# Equivariantly normal varieties
for diagonalizable group actions

Michel Brion

###### Abstract

Given a finite group scheme $G$ over a field and a $G$-variety $X$, we obtain
a criterion for $X$ to be $G$-normal in the sense of [Br24]. When $G$ is
diagonalizable, we describe the local structure of $G$-normal varieties in
codimension $1$ and their dualizing sheaf. As an application, we obtain a
version of the Hurwitz formula for $G$-normal varieties, where $G$ is linearly
reductive.

##  1 Introduction

This paper is a sequel to [Br24], which considered actions of finite group
schemes over fields of positive characteristic. In this setting, an action on
a variety $X$ does not necessarily lift to an action on the normalization
$\tilde{X}$; as a remedy, the notion of equivariant normalization was
introduced and explored. The present paper focuses on $G$-normal varieties,
where $G$ is a finite diagonalizable group scheme, with two main motivations
in mind: firstly, finite diagonalizable group schemes have an especially
simple structure and representation theory, which makes them very accessible.
Secondly, they are quite ubiquitous; for example, the study of $G$-normal
varieties with $G$ linearly reductive reduces to some extent to the
diagonalizable case (see Remark 4.4 for details, and Section 9 for an
illustration).

Given a finite diagonalizable group scheme $G$ acting faithfully on a
$G$-normal variety $X$ with categorical quotient $\pi:X\to Y$, the variety
$Y=X/G$ is known to be normal and $\pi$ is a $G$-torsor over a dense open
subset of $Y$. We may thus view $X$ as a “ramified $G$-cover” of $Y$ (as a
partial converse, every variety obtained as a $G$-torsor over a normal variety
is $G$-normal). When $G$ is constant, $\pi$ is indeed a ramified Galois cover
with group $G$ and $X$ is normal; this gives back the classically studied
abelian covers, see e.g. [Pa91, AP12]. But the general case has quite
different geometric features: if $G$ is infinitesimal, then $\pi$ is purely
inseparable, and hence ramified everywhere. Moreover, $X$ is generally
singular in codimension $1$, with cuspidal singularities only. Still, $X$
satisfies Serre’s property $(S_{2})$, as well as an equivariant version of
$(R_{1})$: every $G$-stable divisor is Cartier at its generic points.

For simplicity, we now state our main results in the setting of curves over an
algebraically closed field $k$. Let $X$ be such a curve, equipped with a
faithful $G$-action with quotient $\pi:X\to Y$. We assume that $X$ is
$G$-normal; this is equivalent to the orbit $G\cdot x\subset X$ being a
Cartier divisor for any $x\in X(k)$, and implies that the curve $Y$ is smooth.
Denote by $\operatorname{Stab}_{G}(x)\subset G$ the stabilizer of $x$; it acts
linearly on the fibre at $x$ of the conormal sheaf to $G\cdot x$ via a
character $\nu(x):\operatorname{Stab}_{G}(x)\to{\mathbb{G}}_{m}$. We may now
describe the equivariant local structure of $X$ at $x$:

######  Theorem 1.

  1. (i)

The group scheme $H=\operatorname{Stab}_{G}(x)$ is cyclic, and $\nu(x)$
generates its character group.

  2. (ii)

There exists an open $G$-stable neighborhood $U=U(x)\subset X$ of $x$ such
that the quotient morphism $U\to U/G=V$ factors as
$U\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}U/H\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}V$,
where $\varphi$ is a cyclic cover of degree $|H|$, and $\psi$ is a
$G/H$-torsor.

By a cyclic cover of degree $n$ in this local setting, we mean a morphism of
affine schemes $\varphi:{\rm Spec}(A)\to{\rm Spec}(B)$, where
$A=B[T]/(T^{n}-g)$ and $g\in B$ is a nonzerodivisor. Then $\varphi$ is the
quotient by the cyclic group scheme $\boldsymbol{\mu}_{n}$ acting on ${\rm
Spec}(A)$ via the ${\mathbb{Z}}/n{\mathbb{Z}}$-grading
$A=\bigoplus_{m=0}^{n-1}B\bar{T}^{m}$; also, $\varphi$ is a
$\boldsymbol{\mu}_{n}$-torsor outside of the zero scheme of $g$.

As a consequence of Theorem 1, every $G$-normal curve is a tamely ramified
$G$-torsor in the sense of [BB22, Def. 2.1]. But equivariant normality imposes
additional conditions; for example, a cyclic cover as above is
$\boldsymbol{\mu}_{n}$-normal if and only if $B$ is an integrally closed
domain and the zero scheme of $g$ is reduced.

As a further consequence, the finite morphism $\pi$ is flat and a local
complete intersection. So its dualizing sheaf is isomorphic to the relative
canonical sheaf $\omega_{X/Y}$, and hence is invertible.

######  Theorem 2.

The sheaf $\omega_{X/Y}$ is equipped with a $G$-linearization and a
$G$-invariant global section $s_{X/Y}$ such that

${\rm div}(s_{X/Y})=\sum\big{(}|\operatorname{Stab}_{G}(x)|-1)\big{)}\,G\cdot
x$

(sum over the $G$-orbits of $k$-rational points of $X$).

For a projective $G$-normal curve $X$, this readily yields the relation
between arithmetic genera

$2p_{a}(X)-2=|G|\,\big{(}2p_{a}(Y)-2\big{)}+\sum\big{(}|\operatorname{Stab}_{G}(x)|-1\big{)},$
(1.1)

where the sum runs again over the orbits. (We have of course $p_{a}(Y)=g(Y)$
as $Y$ is smooth; also, $g(X)=g(Y)$ if $G$ is infinitesimal, see [Br24, Rem.
5.2]).

The above results are used in the preprint [FM24] by P. Fong and M. Maccan,
which studies a remarkable class of smooth projective surfaces in positive
characteristics: relatively minimal surfaces equipped with an isotrivial
elliptic fibration. It turns out that every such surface $S$ is obtained as a
quotient $(E\times X)/G$, where $E$ is an elliptic curve, $G\subset E$ a
finite subgroup scheme, and $X$ a projective $G$-normal curve. Thus, $S$ is
equipped with an action of the algebraic group $E$ for which the quotient is
the elliptic fibration $f:S\to X/G=Y$. When $G$ is diagonalizable, the
multiple fibers and relative canonical bundle of $f$ are described in terms of
the quotient $\pi:X\to Y$ in [FM24, §3], based on Theorems 1 and 2. It would
be very interesting to obtain a global description of $G$-normal curves in
this setting, by gluing their local models in the spirit of [Pa91].

Theorem 1 is well-known if $G$ is constant; then $X$ is a smooth curve, the
morphism $\psi:U/H\to V$ is étale, and $\pi$ is a tamely ramified $G$-cover.
In this setting, Theorem 2 and (1.1) follow from the classical Hurwitz
formula. For an arbitrary finite group scheme $G$, the morphism $\psi$ may
well be purely inseparable. Still, its dualizing sheaf is equipped with a
natural section, and this reduces the proof of Theorem 2 to a local
computation. In view of this theorem, the branch locus of $\pi$ (i.e., the
locus where it fails to be a $G$-torsor) can be read off the relative
canonical sheaf $\omega_{X/Y}$.

In the general setting of a $G$-normal variety $X$ over an arbitrary field,
where $G$ is diagonalizable, we obtain versions of Theorems 1 and 2 over an
open $G$-stable subset of $X$ with complement of codimension at least $2$; see
Theorems 6.2 and 7.4. In this setting, the relative dualizing sheaf
$\omega_{\pi}$ is torsion-free and satisfies $(S_{2})$, so that it suffices to
determine it in codimension $1$; see [Kol22] for further aspects of duality
for torsion-free $(S_{2})$ sheaves.

The layout of this paper is as follows. Sections 2 and 3 collect preliminary
notions and results on actions of finite group schemes over a field,
especially diagonalizable ones. For these, we discuss torsors and uniform
cyclic covers in the sense of [AV04] (Remarks 3.2 and 3.3), which form the
main ingredients of our local structure results. Our presentation has some
overlap with [Hau20, Sec. 3]; we have provided details in order to be self-
contained. The main result of Section 4 is a $G$-normality criterion for a
finite group scheme $G$ (Proposition 4.5), which takes a simpler form when $G$
is diagonalizable. Under this assumption, we obtain our local structure
theorem by steps: in Section 5, we first describe the fibres of the quotient
morphism $\pi$ at points of codimension $1$ for a $G$-normal variety $X$, and
then the corresponding semi-local rings of $X$; these may be viewed as
equivariant analogs of discrete valuation rings (see Propositions 5.1 and
5.4). From this, we deduce a more global description of $X$ in codimension $1$
(Theorem 6.2, the main result of Section 6). In turn, this yields in Section 7
a formula à la Hurwitz for the dualizing sheaf of the quotient morphism $\pi$.
Section 8 contains applications to curves; in particular, we determine the
arithmetic genus of projective $G$-normal curves, and its equivariant analog
(Proposition 8.2). In Section 9, we extend our description of the dualizing
sheaf to the setting of linearly reductive group schemes (Theorem 9.1 and
Proposition 9.3).

Acknowledgements. I am grateful to Pascal Fong, Qing Liu, Matilde Maccan,
David Rydh, and Dajano Tossici for interesting discussions and helpful
comments on earlier versions of this paper.

##  2 Finite group schemes and their actions

Throughout this paper, we work over a field $k$ of characteristic $p>0$.
Schemes are assumed to be separated and of finite type over $k$ unless
explicitly mentioned. A _variety_ is a geometrically integral scheme. We say
that an open subset $U$ of a variety $X$ is _big_ if its complement
$X\setminus U$ has codimension at least $2$.

We denote by $G$ a finite group scheme and by $|G|$ its order, i.e., the
dimension of the algebra ${\mathcal{O}}(G)$ viewed as a $k$-vector space.
Recall that $G$ lies in a unique exact sequence

$1\longrightarrow G^{0}\longrightarrow
G\longrightarrow\pi_{0}(G)\longrightarrow 1,$ (2.1)

where $G^{0}$ is infinitesimal and $\pi_{0}(G)$ is finite and étale. If $k$ is
perfect, then this sequence has a unique splitting (see e.g. [DG70, II.5.2.4].

