0}}$ containing $x_{0}$ such that:

  1. (1)

$X_{0}$ is a generically free primitive $G$-variety, and

  2. (2)

$f$ is flat at $x_{0}$.

Then, for a very general $b\in B(k)$, $\mathcal{X}_{b}$ is generically free
and primitive, and
$\operatorname{ed}_{k}(\mathcal{X}_{b})\geqslant\operatorname{ed}_{k}(X_{0})$.
Furthermore, if $k$ is of infinite transcendence degree over its prime field
(in particular, if $k$ is uncountable), then set of those $b\in B(k)$ such
that
$\operatorname{ed}_{k}(\mathcal{X}_{b})\geqslant\operatorname{ed}_{k}(X_{0})$
is Zariski dense in $B$.

Here, as usual, “very general” means “away from a countable union of proper
subvarieties”. If $k$ has infinite transcendence degree over its prime field,
one may replace the assumption that
$\mathcal{X}_{k(B)}\to\operatorname{Spec}k(B)$ is generically free and
primitive with the assumption that a general fiber of $f$ is generically free
and primitive.

###### Proof.

Let $Z:=\mathcal{X}_{b_{0}}\setminus X_{0}$. Then $Z$ is a $G$-invariant
closed subscheme of $X$, and the restriction $f|_{\mathcal{X}\setminus
Z}:\mathcal{X}\setminus Z\to B$ is a flat $G$-equivariant morphism, whose
fiber at $b_{0}$ is a generically free irreducible $G$-variety. Replacing
$\mathcal{X}$ by $\mathcal{X}\setminus Z$, we may thus assume that
$\mathcal{X}_{b_{0}}$ is a generically free primitive $G$-variety.

By generic flatness [SP, Tag 0529], there exists a dense open subscheme
$U\subset B$ such that $f$ is flat over $U$. If $b_{0}\in U(k)$, the
conclusion follows from 1.2. We may thus assume that $b_{0}$ does not lie in
$U$, and let $u\in U$ be the generic point of $U$. Since $U$ is dense in $B$,
$b_{0}$ is a specialization of $u$. By [GD61, Proposition 7.1.9], there exist
a discrete valuation ring $R$ and a separated morphism
$\operatorname{Spec}R\to B$ mapping the closed point $s$ of
$\operatorname{Spec}R$ to $b_{0}$ and the generic point $\eta$ of
$\operatorname{Spec}R$ to $u$.

Since $f$ is flat at $x_{0}$ and $G$ acts primitively on on
$\mathcal{X}_{b_{0}}$, by the openness of the flat locus [SP, Tag 0399], the
flat locus of the base change
$f_{R}\colon\mathcal{X}_{R}\to\operatorname{Spec}R$ is dense in the component
of the special fiber of $f_{R}$ containing (the preimage of) $x_{0}$. Since we
are assuming that the generic fiber is primitive, we conclude that the flat
locus of $f_{R}$ is dense in the generic fiber. Therefore, after removing the
complement of the flat locus from $\mathcal{X}_{R}$, we may thus assume that
$f_{R}$ is flat. Let $\overline{\eta}$ and $\overline{s}$ be geometric points
lying above $\eta$ and $s$, respectively. By 3.1(b), we have

$\operatorname{ed}_{k(\overline{\eta})}(\mathcal{X}_{\overline{\eta}})\geqslant\operatorname{ed}_{k(\overline{s})}((X_{0})_{\overline{s}}).$

On the other hand, by Lemma 4.2,

$\operatorname{ed}_{k(\overline{s})}((X_{0})_{\overline{s}})\geqslant\operatorname{ed}_{k}(X_{0}).$

Combining these two inequalities, we see that
$\operatorname{ed}_{k(\overline{\eta})}(X_{\overline{\eta}})\geqslant\operatorname{ed}_{k}(X_{0})$.
The conclusion now follows from an application of 1.2 to the restriction of
$f$ to $f^{-1}(U)$. ∎

###### Conclusion of the proof of 1.1.

As we mentioned at the beginning of this section, in the course of proving 1.1
we may assume that $e\geqslant 1$.

We will now reduce the theorem to the case, where $X$ is incompressible, i.e.,
$d=e+\dim(G)$. Indeed, by the definition of essential dimension there exists a
$G$-equivariant dominant rational map $X\dasharrow X^{\prime}$ such that
$\dim(X^{\prime})=d^{\prime}:=e+\dim(G)$. Suppose we know that 1.1 holds for
$X^{\prime}$. In other words, there exists a $G$-variety $Y^{\prime}$ such
that $\dim(Y^{\prime})=d^{\prime}$, $\operatorname{ed}(Y^{\prime})=e$, and
$Y^{\prime}$ is the total space of a $G$-torsor $t\colon Y^{\prime}\to
P^{\prime}$ over a smooth projective variety $P$. Clearly $d\geqslant
d^{\prime}$. Let $Y=\mathbb{P}^{d-d^{\prime}}\times Y^{\prime}$, where $G$
acts trivially. Then $\dim(Y)=d$,
$\operatorname{ed}(Y)=\operatorname{ed}(Y^{\prime})=e$ (see, e.g., [Rei00,
Lemma 3.3(d)] and $Y$ is a $G$-torsor ${\rm id}\times t\colon
Y=\mathbb{P}^{d-d^{\prime}}\times Y^{\prime}\to\mathbb{P}^{d-d^{\prime}}\times
P$ over the smooth projective variety $\mathbb{P}^{d-d^{\prime}}\times P$, as
desired.

From now on we will assume that $d=e+\dim(G)$ and $e\geqslant 1$. By [CGR06,
Theorem 1.1], there exists a finite subgroup $S$ of $G$, such that for every
field extension $K/k$ the natural map $H^{1}(K,S)\to H^{1}(K,G)$ is
surjective. In particular, there exists a generically free primitive
$S$-variety $X_{0}$ such that $X_{0}\times^{S}G$ is birationally equivalent to
$X$. Note that $\dim(X_{0})=\dim(X)-\dim(G)=d-\dim(G)$.

Let $f\colon\mathcal{X}\to B$ be a family obtained by applying 1.4 to the
finite group $S$ and the $S$-variety $X_{0}$. This is possible because $S$ is
a finite group, $X_{0}$ is primitive, hence equidimensional, and $e\geqslant
1$. Note also that since $k$ is algebraically closed, the assumption of 1.4
that $k$-points should be dense in $X_{0}$ is automatically satisfied here.

Recall that the family $f\colon\mathcal{X}\to B$ supplied to us by 1.4 has the
following properties: $f$ is smooth of constant relative dimension
$e=\dim(X_{0})$, $S$ acts freely on $\mathcal{X}$, and there is a dense open
subscheme $U\subset B$ for any $b\in U(k)$ the fiber $\mathcal{X}_{b}$ is
smooth, projective and irreducible. Moreover, $X_{0}$ is $S$-equivariantly
birationally isomorphic to a union of components of the fiber
$\mathcal{X}_{b_{0}}$ for some $b_{0}\in B(k)$.

Set $\mathcal{X}^{\prime}:=\mathcal{X}\times^{S}G$ and let
$f^{\prime}\colon\mathcal{X}^{\prime}\to B$ be the natural projection induced
by $f$. Then $f^{\prime}$ is a smooth morphism of constant relative dimension
$d=e+\dim(G)$, $G$ acts freely on $\mathcal{X}^{\prime}$,
$\mathcal{X}^{\prime}_{b}$ is a primitive $G$-variety for every $b\in U(k)$,
the $G$-action on $\mathcal{X}^{\prime}_{b}$ is strongly unramified, and $X$
is $G$-equivariantly birationally isomorphic to a union of irreducible
components of $\mathcal{X}^{\prime}_{b_{0}}$. Clearly

$\operatorname{ed}(\mathcal{X}^{\prime}_{b})\leqslant\dim(\mathcal{X}^{\prime}_{b})-\dim(G)=\dim(X)=d-\dim(G)=e.$