We say that $G$ is _linearly reductive_ if every $G$-module is semi-simple. By
Nagata’s theorem, this is equivalent to $G^{0}$ being of multiplicative type
and $|\pi_{0}(G)|$ being prime to $p$ (see [DG70, IV.3.3.6]).

A $G$-_scheme_ is a scheme $X$ equipped with a $G$-action

$\alpha:G\times X\longrightarrow X,\quad(g,x)\longmapsto g\cdot x.$

This action is said to be _faithful_ if every nontrivial subgroup scheme of
$G$ acts nontrivially.

For any $G$-scheme $X$ and any closed subscheme $X^{\prime}$, the morphism
$G\times X^{\prime}\to X$, $(g,x)\mapsto g\cdot x$ is finite. We denote its
schematic image by $G\cdot X^{\prime}$; this is the smallest closed $G$-stable
subscheme of $X$ containing $X^{\prime}$. Also, we denote by
$C_{G}(X^{\prime})$ the _centralizer_ of $X^{\prime}$, i.e., the largest
subgroup scheme of $G$ that acts trivially on $X^{\prime}$. In particular, for
any closed point $x\in X$, we obtain the _orbit_ $G\cdot x$ and the
centralizer $C_{G}(x)$. These will be discussed at the end of the next
paragraph.

Given a field extension $K/k$ and a scheme $X$, we denote by $X_{K}$ the
$K$-scheme obtained from $X$ by the base change ${\rm Spec}(K)\to{\rm
Spec}(k)$. Then $G_{K}$ is a finite $K$-group scheme and the formation of
(2.1) commutes with such base change. Moreover, every $G$-action $\alpha$ on
$X$ yields a $G_{K}$-action $\alpha_{K}$ on $X_{K}$, which may be identified
with a $G$-action on the $k$-scheme $X_{K}$ via the isomorphism

$G_{K}\times_{{\rm
Spec}(K)}X_{K}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}G\times X_{K}.$

The canonical projection ${\rm pr}:X_{K}\to X$ is equivariant relative to the
homomorphism $G_{K}\to G$. If $x\in X$ is a closed point with residue field
$K=\kappa(x)$, then viewing $x$ as a $K$-point of $X_{K}$, the orbit
$G_{K}\cdot x\subset X_{K}$ has schematic image $G\cdot x$ under the above
projection. Moreover, we have an isomorphism of $G_{K}$-schemes $G_{K}\cdot
x\simeq G_{K}/C_{G_{K}}(x)$ and the equality
$C_{G_{K}}(x)=\operatorname{Stab}_{G}(x)$, the fibre at $x$ of the stabilizer
$\operatorname{Stab}_{G}\subset G\times X$.

Next, we discuss quotients by $G$; for this, we only consider $G$-schemes $X$
that admit a covering by open affine $G$-stable subsets. This assumption is
satisfied if $G$ is infinitesimal (then every open subset is $G$-stable), or
if $X$ is quasi-projective (e.g., a curve). Then $X$ admits a categorical
quotient $\pi:X\to Y=X/G$, where $Y$ is a scheme, $\pi$ is finite and
surjective, and the morphism

$\gamma=(\alpha,{\rm pr}_{2}):G\times X\longrightarrow
X\times_{Y}X,\quad(g,x)\longmapsto(g\cdot x,x)$ (2.2)

is surjective as well. Therefore, ${\mathcal{A}}=\pi_{*}({\mathcal{O}}_{X})$
is a coherent algebra over ${\mathcal{B}}={\mathcal{O}}_{Y}$, equipped with a
$G$-linearization. Moreover, the natural map
${\mathcal{B}}\to{\mathcal{A}}^{G}$ is an isomorphism (see [DG70, III.2.6.1]
for these results). If $X$ is a variety, then so is $Y$.

Also, recall that $X$ has a largest open $G$-stable subset $X_{{\rm fr}}$ on
which $G$ acts freely. Moreover, $\pi$ restricts to a $G$-torsor $X_{{\rm
fr}}\to Y_{{\rm fr}}$, where $Y_{{\rm fr}}$ is open in $Y$ (see loc. cit.). We
say that a $G$-variety $X$ is _generically free_ if $X_{{\rm fr}}$ is
nonempty.

The quotient morphism $\pi$ is the composition

$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/G^{0}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y,$
(2.3)

where $\varphi$ (resp. $\psi$) denotes the quotient by $G^{0}$ (resp.
$\pi_{0}(G)$). Moreover, $\varphi$ is purely inseparable, and hence a
universal homeomorphism. The formation of $\pi$, $\varphi$ and $\psi$ commutes
with flat base change $Y^{\prime}\to Y$; in particular, with field extensions.

######  Lemma 2.1.

Assume that $X$ is a $G$-variety. Then the $G$-action on $X$ is faithful
(resp. generically free, free) if and only if the $G^{0}$-action on $X$ is
faithful (resp. generically free, free) and the $\pi_{0}(G)$-action on
$X/G^{0}$ is faithful (resp. free).

###### Proof.

By fpqc descent, we may assume $k$ algebraically closed. Then
$G=G^{0}\rtimes\pi_{0}(G)$ and hence
$G(k)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\pi_{0}(G)(k)$; moreover,
$\pi_{0}(G)$ is constant.

Denote by $H$ the kernel of the $G$-action on $X$. Then also $H\simeq
H^{0}\rtimes\pi_{0}(H)$, and $\pi_{0}(H)$ is constant. Moreover, $H^{0}$
(resp. $\pi_{0}(H)$) is the kernel of the action of $G^{0}$ (resp.
$\pi_{0}(G)$) on $X$. Since $\varphi$ is bijective, it follows that
$\pi_{0}(H)$ is the kernel of the $\pi_{0}(G)$-action on $Z$. This readily
implies the assertion on faithful actions.

The assertion on free actions is obtained similarly by considering the
stabilizers of $k$-rational points of $X$. This assertion implies that the
$G$-action on $X$ is generically free if and only if so are the $G^{0}$-action
on $X$ and the $\pi_{0}(G)$-action on $Z$. But for an action of a finite group
(that is, a finite constant group scheme) on a variety, being generically free
is equivalent to being faithful; this completes the proof. ∎

######  Lemma 2.2.

Let $X$ be a $G$-variety with quotient $\pi:X\to Y$, and let
$X^{\prime}\subset X$ be a closed $G$-stable subscheme.

  1. (i)

The quotient morphism $\pi^{\prime}:X^{\prime}\to Y^{\prime}=X^{\prime}/G$
exists and lies in a commutative square

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\pi^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{Y,}$

where $i$ denotes the inclusion, and $j$ is finite and purely inseparable. If
$G$ is linearly reductive, then $j$ is a closed immersion.

  2. (ii)

For any open subset $U\subset X$ such that $U\cap X^{\prime}$ is dense in
$X^{\prime}$, there exists an open $G$-stable subset $V\subset U$ such that
$V\cap X^{\prime}$ is dense in $X^{\prime}$.

###### Proof.

(i) Since $X^{\prime}\subset X$ is covered by $G$-stable open affine subsets,
it admits a categorical quotient $\pi^{\prime}:X^{\prime}\to Y^{\prime}$.
Moreover, the universal property of $\pi$ yields a unique morphism
$j:Y^{\prime}\to Y$ such that $j\circ\pi^{\prime}=i\circ\pi$; as a
consequence, $j\circ\pi^{\prime}$ is finite. Since $\pi^{\prime}$ is dominant,
it follows that $j$ is finite. Also, for any algebraically closed field
extension $K/k$, the map $j(K):Y^{\prime}(K)\to Y(K)$ is injective, since
$Y^{\prime}(K)$ is the orbit space $X^{\prime}(K)/G(K)$ and likewise
$Y(K)=X(K)/G(K)$. So $j$ is purely inseparable.

To show the final assertion, we may assume $X$ affine. Then the restriction
map ${\mathcal{O}}(X)\to{\mathcal{O}}(X^{\prime})$ is surjective, and hence so
is ${\mathcal{O}}(X)^{G}\to{\mathcal{O}}(X^{\prime})^{G}$ by linear
reductivity. Equivalently,
$j^{\\#}:{\mathcal{O}}(Y)\to{\mathcal{O}}(Y^{\prime})$ is surjective as
desired.

(ii) If $G$ is constant, then it permutes the generic points
$x_{1},\ldots,x_{n}$ of $X^{\prime}$. By assumption, $U\cap
X^{\prime}\supset\\{x_{1},\ldots,x_{n}\\}$ and hence $V=\bigcap_{g\in G}g\cdot
U$ is an open $G$-stable subset of $X$ such that $V\cap
X^{\prime}\supset\\{x_{1},\ldots,x_{n}\\}$. This yields our assertion in this
case, and hence in the case where $G$ is étale by Galois descent. The general
case follows from this by using the factorization (2.3) of $\pi$. ∎

With the assumption of Lemma 2.2, we denote by $x_{1},\ldots,x_{n}$ the
generic points of $X^{\prime}$ and let

${\mathcal{O}}_{X,X^{\prime}}={\mathcal{O}}_{X,x_{1}}\cap\cdots\cap{\mathcal{O}}_{X,x_{n}}.$
(2.4)

This is a semi-local ring, and its maximal ideals correspond bijectively to
$x_{1},\ldots,x_{n}$.

######  Lemma 2.3.

With the above notation, the ring ${\mathcal{O}}_{X,X^{\prime}}$ is equipped
with a (functorial) $G$-action, and is a union of finite-dimensional
$G$-submodules.

###### Proof.

We have ${\mathcal{O}}_{X,X^{\prime}}=\varinjlim{\mathcal{O}}(U)$, where $U$
runs over the open subsets of $X$ containing $x_{1},\ldots,x_{n}$. By Lemma
2.2, it follows that
${\mathcal{O}}_{X,X^{\prime}}=\varinjlim{\mathcal{O}}(V)$, where $V$ runs over
the above open subsets which are $G$-stable. This readily yields the
assertions. ∎

In what follows, we will consider $G$-varieties with a prescribed quotient
$Y$, that is, pairs $(X,\pi)$, where $X$ is a $G$-variety and $\pi:X\to Y$ is
the quotient morphism. Assuming in addition that the $G$-action is faithful,
we say that $X$ is a _$G$ -variety over $Y$_. Morphisms of such pairs are
$G$-equivariant morphisms of varieties over $Y$; in particular, the
automorphism group of $(X,\pi)$ is the group ${\rm Aut}^{G}_{Y}(X)$ of
equivariant relative automorphisms. We now obtain a description of this group,
under assumptions that will be fulfilled by diagonalizable groups.

######  Lemma 2.4.

Let $Y$ be a variety, and $X$ a $G$-variety over $Y$.

  1. (i)

The quotient morphism $\varphi:X\to X/G^{0}=Z$ induces an injective
homomorphism $\varphi_{*}:{\rm Aut}_{Y}^{G^{0}}(X)\to{\rm Aut}_{Y}(Z)$.

  2. (ii)

The natural homomorphism $\pi_{0}(G)(k)\to{\rm Aut}_{Y}(Z)$ is bijective.

  3. (iii)

If $G$ is commutative and the exact sequence (2.1) splits, then

$G(k)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Aut}_{Y}^{G^{0}}(X)={\rm Aut}_{Y}^{G}(X).$

###### Proof.

The existence of the homomorphism $\varphi_{*}$ follows from the universal
property of the categorical quotient. Clearly, the kernel of $\varphi_{*}$ is
isomorphic to ${\rm Aut}_{Z}(X)$, and hence is a subgroup of ${\rm
Aut}_{k(Z)}(k(X))$. Since the extension of function fields $k(X)/k(Z)$ is
purely inseparable, it follows that $\varphi_{*}$ is injective, proving (i).

By Lemma 2.1, the action of $\pi_{0}(G)$ on $Z$ is generically free.
Therefore, we may identify ${\rm Aut}_{Y}(Z)$ with a subgroup of ${\rm
Aut}_{Y_{{\rm fr}}}(Z_{{\rm fr}})$, where $Z_{{\rm fr}}\to Y_{{\rm fr}}$ is a
$\pi_{0}(G)$-torsor. Moreover, we have natural isomorphisms

${\rm Aut}_{Y_{{\rm fr}}}(Z_{{\rm fr}})\simeq\operatorname{Hom}(Z_{{\rm
fr}},\pi_{0}(G))\simeq\operatorname{Hom}(\pi_{0}(Z_{{\rm
fr}}),\pi_{0}(G))\simeq\pi_{0}(G)(k),$

where the second one follows from [DG70, I.4.6.5], and the third one holds
since $Z_{{\rm fr}}$ is a variety and hence $\pi_{0}(Z_{{\rm fr}})={\rm
Spec}(k)$. This implies (ii).

Under the assumptions of (iii), the group $G(k)$ acts on $X$ by
$G$-equivariant automorphisms over $Y$, and
$G(k)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\pi_{0}(G)(k)$. This
yields the desired assertion by combining (i) and (ii). ∎

##  3 Diagonalizable group schemes

We assume that $G$ is diagonalizable, and denote by
$\Lambda=\Lambda(G)=\operatorname{Hom}_{{\rm gp}}(G,{\mathbb{G}}_{m})$ its
character group. Recall that $\Lambda$ is a finite abelian group, and we have
a $\Lambda$-grading

${\mathcal{O}}(G)=\bigoplus_{\lambda\in\Lambda}k\lambda.$ (3.1)

We say that $G$ is _cyclic_ if $\Lambda={\mathbb{Z}}/n{\mathbb{Z}}$ for some
positive integer $n$; equivalently, $G$ is the group scheme
$\boldsymbol{\mu}_{n}$ of $n$th roots of unity.

For an arbitrary diagonalizable group $G$, the exact sequence (2.1) has a
unique splitting: $G=G^{0}\times\pi_{0}(G)$, where $G^{0}$ (resp.
$\pi_{0}(G)$) is a product of finitely many cyclic groups of order a power of
$p$ (resp. prime to $p$). Moreover, the subgroup schemes $H$ of $G$ correspond
bijectively to the subgroups of $\Lambda$ via
$H\mapsto\Lambda(G/H)=\Lambda^{H}$. As a consequence, for any field extension
$K/k$, the map $H\mapsto H_{K}$ yields a bijection between the subgroup
schemes of $G$ and those of $G_{K}$.

Given a $G$-scheme $X$ with quotient $\pi:X\to Y$, the $G$-linearized sheaf
${\mathcal{A}}$ decomposes into eigenspaces:

${\mathcal{A}}=\bigoplus_{\lambda\in\Lambda}{\mathcal{A}}_{\lambda},$ (3.2)

where ${\mathcal{A}}_{0}={\mathcal{B}}$ and each ${\mathcal{A}}_{\lambda}$ is
a coherent ${\mathcal{B}}$-module. This yields again a $\Lambda$-grading,
i.e., the multiplication of ${\mathcal{A}}$ induces morphisms of
${\mathcal{B}}$-modules

${\rm
mult}_{\lambda,\mu}:{\mathcal{A}}_{\lambda}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{\mu}\longrightarrow{\mathcal{A}}_{\lambda+\mu}$
(3.3)

for all $\lambda,\mu\in\Lambda$, and

$\sigma_{\lambda,n}:{\mathcal{A}}_{\lambda}^{\otimes
n}\longrightarrow{\mathcal{A}}_{n\lambda}$ (3.4)

for all such $\lambda$ and all integers $n\geq 1$. The formation of
${\mathcal{A}}_{\lambda}$, ${\rm mult}_{\lambda,\mu}$ and $\sigma_{\lambda,n}$
commutes with arbitrary base change $Y^{\prime}\to Y$.

######  Lemma 3.1.

With the above notation, $\pi$ is a $G$-torsor if and only if ${\rm
mult}_{\lambda,\mu}$ is an isomorphism for all $\lambda,\mu\in\Lambda$. Under
these assumptions, each $\sigma_{\lambda,n}$ is an isomorphism as well.

###### Proof.

The first assertion follows from [SGA3, Exp. VIII, Prop. 4.1]; we present a
proof for completeness. If $\pi$ is the trivial torsor ${\rm pr}_{Y}:G\times
Y\to Y$, then ${\mathcal{A}}$ is identified with
${\mathcal{B}}\otimes_{k}{\mathcal{O}}(G)$ as a ${\mathcal{B}}$-$G$-algebra,
where $G$ acts on ${\mathcal{O}}(G)$ via its action on itself by
multiplication. Using the decomposition (3.1), this identifies each
multiplication map ${\rm mult}_{\lambda,\mu}$ with the map
${\mathcal{B}}\otimes_{k}k\lambda\otimes_{k}k\mu\to{\mathcal{B}}\otimes_{k}k(\lambda+\mu)$
induced by the multiplication in ${\mathcal{O}}(G)$. As a consequence, ${\rm
mult}_{\lambda,\mu}$ is an isomorphism. By fpqc descent, this still holds if
$\pi$ is an arbitrary $G$-torsor.

Conversely, assume that each ${\rm mult}_{\lambda,\mu}$ is an isomorphism.
Then the multiplication map induces an isomorphism
${\mathcal{A}}_{\lambda}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{-\lambda}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{B}}$
for any $\lambda\in\Lambda$. In particular, every ${\mathcal{A}}_{\lambda}$ is
invertible, and hence $\pi$ is finite and flat. To show that it is a
$G$-torsor, it suffices to check that the morphism (2.2) is an isomorphism. As
in the proof of Lemma 2.4, we may assume that $X={\rm Spec}(A)$ and $Y={\rm
Spec}(B)$. Then $\gamma$ corresponds to the algebra homomorphism

$\gamma^{\\#}:A\otimes_{B}A\longrightarrow{\mathcal{O}}(G)\otimes_{k}A,\quad
a_{1}\otimes a_{2}\longmapsto(g\mapsto(g\cdot a_{1})a_{2}),$

where ${\mathcal{O}}(G)\otimes_{k}A$ is identified with the space of morphisms
$G\to A$. Therefore,
$\gamma^{\\#}=\bigoplus_{\lambda,\mu\in\Lambda}\gamma^{\\#}_{\lambda,\mu}$,
where $\gamma^{\\#}_{\lambda,\mu}:A_{\lambda}\otimes_{B}A_{\mu}\to
k\lambda\otimes_{k}A_{\lambda+\mu}$ satisfies
$\gamma^{\\#}_{\lambda,\mu}=\lambda\otimes{\rm mult}_{\lambda,\mu}$. Thus,
each $\gamma^{\\#}_{\lambda,\mu}$ is an isomorphism, and hence so is $\gamma$.

This completes the proof of the first assertion ; the second assertion is
obtained by a similar descent argument. ∎

######  Remark 3.2.

The above lemma takes a much more concrete form when $G=\boldsymbol{\mu}_{n}$.
Then for any $G$-torsor $\pi:X\to Y$, we have

${\mathcal{A}}=\bigoplus_{m=0}^{n-1}{\mathcal{A}}_{m},$

where each ${\mathcal{A}}_{m}$ is an invertible ${\mathcal{B}}$-module. The
multiplication map yields isomorphisms
$\sigma_{1,m}:{\mathcal{A}}_{1}^{\otimes
m}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}_{m}$ for
$m=1,\ldots,n-1$, together with an isomorphism
$\sigma=\sigma_{1,n}:{\mathcal{A}}_{1}^{\otimes
n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{B}}$. Moreover,
one may easily check that the resulting homomorphism of
${\mathcal{B}}$-algebras ${\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{1})\to{\mathcal{A}}$ factors through an
isomorphism

${\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{1})/{\mathcal{I}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}},$

where ${\mathcal{I}}$ denotes the ideal of ${\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{1})$ generated by the
$a^{n}-\sigma(a^{\otimes n})$ for $a\in{\mathcal{A}}_{1}$.