On the other hand, since $k$ is algebraically closed and of infinite
transcendence degree over its prime field, by 8.1 there exists a $k$-rational
point $b$ of $B$ such that
$\operatorname{ed}_{k}(\mathcal{X}^{\prime}_{b})\geqslant\operatorname{ed}_{k}(X)=e$.
Setting $Y:=\mathcal{X}^{\prime}_{b}$, we obtain a generically free primitive
$G$-variety $Y$ such that $\dim(Y)=d$, $\operatorname{ed}_{k}(Y)=e$, and the
$G$-action on $Y$ is strongly unramified, as desired. ∎

######  Corollary 8.2.

Let $G$ be a linear algebraic group over an algebraically closed field $k$ of
good characteristic and of infinite transcendence degree over its prime field.
Then there exists an unramified generically free $G$-variety $Y$ such that

  1. (1)

$\operatorname{dim}(Y)=\operatorname{ed}_{k}(G)+\dim G$, and

  2. (2)

$\operatorname{ed}_{k}(Y)=\operatorname{ed}_{k}(G)$.

###### Proof.

Let $V$ be a generically free $G$-variety over $k$ of essential dimension
$e=\operatorname{ed}_{k}(G)$. Then there is a $G$-compression $V\dasharrow X$,
where $X$ is a generically free $G$-variety of dimension $e+\dim(G)$. Clearly
$\operatorname{ed}_{k}(X)\leqslant e$; on the other hand,
$\operatorname{ed}_{k}(X,G)\leqslant\operatorname{ed}_{k}(V,G)=e=\dim(X)$. We
conclude that $\dim(X)=e+\dim(G)$ and $\operatorname{ed}_{k}(X)=e$. By Theorem
1.1 there exists an unramified generically free variety $Y$ of dimension
$e+\dim(G)$ and essential dimension $e$. ∎

##  9\. Finite group actions on hypersurfaces

We conclude this section with yet another application of 8.1.

######  Proposition 9.1.

Let $k$ be an algebraically closed field of infinite transcendence degree over
its prime field, $G$ be a finite group, $W$ be a faithful $k$-representation
of $G$, and set $V:=W\oplus k$, where $G$ acts trivially on $k$. Then for
every $d\geqslant 1$, a very general polynomial $f\in(k[V]_{d})^{G}$ cuts out
an affine hypersurface $Z(f)$ of essential dimension
$\operatorname{ed}_{k}(Z(f);G)=\operatorname{ed}_{k}(G)$.

Here as usual, $k[V]_{d}$ denotes the vector space of homogeneous polynomials
of degree $d$ on $V$, and $(k[V]_{d})^{G}$ denotes the subspace of
$G$-invariant polynomials in $k[V]_{d}$.

###### Proof.

Let $n:=\dim W$, let $x_{1},\dots,x_{n}$ be coordinates on $W$, and let
$x_{n+1}$ be a coordinate on $k$. We fix a polynomial function $f_{0}$ of
degree $d$ on $W\oplus k$ which only depends on $x_{n+1}$,
$f_{0}=f_{0}(x_{n+1})$, and which has distinct roots. Then
$f_{0}\in(k[V]_{d})^{G}\setminus\\{0\\}$ and $H_{f_{0}}$ is a disjoint union
of $d$ translates of $W$.

Let $B$ be the complement of the origin in $\mathbb{A}((k[V]_{d})^{G})$.
Consider the commutative triangle

${\mathcal{X}}$${\mathbb{A}(V)\times
B}$${B,}$$\scriptstyle{\operatorname{pr}_{2}}$

where $\mathcal{X}$ is the reduced closed subscheme whose $k$-points
parametrize pairs $(v,f)\in V\times k[V]^{G}_{d}$ such that $f\neq 0$ and
$f(v)=0$. By definition the fiber $\mathcal{X}_{f}$ over $f\in B$ is the
hypersurface $Z(f)$ in the affine space $\mathbb{A}(V)$. In other words, the
morphism $\mathcal{X}\to B$ is the restriction to $B$ of the universal family
of affine hypersurfaces in $V$ of degree $d$. In particular, it is flat.

Every irreducible component of $\mathcal{X}_{[f_{0}]}$ is $G$-equivariantly
isomorphic to $W$ and so has essential dimension equal to
$\operatorname{ed}_{k}(G)$; see [Mer13, Proposition 3.11]. Let
$\mathcal{X}^{\prime}\subset\mathcal{X}$ be the Zariski open subvariety
obtained by removing all irreducible components of $\mathcal{X}_{[f_{0}]}$ but
one. Since the inclusion map $\mathcal{X}^{\prime}\hookrightarrow\mathcal{X}$
is an open embedding, and open embeddings are flat, we conclude that the
composition $\mathcal{X}^{\prime}\hookrightarrow\mathcal{X}\to B$ is also
flat. Thus for a very general $f\in(k[V]_{d})^{G}$, we have

$\operatorname{ed}_{k}(Z(f))=\operatorname{ed}_{k}(\mathcal{X}_{[f]})=\operatorname{ed}_{k}(\mathcal{X}^{\prime}_{[f]})\geqslant\operatorname{ed}_{k}(\mathcal{X}^{\prime}_{[f_{0}]})=\operatorname{ed}_{k}(G)$

where the inequality follows from 8.1. The opposite inequality,
$\operatorname{ed}_{k}(Z(f))\leqslant\operatorname{ed}_{k}(G)$ is immediate
from the definition of $\operatorname{ed}_{k}(G)$. ∎

##  10\. Appendix: Essential dimension at a prime

The purpose of this appendix is to point out that 3.1, 1.2 and 1.1 continue to
hold if we replace essential dimension with essential dimension at a prime
$q$. The proofs are largely unchanged. One notable feature of these results is
that they hold for any prime $q\neq\operatorname{char}(k)$, where $k$ is a
base field. 3.1 and Theorems 1.1 and 1.2 assume that $k$ is a field of good
characteristic; this assumption is not needed here.

######  Proposition 10.1.

Let $k$ be an algebraically closed field, $R$ be a discrete valuation ring
containing $k$ and with residue field $k$, and $l$ be the fraction field of
$R$. Let $G$ be a linear algebraic group over $k$ and $q$ be a prime number
invertible in $k$. Let $X$ be a flat separated $R$-scheme of finite type
endowed with a $G$-action over $R$, whose fibers are generically free and
primitive $G$-varieties. Then
$\operatorname{ed}_{\overline{l},q}(X_{\overline{l}})\geqslant\operatorname{ed}_{k,q}(X_{k})$.

###### Proof.

The proof is the same as that of 3.1, except that instead of the [RS21,
Theorem 1.2], one should use [RS21, Theorem 11.1] which gives an analogous
assertion for essential $q$-dimension. ∎

######  Theorem 10.2.

Let $G$ be a linear algebraic group over an algebraically closed field $k$ and
let $q$ be a prime number invertible in $k$. Let $B$ be a noetherian
$k$-scheme, $f\colon\mathcal{X}\to B$ be a flat separated $G$-equivariant
morphism of finite type such that $G$ acts trivially on $B$ and the geometric
fibers of $f$ are generically free and primitive $G$-varieties (in particular,
reduced). Then for any fixed integer $n\geqslant 0$ the subset of $b\in B$
such that
$\operatorname{ed}_{k(\overline{b}),q}(\mathcal{X}_{\overline{b}};G_{k(\overline{b})})\leqslant
n$ for every (equivalently, some) geometric point $\overline{b}$ above $b$ is
a countable union of closed subsets of $B$.

###### Proof.

The proof is analogous to that of 1.2, replacing 3.1 by 10.1. ∎

######  Theorem 10.3.

Let $G$ be a linear algebraic group over an algebraically closed field $k$ of
infinite transcendence degree over its prime field, $q$ be a prime number
invertible in $k$, and $X$ be a generically free primitive $G$-variety. Then
there exists an irreducible smooth projective variety $Z$ and a $G$-torsor
$Y\to Z$ such that $\dim(Y)=\dim(X)$ and
$\operatorname{ed}_{k,q}(Y)=\operatorname{ed}_{k,q}(X)$.

###### Proof.

Analogous to that of 1.1, replacing 1.2 by 10.2. ∎

## Acknowledgements

We are grateful to Najmuddin Fakhruddin for helpful discussions. The second-
named author thanks Kestutis Česnavičius and the Département de Mathématiques
d’Orsay (Université Paris-Saclay) for hospitality during Summer 2021, and the
Institut des Hautes Études Scientifiques for hospitality in the Fall 2021.

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