Conversely, given an invertible sheaf ${\mathcal{L}}$ on $Y$ together with a
trivializing section $\sigma\in\Gamma(Y,{\mathcal{L}}^{\otimes
n})=\operatorname{Hom}_{{\mathcal{O}}_{Y}}({\mathcal{L}}^{\otimes-n},{\mathcal{O}}_{Y})$,
the ${\mathcal{B}}$-algebra ${\mathcal{A}}={\rm
Sym}_{{\mathcal{B}}}({\mathcal{L}}^{\otimes-1})/{\mathcal{I}}$ (where
${\mathcal{I}}$ is defined as above) corresponds to a
$\boldsymbol{\mu}_{n}$-torsor satisfying
${\mathcal{A}}_{1}={\mathcal{L}}^{\otimes-1}$. This gives back a well-known
classification of $\boldsymbol{\mu}_{n}$-torsors, see e.g. [DG70, III.4.5.6].
It may be extended to torsors under diagonalizable group schemes with only
notational difficulties.

######  Remark 3.3.

More generally, consider a scheme $Y$ equipped with an invertible sheaf
${\mathcal{L}}$ and a section $\sigma\in\Gamma(Y,{\mathcal{L}}^{\otimes n})$,
where $n$ is a positive integer (we no longer assume that $\sigma$ trivializes
${\mathcal{L}}$). Denote again by ${\mathcal{I}}$ the ideal of ${\rm
Sym}_{{\mathcal{B}}}({\mathcal{L}}^{\otimes-1})$ generated by the
$a^{n}-\sigma(a^{\otimes n})$, where $a\in{\mathcal{L}}^{\otimes-1}$, and let
${\mathcal{A}}={\rm
Sym}_{{\mathcal{B}}}({\mathcal{L}}^{\otimes-1})/{\mathcal{I}}$. Then
${\mathcal{A}}$ is ${\mathbb{Z}}/n{\mathbb{Z}}$-graded and
${\mathcal{A}}_{m}={\mathcal{L}}^{\otimes-m}$ for $m=0,\ldots,n-1$. So $X={\rm
Spec}_{{\mathcal{B}}}({\mathcal{A}})$ is a $\boldsymbol{\mu}_{n}$-scheme with
quotient $Y$, and $\sigma=\sigma_{1,n}$; moreover, the quotient morphism
$\pi:X\to Y$ is flat.

Denote by $E$ the zero scheme of $\sigma$; this is an effective Cartier
divisor on $Y$. Moreover, the ideal sheaf ${\mathcal{J}}\subset{\mathcal{A}}$
generated by ${\mathcal{L}}^{\otimes-1}$ is homogeneous and satisfies
${\mathcal{A}}/{\mathcal{J}}\simeq{\mathcal{O}}_{Y}/\sigma{\mathcal{O}}_{Y}={\mathcal{O}}_{Y}/{\mathcal{O}}_{Y}(-E)$.
Thus, ${\mathcal{J}}$ corresponds to an effective
$\boldsymbol{\mu}_{n}$-stable Cartier divisor $D$ on $X$, sent isomorphically
to $E$ by $\pi$. Moreover, $D$ is the fixed point subscheme
$X^{\boldsymbol{\mu}_{n}}$, and $\pi$ restricts to a
$\boldsymbol{\mu}_{n}$-torsor $X\setminus\operatorname{Supp}(D)\to
Y\setminus\operatorname{Supp}(E)$. We have
$\pi^{*}{\mathcal{O}}_{Y}(-E)={\mathcal{O}}_{X}(-nD)$ in view of the
definition of ${\mathcal{A}}$; equivalently, $\pi^{*}(E)=nD$.

The above construction is classical if $n$ is prime to $p$; then $\pi:X\to Y$
is étale outside of $D$, and totally ramified along $D$. For an arbitrary $n$,
this yields exactly the _uniform cyclic covers_ considered in [AV04].

######  Lemma 3.4.

Let $X$ be a $G$-variety.

  1. (i)

Each ${\mathcal{A}}_{\lambda}$ is a torsion-free ${\mathcal{B}}$-module of
rank $1$, and
$\Lambda(X)=\\{\lambda\in\Lambda~{}|~{}{\mathcal{A}}_{\lambda}\neq 0\\}$ is a
subgroup of $\Lambda$.

  2. (ii)

The $G$-action on $X$ is faithful if and only if $\Lambda(X)=\Lambda$; under
these assumptions, the action is generically free.

###### Proof.

(i) We may assume again that $X={\rm Spec}(A)$ and $Y={\rm Spec}(B)$. Then
$A=\bigoplus_{\lambda\in\Lambda}A_{\lambda}$ is a $\Lambda$-graded domain,
hence $B$ is a domain and each $A_{\lambda}$ is a torsion-free $B$-module.
Denoting by $K$ (resp. $L$) the fraction field of $A$ (resp. $B$), we have
$K=A\otimes_{B}L$ (see e.g. [Br24, Lem. 2.3 (ii)]) and hence
$K=\bigoplus_{\lambda\in\Lambda}K_{\lambda}$, where $K_{0}=L$ and
$K_{\lambda}=A_{\lambda}\otimes_{B}L$. Thus, $\lambda\in\Lambda(X)$ if and
only if $K_{\lambda}\neq 0$; then $K_{\lambda}$ is a $1$-dimensional
$L$-vector space. Moreover, if $K_{\lambda}$ and $K_{\mu}$ are nonzero, then
also $K_{\lambda-\mu}$ is nonzero. This implies the assertion.

(ii) Since $X$ is affine, the $G$-action is faithful if and only if so is its
linear representation in $A$; equivalently, the intersection of the kernels of
the characters $\lambda\in\Lambda(X)$ is trivial. In turn, this is equivalent
to the equality $\Lambda(X)=\Lambda$. We may then choose finitely many
homogeneous elements $f_{1},\ldots,f_{n}\in A$ of respective weights
$\lambda_{1},\ldots,\lambda_{n}$ which generate the group $\Lambda$. Then
$U=X_{f_{1}}\cap\cdots\cap X_{f_{n}}$ is a dense open $G$-stable subset of
$X$, equipped with a $G$-equivariant morphism
$(f_{1},\ldots,f_{n}):U\to{\mathbb{G}}_{m}^{n}$ where $G$ acts on
${\mathbb{G}}_{m}^{n}$ via
$g\cdot(t_{1},\dots,t_{n})=(\lambda_{1}(g)t_{1},\ldots,\lambda_{n}(g)t_{n})$.
Since the latter action is free, we see that $G$ acts freely on $U$. ∎

##  4 Equivariantly normal varieties

Let $G$ be a finite group scheme, and $X$ a $G$-variety. Following [Br24, Def.
4.1], we say that $X$ is $G$-_normal_ if every finite birational morphism of
$G$-varieties $X^{\prime}\to X$ is an isomorphism.

By loc. cit., §4, this notion of equivariant normality is related to the usual
notion of normality as follows: we may view $G$ as a closed subgroup scheme of
a smooth connected algebraic group $G^{\\#}$ (for example, we may embed $G$ in
${\rm GL}_{n}$ via the regular representation, where $n=|G|$). Given a
$G$-variety $X$, the product $G^{\\#}\times X$ is a variety equipped with an
action of $G^{\\#}\times G$ via $(a,g)\cdot(b,x)=(abg^{-1},g\cdot x)$. Since
$G^{\\#}$ and $X$ are covered by open affine $G$-stable subsets, the quotient
$X^{\\#}=(G^{\\#}\times X)/G$ exists; this is a variety equipped with an
action of $G^{\\#}$ and a $G^{\\#}$-equivariant morphism $\varphi:X^{\\#}\to
G^{\\#}/G$. The fibre of $\varphi$ at the base point of the homogeneous space
$G^{\\#}/G$ is $G$-equivariantly isomorphic to $X$. We say that $X^{\\#}$ is
the _induced variety_ $G^{\\#}\times^{G}X$.

######  Lemma 4.1.

With the above notation, $X$ is $G$-normal if and only if $X^{\\#}$ is normal.

This is proved by the same argument as [Br24, Lem. 4.7], which deals with the
quotient $(G^{\\#}\times X)/G^{0}$. As an application, we show:

######  Lemma 4.2.

Let $Y$ be a variety, and $u:X\to X^{\prime}$ a morphism of $G$-varieties over
$Y$. If $X^{\prime}$ is $G$-normal and generically free, then $u$ is an open
immersion.

###### Proof.

Denote by $\eta$ the generic point of $Y$; then $u$ induces a
$G_{\kappa(y)}$-equivariant morphism $u_{\eta}:X_{\eta}\to X^{\prime}_{\eta}$
between generic fibres. Moreover, $X^{\prime}_{\eta}$ is a
$G_{\kappa(y)}$-torsor by assumption, and $X_{\eta}$ is
$G_{\kappa(y)}$-homogeneous in view of [Br24, Cor. 2.5]. Thus, $u_{\eta}$ is
an isomorphism, and hence $u$ is birational. Also, $u$ is quasi-finite, since
$\pi=\pi^{\prime}\circ u$ is finite. We now consider the induced varieties
$X^{\\#}$, $X^{\prime\\#}$ and the induced morphism $u^{\\#}:X^{\\#}\to
X^{\prime\\#}$, which is still birational and quasi-finite. Since
$X^{\prime\\#}$ is normal (Lemma 4.1), $u^{\\#}$ is an open immersion by
Zariski’s Main Theorem. Thus, $u$ is an open immersion. ∎

######  Remark 4.3.

A finite group scheme $G$ is diagonalizable if and only if it is isomorphic to
a subgroup scheme of a torus $T\simeq{\mathbb{G}}_{m}^{n}$. Then the
assignement $X\mapsto X^{\\#}=T\times^{G}X$ defines a bijection between
$G$-normal varieties and normal $T$-varieties equipped with an equivariant
morphism $\varphi$ to the quotient torus $T/G$, whose fibre at the base point
is geometrically integral. The dimension of $X$ is the _complexity_ of the
$T$-variety $X^{\\#}$ (the transcendence degree of the field of $T$-invariant
rational functions).

Given a torus $T$, there is a classification of normal $T$-varieties in terms
of objects mixing convex and algebraic geometry, called proper polyhedral
divisors and divisorial fans, which generalize the cones and fans associated
with toric varieties. Over an algebraically closed field $k$ of characteristic
$0$, this classification is obtained in [AH06] for affine varieties; the
general case is treated in [AHS08]. Over an arbitrary field, normal affine
$T$-varieties of complexity $1$ are also classified by proper polyhedral
divisors, as shown in [Lan15]. Yet we do not know how to deduce from this a
full classification of $G$-normal varieties, which incorporates the additional
datum of the morphism $\varphi$.

######  Remark 4.4.

Returning to an arbitrary finite group scheme $G$, note that a $G$-variety $X$
is $G$-normal if and only if $X_{K}$ is $G^{0}_{K}$-normal for some separable
field extension $K/k$ (see [Br24, Cor. 4.8, Prop. 4.9]). If $G$ is linearly
reductive, we may take for $K/k$ a finite Galois extension such that
$G^{0}_{K}$ is diagonalizable and $\pi_{0}(G)_{K}$ is constant. Then the
quotient morphism $X_{K}\to Y_{K}$ factors uniquely as $X_{K}\to Z_{K}\to
Y_{K}$, where $Z_{K}=X_{K}/G^{0}_{K}$ is a normal variety and $Z_{K}\to Y_{K}$
is a finite Galois cover with group $\pi_{0}(G)(K)$, of order prime to $p$.
This reduces a number of questions on equivariantly normal varieties under
linearly reductive group schemes to the diagonalizable case; see the final
Section 9 for an illustration.

Next, we obtain a criterion for $G$-normality of a $G$-variety $X$ in terms of
the ${\mathcal{B}}$-algebra ${\mathcal{A}}$, with the notation of Section 2.
Clearly, we may assume that $G$ acts faithfully on $X$.

Let $D$ be a prime divisor on $X$. Then $G\cdot D$ is a closed $G$-stable
subscheme of $X$, of pure codimension $1$. We denote by
${\mathcal{O}}_{X,G\cdot D}$ the corresponding semi-local ring; it has Krull
dimension $1$ and is equipped with a (functorial) $G$-action (Lemma 2.3). We
have ${\mathcal{O}}_{X,G\cdot D}^{G}={\mathcal{O}}_{Y,E}$, where $E=\pi(D)$ is
a prime divisor on $Y$. Moreover, ${\mathcal{O}}_{X,G\cdot D}$ has a largest
$G$-stable ideal $I=I_{G\cdot D}$, since $G\cdot D$ is the smallest $G$-stable
subscheme of $X$ containing $D$. (We may think of ${\mathcal{O}}_{X,G\cdot D}$
as a “$G$-local” ring of dimension $1$).

We say that $D$ is _free_ if it meets the free locus $X_{{\rm fr}}$, and
_nonfree_ otherwise. If $G$ acts generically freely on $X$, then there are
only finitely many nonfree divisors: the irreducible components of codimension
$1$ of the nonfree locus $X\setminus X_{{\rm fr}}$.

If $G$ is diagonalizable with character group $\Lambda$, then the algebra
${\mathcal{O}}_{X,G\cdot D}$ is $\Lambda$-graded by Lemma 2.3 again. Thus,
$I_{G\cdot D}$ is the largest $\Lambda$-graded ideal of
${\mathcal{O}}_{X,G\cdot D}$. Moreover, the $G$-action on $X$ is generically
free (Lemma 3.4).

######  Proposition 4.5.

Let $G$ be a finite group scheme, $Y$ a variety, and $X$ a $G$-variety over
$Y$. Then $X$ is $G$-normal if and only if it satisfies the following
conditions:

  1. (i)

The variety $Y$ is normal.

  2. (ii)

The coherent sheaf of ${\mathcal{B}}$-algebras ${\mathcal{A}}$ satisfies
$(S_{2})$.

  3. (iii)

For any nonfree divisor $D$, the ideal $I_{G\cdot D}$ is principal.

If $G$ is diagonalizable, then (ii) is equivalent to

  1. (ii)’

The sheaf ${\mathcal{A}}_{\lambda}$ is divisorial for any $\lambda\in\Lambda$.

Recall that a coherent sheaf ${\mathcal{F}}$ satisfies Serre’s condition
($S_{2}$) if it has depth at least $2$ at every point of codimension at least
$2$ of its support. Before proving Proposition 4.5, we record two preliminary
results which can be extracted from [EGAIV2].

######  Lemma 4.6.

Let ${\mathcal{F}}$ be a coherent torsion-free sheaf on a variety $Z$.

  1. (i)

${\mathcal{F}}$ satisfies ($S_{2}$) if and only if the restriction map
${\mathcal{F}}(V)\to{\mathcal{F}}(U)$ is an isomorphism for any open subsets
$U\subset V\subset Z$ such that $V$ is big in $U$.

  2. (ii)

Given a finite surjective morphism of varieties $\pi:Z\to W,$ the sheaf
${\mathcal{F}}$ satisfies ($S_{2}$) if and only if so does
$\varphi_{*}({\mathcal{F}})$.

###### Proof.

(i) This is a consequence of [EGAIV2, Prop. 5.9.8 and Prop. 5.10.14] (see also
[Kol22, §1]).

(ii) This follows from [EGAIV2, Cor. 5.7.11] (see also [Kol22, Lem. 18]). ∎

_Proof of Proposition 4.5_. By [Br24, Thm. 4.12], $X$ is $G$-normal if and
only if ${\mathcal{O}}_{X}$ satisfies ($S_{2}$) and every $G$-stable ideal is
invertible in codimension $1$. Together with the above lemma, it follows that
(ii) holds for every $G$-normal variety $X$. Moreover, (i) holds by loc. cit.,
Lem. 4.5, and (iii) follows from the fact that every invertible ideal of a
semi-local ring is principal.

Conversely, assume (i), (ii) and (iii). Then ${\mathcal{O}}_{X}$ satisfies
($S_{2}$) by Lemma 4.6 again. So it suffices to check that every $G$-stable
ideal is invertible in codimension $1$. In other words, for any prime divisor
$D$ on $X$, every $G$-stable ideal $I$ of ${\mathcal{O}}_{X,G\cdot D}$ is
principal.

If $D$ is free, we may replace $X$ with its dense open $G$-stable subset
$X_{{\rm fr}}$, and hence assume that $G$ acts freely on $X$. Then by fpqc
descent, the assignement $Z\mapsto Z\times_{Y}X$ yields a bijection between
closed subschemes of $Y$ and closed $G$-stable subschemes of $X$. As a
consequence, we have $I=I^{G}{\mathcal{O}}_{X}$. Moreover, $I^{G}$ is
identified with an ideal of the discrete valuation ring ${\mathcal{O}}_{Y,E}$
(where $E=\pi(D)$), and hence is invertible. This yields our assertion in this
case.

If $D$ is nonfree, then we may choose a generator $f$ of the ideal $I_{G\cdot
D}$. Then $I\subset f{\mathcal{O}}_{X,G\cdot D}$, and hence $I=fJ$ where $J$
is a $G$-stable ideal of ${\mathcal{O}}_{X,G\cdot D}$; in particular,
$I\subsetneq J$. So we have an exact sequence $0\to
J/I\to{\mathcal{O}}_{X,G\cdot D}/I\to{\mathcal{O}}_{X,G\cdot D}/J\to 0$ of
${\mathcal{O}}_{X,G\cdot D}$-modules of finite length. This implies our
assertion by induction on the length of ${\mathcal{O}}_{X,G\cdot D}/I$ (this
length is minimal if and only if $I=I_{G\cdot D}$).

It remains to prove the equivalence (ii)$\Leftrightarrow$(ii)’ when $G$ is
diagonalizable. In view of the decomposition (3.2), ${\mathcal{A}}$ satisfies
$(S_{2})$ if and only if each ${\mathcal{A}}_{\lambda}$ satisfies $(S_{2})$.
As $Y$ is normal and ${\mathcal{A}}_{\lambda}$ has rank $1$, this amounts to
${\mathcal{A}}_{\lambda}$ being divisorial (see e.g. [Har80, Cor. 1.4 and
Prop. 1.6]). ∎

Combining Lemma 3.4 and Proposition 4.5, we readily obtain:

######  Corollary 4.7.

Let $G$ be a finite diagonalizable group scheme acting faithfully on a
$G$-normal variety with quotient morphism $\pi:X\to Y$. Then $\pi$ is flat if
and only if ${\mathcal{A}}_{\lambda}$ is invertible for any
$\lambda\in\Lambda$.

In particular, $\pi$ is flat if $Y$ is regular.

##  5 Local structure in codimension 1

Throughout this section, we consider a finite diagonalizable group $G$ and a
$G$-variety $X$ over a normal variety $Y$. Let again $D$ be a prime divisor on
$X$, with image $E$ in $Y$. Then $E$ is a prime divisor on the normal variety
$Y$, and hence $B={\mathcal{O}}_{Y,E}$ is a discrete valuation ring. Moreover,
the semi-local ring $A={\mathcal{O}}_{X,G\cdot D}$ is a finitely generated,
torsion-free $B$-module, and hence is free of finite rank. Also, $G$ acts on
$A$, and $B=A^{G}$.

Denote by $\mathfrak{m}$ the maximal ideal of $A$ corresponding to $D$; then
$\mathfrak{n}=\mathfrak{m}\cap B$ is the maximal ideal of $B$. The residue
field $A/\mathfrak{m}$ is the function field $k(D)=\kappa(x)$, where $x$
denotes the generic point of $D$. Likewise, $B/\mathfrak{n}=k(E)=\kappa(y)$
for the generic point $y=\pi(x)$ of $D$; the extension $\kappa(x)/\kappa(y)$
is finite. Let $\bar{A}=A/\mathfrak{n}A$; this is an artinian
$\kappa(y)$-algebra, equipped with a action of $G_{\kappa(y)}$. We have
$\bar{A}={\mathcal{O}}(X_{y})$, where $X_{y}$ denotes the fibre of $\pi$ at
$y$. Moreover, the orbit $G_{\kappa(y)}\cdot x$ is a subscheme of the finite
$\kappa(y)$-scheme $X_{y}$ and both have the same underlying topological
space. In particular, $G_{\kappa(y)}\cdot x$ only depends on the point $y\in
Y$. Also, this orbit may be identified with $G\cdot x$, as
${\mathcal{O}}(G\cdot x)$ is the image of the natural homomorphism
${\mathcal{O}}(X_{y})\to{\mathcal{O}}(G)\otimes_{k}\kappa(x)$, and the right-
hand side is identified with
${\mathcal{O}}(G_{\kappa(y)})\otimes_{\kappa(y)}\kappa(x)$.

We will obtain a description of the $G$-algebra $\bar{A}$, or equivalently of
the $G$-scheme $X_{y}$, under the assumption that $G$ is diagonalizable. Then
$A$ is a $\Lambda$-graded $B$-module:

$A=\bigoplus_{\lambda\in\Lambda}A_{\lambda},$ (5.1)

where $A_{0}=B$ and each $A_{\lambda}$ is a free $B$-module of rank $1$ (Lemma
3.4). This yields a $\Lambda$-grading

$\bar{A}=\bigoplus_{\lambda\in\Lambda}\bar{A}_{\lambda}$ (5.2)

such that each $\bar{A}_{\lambda}$ is a one-dimensional $\kappa(y)$-vector
space. Let $I=I_{G\cdot D}$ be the largest $\Lambda$-graded ideal of $A$.
Clearly, we have $\mathfrak{n}A\subset I$, and hence $I$ is the preimage of
the largest $\Lambda$-graded ideal $\bar{I}$ of $\bar{A}$ under the quotient
map $A\to\bar{A}$. Moreover, there is an exact sequence of
$G_{\kappa(y)}$-modules

$0\longrightarrow\bar{I}\longrightarrow\bar{A}\longrightarrow{\mathcal{O}}(G_{\kappa(y)}\cdot
x)\longrightarrow 0,$ (5.3)

and $\bar{I}$ is nilpotent.

######  Proposition 5.1.

  1. (i)

There exists a unique subgroup scheme $H\subset G$ such that
$\bar{A}=\bar{A}^{H_{\kappa(y)}}\oplus\bar{I}$.

  2. (ii)

The morphism $X_{y}/H_{\kappa(y)}={\rm Spec}(\bar{A}^{H_{\kappa(y)}})\to{\rm
Spec}(\kappa(y))$ is a torsor under $(G/H)_{\kappa(y)}$.

  3. (iii)

We have $H_{\kappa(y)}=C_{G_{\kappa(y)}}(x)$ and ${\rm
Spec}(\bar{A}^{H_{\kappa(y)}})\simeq G_{\kappa(y)}\cdot x$ as
$(G/H)_{\kappa(y)}$-schemes.

######  Remark 5.2.

The subgroup scheme $H$ only depends on $y\in Y$, or equivalently on the prime
divisor $E$; we will denote it by $H(E)$ or by $H(D)$ according to the setup.

If the restriction $\pi|_{D}:D\to E$ is birational, then $\kappa(y)=\kappa(x)$
and hence $H_{\kappa(x)}$ is just the stabilizer $\operatorname{Stab}_{G}(x)$.
In the general case, as the above proposition is rather technical, we
illustrate it with two examples before giving the proof.

Let $G=\boldsymbol{\mu}_{p^{2}}$ so that
$\Lambda={\mathbb{Z}}/p^{2}{\mathbb{Z}}$. Consider the polynomial ring
$k[T_{0},T_{1},T_{2}]$ equipped with the $\Lambda$-grading such that
$T_{0},T_{1},T_{2}$ are homogeneous of respective weights $0,1,p$. Let

$X={\rm Spec}\big{(}k[T_{0},T_{1},T_{2}]/(T_{1}^{p}-T_{0}T_{2})\big{)}.$

This is a normal affine surface containing the origin as its unique nonsmooth
point. Moreover, $G$ acts on $X$ (since $T_{1}^{p}-T_{0}T_{2}$ is homogeneous
of weight $p$) with quotient

$Y={\rm Spec}(k[T_{0},T_{2}^{p}])\simeq{\mathbb{A}}^{2}.$

The zero subscheme of $(T_{0},T_{1})$ in $X$ is a $G$-stable nonfree divisor
$D$, and its image $E$ in $Y$ is the zero subscheme of $T_{0}$. With the above
notation, we obtain

$\kappa(x)=k(T_{2})\supset
k(T_{2}^{p})=\kappa(y),\quad\bar{A}=k(T_{2})[T_{1}]/(T_{1}^{p}),$

so that $H=\boldsymbol{\mu}_{p}$. Note that $G_{\kappa(y)}\cdot x=x$ but $x$
is not fixed by $G_{\kappa(y)}$ (or equivalently, by $G$).

Next, let $X^{\prime}\subset X$ be the zero subscheme of
$T_{2}^{p}-T_{0}-1\in{\mathcal{O}}(X)^{G}={\mathcal{O}}(Y)$. Then $X^{\prime}$
is a $G$-stable curve containing $x^{\prime}=(0,0,1)$ as its unique nonsmooth
point. Moreover, $X^{\prime}$ has quotient $Y^{\prime}={\rm
Spec}(k[T_{0}])\simeq{\mathbb{A}}^{1}$, and $x^{\prime}$ has image
$y^{\prime}=0$. We obtain with an obvious notation

$\kappa(x^{\prime})=\kappa(y^{\prime}),\quad\bar{A}^{\prime}=k[T_{1},T_{2}]/(T_{1}^{p},T_{2}^{p}-1),$

so that $H^{\prime}=\boldsymbol{\mu}_{p}$ again; also, $x^{\prime}$ is neither
free nor $G$-stable. One may check that $X^{\prime}$ is $G$-normal.

The latter example will be generalized in Section 8 to all $G$-normal curves,
where $G$ is diagonalizable and infinitesimal.

_Proof of Proposition 5.1_. (i) For any $\lambda\in\Lambda$, we have either
$\bar{A}_{\lambda}\subset\bar{I}$, or every nonzero element of
$\bar{A}_{\lambda}$ is invertible. (Indeed, the ideal
$\bar{A}_{\lambda}\bar{A}$ is $\Lambda$-graded, and hence is either contained
in $\bar{I}$ or equal to $\bar{A}$). Also, note that every nonzero element of
$\bar{A}_{\lambda}$ is invertible if and only if $\bar{A}_{\lambda}$ meets
$\bar{A}^{\times}$. As a consequence, the characters satisfying this condition
form a subgroup of $\Lambda$. Thus, there exists a unique subgroup scheme $H$
of $G$ such that $\bar{A}_{\lambda}$ meets $\bar{A}^{\times}$ if and only if
$\bar{A}_{\lambda}\subset\bar{A}^{H_{\kappa(y)}}$. In view of (5.2), this
yields $\bar{A}=\bar{A}^{H_{\kappa(y)}}\oplus\bar{I}$.

(ii) This follows from the definition of $H$ in view of Lemma 3.1.

(iii) By (i) together with the exact sequence (5.3), we obtain a
$G_{\kappa(y)}$-equivariant isomorphism
$\bar{A}^{H_{\kappa(y)}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}(G_{\kappa(y)}\cdot
x)$. As a consequence, $H_{\kappa(y)}$ acts trivially on $G_{\kappa(y)}\cdot
x$. In particular, $H_{\kappa(y)}\subset C_{G_{\kappa(y)}}(x)$. On the other
hand, $C_{G_{\kappa(y)}}(x)$ fixes the image of $x$ in ${\rm
Spec}(\bar{A}^{H_{\kappa(y)}})$. In view of (ii), this gives the opposite
inclusion $C_{G_{\kappa(y)}}(x)\subset H_{\kappa(y)}$. ∎

We may reformulate the above proposition in geometric terms by considering the
factorization of $\pi:X\to Y$ as

$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/H\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}X/G=Y,$

where $\varphi$ (resp. $\psi$) is the quotient by $H$ (resp. $G/H$). Then
$\varphi$ yields a morphism between fibres $\varphi_{y}:X_{y}\to Z_{y}$.
Moreover, $Z_{y}=X_{y}/H_{\kappa(y)}$ by Lemma 2.2 together with the linear
reductivity of $H$.

######  Corollary 5.3.

  1. (i)

The restriction of $\varphi$ to $D$ is birational to its image.

  2. (ii)

The morphism $\psi$ is a $G/H$-torsor in a neighborhood of $y$.

  3. (iii)

The fixed point subscheme $X_{y}^{H_{\kappa(y)}}$ yields a section of
$\varphi_{y}$.

  4. (iv)

The divisor $D$ is free if and only if $H$ is trivial.

###### Proof.

(i) The decomposition $\bar{A}=\bar{A}^{H}\oplus\bar{I}$ induces a
decomposition $\bar{\mathfrak{m}}=\bar{\mathfrak{m}}^{H}\oplus\bar{I}$, so
that
$\kappa(x)=A/\mathfrak{m}=\bar{A}/\bar{\mathfrak{m}}=\bar{A}^{H}/\bar{\mathfrak{m}}^{H}=A^{H}/\mathfrak{m}^{H},$
where the latter equality follows from the inclusion
$\mathfrak{n}A^{H}\subset\mathfrak{m}^{H}$. This yields our assertion.

(ii) This follows from Proposition 5.1 together with the openness of the free
locus of the $G/H$-variety $Z$.

(iii) We identify the algebra homomorphism
$\varphi_{y}^{\\#}:{\mathcal{O}}(Z_{y})\to{\mathcal{O}}(X_{y})$ with the
inclusion $\bar{A}^{H_{\kappa(y)}}\subset\bar{A}$. The latter inclusion has a
retraction $\bar{A}\to\bar{A}/\bar{I}=\bar{A}^{H_{\kappa(y)}}$, where
$\bar{I}$ is the largest $H_{\kappa(y)}$-stable ideal of $\bar{A}$ such that
$H_{\kappa(y)}$ acts trivially on $\bar{A}/\bar{I}$, i.e., the ideal of
$X_{y}^{H_{\kappa(y)}}$.

(iv) If $H$ is trivial, then $X_{y}\to{\rm Spec}(\kappa(y))$ is a
$G_{\kappa(y)}$-torsor, and hence $x\in X_{{\rm fr}}$. Conversely, if $H$ is
nontrivial then $D$ is nonfree by (iii). ∎

For a $G$-normal variety $X$, the above description of
$\bar{A}={\mathcal{O}}(X_{y})$ lifts to a description of
$A={\mathcal{O}}_{X,G\cdot x}$:

######  Proposition 5.4.

Assume that $X$ is $G$-normal.

  1. (i)

The largest $\Lambda$-graded ideal $I$ of $A$ is generated by a homogeneous
element $f$ of weight $\nu\in\Lambda$ such that the restriction $\nu|_{H}$
generates $\Lambda(H)$. Moreover, $\nu|_{H}$ is uniquely determined by $I$.

  2. (ii)

We have $H\simeq\boldsymbol{\mu}_{n}$ for some $n\geq 1$, and
$A=\bigoplus_{m=0}^{n-1}A^{H}f^{m}\simeq A^{H}[T]/(T^{n}-g)$, where $g=f^{n}$
satisfies $g\in A^{H}$ and $gA=\mathfrak{n}A$.

  3. (iii)

We have the equality of $H$-fixed subschemes $X_{y}^{H}=G_{\kappa(y)}\cdot x$,
and the inclusion $G\cdot D\subset X^{H}$ with equality in a $G$-stable
neighborhood of $x$.

###### Proof.

We show (i) and (ii) simultaneously. The largest $\Lambda$-graded ideal $I$ is
principal (Proposition 4.5). Let $f\in A$ such that $I=Af$ and write
$f=\sum_{\lambda}f_{\lambda}$, where $f_{\lambda}\in I_{\lambda}$ for all
$\lambda$. We claim that $I=Af_{\nu}$ for some $\nu$.

Indeed, $f\notin\mathfrak{m}I$, where $\mathfrak{m}$ denotes the maximal ideal
of $x$ in $A$. So there exists $\nu$ such that $f_{\nu}\notin\mathfrak{m}I$.
We have $Af_{\nu}\subset I$; if this inclusion is strict, then the annihilator
$J$ of the $\Lambda$-graded $A$-module $I/Af_{\nu}$ is a proper
$\Lambda$-graded ideal, and hence is contained in $I$. But we have the
equality of localizations $I_{\mathfrak{m}}=(Af_{\nu})_{\mathfrak{m}}$ and
hence $J_{\mathfrak{m}}=A_{\mathfrak{m}}$, which contradicts the fact that
$I_{\mathfrak{m}}=\mathfrak{m}_{\mathfrak{m}}\neq A_{\mathfrak{m}}$. This
proves our claim.

Let $f=f_{\nu}$ as in that claim and let $g=f^{n}$, where $n=|H|$. Then $g\in
A^{H}$, since the character group $\Lambda(H)$ is killed by $n$. Thus, the
image $\bar{f}$ of $f$ in $\bar{A}$ satisfies
$\bar{f}^{n}\in\bar{A}^{H_{\kappa(y)}}$. But $\bar{f}\in\bar{I}$ and
$\bar{A}=\bar{A}^{H_{\kappa(y)}}\oplus\bar{I}$, so that $\bar{f}^{n}=0$;
equivalently, $g\in\mathfrak{n}A$. Moreover, Proposition 5.1 yields the
equalities
$\bar{A}=\bar{A}^{H_{\kappa(y)}}\oplus\bar{I}=\bar{A}^{H_{\kappa(y)}}\oplus\bar{f}\bar{A}$
and hence

$\bar{A}=\bar{A}^{H_{\kappa(y)}}+\bar{f}\bar{A}=\bar{A}^{H_{\kappa(y)}}+\bar{f}\bar{A}^{H_{\kappa(y)}}+\bar{f}^{2}\bar{A}=\cdots=\sum_{m=0}^{n-1}\bar{f}^{m}\bar{A}^{H_{\kappa(y)}}.$

Taking weights, we obtain $\Lambda=\Lambda(G/H)+{\mathbb{Z}}\nu$, where
$\Lambda(G/H)=\Lambda^{H}$. Also, recall that
$\Lambda(H)=\Lambda/\Lambda^{H}$; thus, this group is generated by $\nu|_{H}$.
As a consequence, $\nu|_{H}$ has order $n$, and hence
$H\simeq\boldsymbol{\mu}_{n}$; moreover, $\Lambda$ is the disjoint union of
the translates $\Lambda^{H}+m\nu$ where $m=0,\ldots,n-1$. Also, $\bar{f}$ is
uniquely determined up to multiplication by a homogeneous unit of $\bar{A}$,
so that $\nu$ is unique up to translation by $\Lambda^{H}$; equivalently,
$\nu|_{H}$ is unique.

For any $\lambda\in\Lambda^{H}$ and $m=0,\ldots,n-1$, we have
$\bar{A}_{\lambda+m\nu}=\bar{A}_{\lambda}\bar{f}^{m}$, and hence
$A_{\lambda+m\nu}=A_{\lambda}f^{m}$ by Nakayama’s lemma. As a consequence,
$A=\sum_{m=0}^{n-1}A^{H}f^{m}$. Moreover, this sum is direct in view of the
above decomposition of $\Lambda$ into cosets of $\Lambda^{H}$.

It remains to show that $gA=\mathfrak{n}A$, or equivalently,
$gA^{H}=\mathfrak{n}A^{H}$. Clearly, we have $gA^{H}\subset\mathfrak{n}A^{H}$.
Thus, it suffices to check that
$\dim_{\kappa(y)}A^{H}/gA^{H}=\dim_{\kappa(y)}A^{H}/\mathfrak{n}A^{H}$. But we
have isomorphisms of $\kappa(y)$-algebras

$A^{H}/\mathfrak{n}A^{H}\simeq\bar{A}^{H_{\kappa(y)}}\simeq\bar{A}/\bar{I}\simeq
A/I=A/fA\simeq A^{H}/gA^{H},$

where the latter isomorphism follows from the decomposition of $A$ as an
$A^{H}$-module. This yields the desired equality of dimensions.

(iii) Note that $X^{H}$ is $G$-stable, and contains $D$ by Proposition 5.1;
therefore, $G\cdot D\subset X^{H}$. Denote by
${\mathcal{J}}\subset{\mathcal{O}}_{X}$ the ideal sheaf of $X^{H}$, and by
$J\subset A$ the corresponding ideal. Then $f\in J$, as it is an
$H$-eigenvector of nonzero weight. Thus, we have $I(G\cdot D)\subset J$; this
implies our assertions. ∎

######  Remark 5.5.

Given $\lambda\in\Lambda$, consider the multiplication map

$\sigma_{\lambda,|G|}:A_{\lambda}^{\otimes|G|}\longrightarrow
A_{|G|\lambda}=A_{0}=B$

as in (3.4). With the notation of Proposition 5.4, write $\lambda=\mu+m\nu$,
where $\mu\in\Lambda^{H}$ and $0\leq m\leq n-1$. Then the image of
$\sigma_{\lambda,|G|}$ is the ideal of $B$ generated by $f^{|G|m}=g^{[G:H]m}$.

Indeed, we have $A_{\lambda}=A_{\mu}f^{m}$ by the above proposition, and the
multiplication map $\sigma_{\mu,|G|}:A_{\mu}^{\otimes|G|}\to B$ is an
isomorphism in view of Remark 3.2 and Corollary 5.3.

##  6 Local structure in codimension 1 (continued)

We still consider a finite diagonalizable group scheme $G$ and a $G$-normal
variety $X$ over $Y$ (a normal variety). To describe the local structure of
$X$ in codimension $1$, we may freely replace it by any big open subset $U$
which is $G$-stable and satisfies some additional assumptions. Here are those
we will need at this stage.

######  Lemma 6.1.

There exists a largest open subset $U\subset X$ satisfying the following
properties:

  1. (i)

The quotient $U/G$ is regular.

  2. (ii)

The nonfree locus $U\setminus U_{{\rm fr}}$ is a disjoint union of prime
divisors $D$.

  3. (iii)

For any such $D$, the ideal ${\mathcal{I}}_{G\cdot D}$ is invertible.

Moerover, $U$ is $G$-stable and big in $X$.

###### Proof.

These properties are open, and hold in codimension $1$; this readily yields
that $U$ exists and is big in $X$. To complete the proof, it suffices to show
that each of these properties defines a $G$-stable open subset. This clearly
holds for (i) and (ii). For (iii), this follows from fpqc descent as this
property is stable by automorphisms arising from $k$-points of $G$. ∎

We now assume that $X$ satisfies the above properties (i), (ii), (iii); we
then say that $X$ is _strongly $G$-normal_.

To every prime divisor $D$ on $X$, we associate a cyclic subgroup scheme
$H(D)\subset G$ (where $x$ denotes the generic point of $D$) and a weight
$\nu(D)\in\Lambda$ (the weight of a homogeneous generator of the largest
$\Lambda$-graded ideal of ${\mathcal{O}}_{X,G\cdot D}$), as in Proposition
5.4; recall that $\nu(D)|_{H(D)}$ generates $\Lambda(H(D))$.

Given a cyclic subgroup scheme $H$ of $G$ and a weight $\nu\in\Lambda$ such
that $\nu|_{H}$ generates $\Lambda(H)$, let

$U(H,\nu)=X_{{\rm fr}}\cup\bigcup_{D}G\cdot D,$

where the union runs over the nonfree divisors $D$ on $X$ such that $H(D)=H$
and $\nu(D)|_{H(D)}=\nu|_{H}$. Then $U(H,\nu)$ is an open $G$-stable subset of
$X$, and its quotient $V(H,\nu)=U(H,\nu)/G$ is identified with an open subset
of $Y$. Also, note that $\nu|_{H}$ identifies $H$ with the cyclic group scheme
$\boldsymbol{\mu}_{n}$, where $n=n(H,\nu)$; moreover, we have
$\Lambda=\Lambda^{H}+\sum_{m=0}^{n-1}m\nu$ and $n\nu\in\Lambda^{H}$. The
multiplication in ${\mathcal{A}}$ yields a morphism of ${\mathcal{B}}$-modules

$\sigma=\sigma_{\nu,n}:{\mathcal{A}}_{\nu}^{\otimes
n}\longrightarrow{\mathcal{A}}_{n\nu}.$

The restriction $\sigma|_{V(H,\nu)}$ may be viewed as a section of the
invertible sheaf
${\mathcal{A}}_{\nu}^{\otimes-n}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{n\nu}$
on $V(H,\nu)$. We denote its divisor of zeroes by $\Delta(H,\nu)$; this is an
effective divisor on the regular variety $V(H,\nu)$, that we call the _branch
divisor_.

######  Theorem 6.2.

Let $X$ be a strongly $G$-normal variety.

  1. (i)

We have $X=\bigcup_{H,\nu}U(H,\nu)$, and $U(H,\nu)\cap
U(H^{\prime},\nu^{\prime})=X_{{\rm fr}}$ unless $H^{\prime}=H$ and
$\nu^{\prime}|_{H}=\nu|_{H}$; then $U(H^{\prime},\nu^{\prime})=U(H,\nu)$.

  2. (ii)

The quotient $U=U(H,\nu)\to V=V(H,\nu)$ factors as
$U\stackrel{{\scriptstyle\varphi_{U}}}{{\longrightarrow}}U/H\stackrel{{\scriptstyle\psi_{U}}}{{\longrightarrow}}U/G$,
where $\psi_{U}$ is a $G/H$-torsor. Moreover, the natural homomorphism of
${\mathcal{A}}^{H}$-algebras

$\Phi:\big{(}{\mathcal{A}}^{H}\otimes_{{\mathcal{B}}}{\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{\nu})\big{)}/{\mathcal{I}}\longrightarrow{\mathcal{A}}$

restricts to an isomorphism on $V$, where ${\mathcal{I}}$ denotes the ideal of
${\mathcal{A}}^{H}\otimes_{{\mathcal{B}}}{\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{\nu})$ generated by the
$a^{n}-\sigma(a^{\otimes n})$ with $a\in{\mathcal{A}}_{\nu}$.

  3. (iii)

The branch divisor $\Delta=\Delta(H,\nu)$ is reduced and its support is
$V\setminus V_{{\rm fr}}$.

  4. (iv)

The fixed point subscheme $U^{H}$ is an effective Cartier divisor on $U$, and
we have $U^{H}=\bigcup G\cdot D$ (union over the $G$-orbits of nonfree
divisors of $U$). Moreover, $\pi_{U}^{*}(\Delta)=nU^{H}$ as Cartier divisors
on $U$.

###### Proof.

(i) All of this follows readily from the definition of $U(H,\nu)$.

(ii) For any $\lambda,\mu\in\Lambda$, the multiplication map ${\rm
mult}_{\lambda,\mu}$ (3.3) restricts to a nonzero homomorphism of invertible
${\mathcal{B}}_{V}$-modules, and hence yields an isomorphism

$\left({\mathcal{A}}_{\lambda}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{\mu}\right)|_{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}_{\lambda+\mu}|_{V}(-D_{\lambda,\mu}),$

where $D_{\lambda,\mu}$ is an effective divisor on $V$. By Lemma 3.1,
$D_{\lambda,\mu}$ is a sum of nonfree divisors. Combining this lemma with
Proposition 5.4, it follows that $D_{\lambda,\mu}=0$ for all
$\lambda,\mu\in\Lambda^{H}$. This yields the assertion on $\psi_{U}$ by using
again Lemma 3.1.

We now show that $\Phi_{V}$ is an isomorphism. Note that the natural map

$\bigoplus_{m=0}^{n-1}{\mathcal{A}}^{H}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{m\nu}\longrightarrow{\mathcal{A}}^{H}\otimes_{{\mathcal{B}}}{\rm
Sym}_{{\mathcal{B}}}({\mathcal{A}}_{\nu})/{\mathcal{I}}$

is an isomorphism on $V$. So it suffices to show that the multiplication map

${\rm
mult}_{\lambda,m\nu}:{\mathcal{A}}_{\lambda}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{m\nu}\longrightarrow{\mathcal{A}}_{\lambda+m\nu}$

is an isomorphism on $V$ for all $\lambda\in\Lambda^{H}$ and $m=0,\ldots,n-1$.
But this map is an isomorphism at the generic point of every prime divisor $D$
meeting $U$, in view of Proposition 5.4. This yields the desired assertion by
arguing as in the above paragraph.

(iii) This follows from Proposition 5.4, since $\sigma$ is identified with $g$
at the generic point of every $D$ as above.

(iv) The ideal of $U^{H}$ is generated by the homogeneous components of
${\mathcal{A}}^{H}$ of nonzero weight relative to $H$, and hence by
${\mathcal{A}}_{\nu}$ in view of (ii). In particular, this ideal is
invertible; this yields the first assertion.

By Proposition 5.4 again, we have $U^{H}\subset\bigcup\,G\cdot D$ with
equality at the generic points. As $U^{H}$ and $\bigcup_{D}\,G\cdot D$ are
Cartier divisors, they must coincide. The equality of Cartier divisors
$\pi_{U}^{*}(\Delta)=nU^{H}$ follows similarly from Proposition 5.4. ∎

######  Remark 6.3.

The above theorem has a partial converse: let $U$ be a $G$-variety with
quotient $\pi:U\to V$, where $V$ is regular. If $U$ satisfies the conditions
(ii) and (iii) for some $H$, $\nu$, $\sigma$ and $\Delta$, then $U$ is
$G$-normal.

Indeed, by Proposition 4.5, it suffices to show that the largest
$\Lambda$-graded ideal of $A={\mathcal{O}}_{U,G\cdot D}$ is principal for any
nonfree divisor $D$. But we have $A=A^{H}[f]$, where $f\in A$ is homogeneous
and satisfies $f^{n}\in A^{H}$. Thus, $\bar{A}=\bar{A}^{H}[\bar{f}]$, where
$\bar{f}^{n}=0$. Moreover, the natural morphism ${\rm Spec}(A^{H})\to{\rm
Spec}(B)$ is a $G/H$-torsor, hence the morphism ${\rm
Spec}(\bar{A}^{H})\to{\rm Spec}(B/\mathfrak{n})={\rm Spec}(\kappa(y))$ is a
$(G/H)_{\kappa(y)}$-torsor. As a consequence, $\bar{f}$ generates the largest
$\Lambda$-graded ideal of $\bar{A}=A/\mathfrak{n}A$. Since $fA\supset
f^{n}A=\mathfrak{n}A$, it follows that $f$ generates the largest
$\Lambda$-graded ideal of $A$.

######  Remark 6.4.

For any $\lambda\in\Lambda$, the multiplication map
$\sigma_{\lambda,|G|}:{\mathcal{A}}_{\lambda}^{\otimes|G|}\to{\mathcal{B}}$
restricts to a global section of $({\mathcal{A}}_{\lambda}|_{Y_{{\rm
reg}}})^{\otimes-|G|}$. By Remark 5.5, the divisor of this section on each
$U(H,\nu)$ equals $[G:H]m\Delta(H,\nu)$, where $m=m(H,\nu)$ is the unique
integer such that $0\leq m\leq|H|-1$ and $\lambda-m\nu\in\Lambda^{H}$. As a
consequence, the class of the divisorial sheaf ${\mathcal{A}}_{\lambda}$ in
$\operatorname{{\rm Cl}}(Y)_{{\mathbb{Q}}}$ (the divisor class group of $Y$
with rational coefficients) satisfies

$[{\mathcal{A}}_{\lambda}]=-\sum_{H,\nu}\frac{m(H,\nu)}{|H|}\Delta(H,\nu)$
(6.1)

with the above notation.

With the notation of Theorem 6.2 (ii), the morphism $\varphi_{U}:U\to U/H$ is
a uniform cyclic cover with branch divisor $\Delta$, as defined in [AV04]. We
say that a $G$-normal variety $X$ is _uniform_ , if it is the closure of some
$U=U(H,\nu)$; equivalently, the equality $X=U(H,\nu)$ holds in codimension
$1$. (We do not assume that $X$ is strongly $G$-normal). Then the closure of
the branch divisor $\Delta$ in $Y$ is a reduced effective divisor that we will
still call the branch divisor, and denote by $\Delta$ for simplicity.

######  Proposition 6.5.

Let $X=\overline{U(H,\nu)}$ be a uniform $G$-normal variety over $Y$. Assume
that ${\mathcal{O}}(Y)^{\times}=k^{\times}$ and the divisor class group
$\operatorname{{\rm Cl}}(Y)$ has no $n$-torsion, where $n=|G|$. Then
$G=H\simeq\boldsymbol{\mu}_{n}$, the $G$-variety $X$ over $Y$ is uniquely
determined by $\Delta$, and the class $[\Delta]$ is divisible by $n$ in
$\operatorname{{\rm Cl}}(Y)$. Moreover, every reduced effective divisor on $Y$
with class divisible by $n$ is obtained from a uniform
$\boldsymbol{\mu}_{n}$-normal variety.

###### Proof.

If $H\neq G$ then $G/H$ has a quotient isomorphic to $\boldsymbol{\mu}_{d}$
for some $d$ dividing $n$, and hence the $G/H$-torsor $\psi:U/H\to V$ factors
through a $\boldsymbol{\mu}_{d}$-torsor $\phi:W\to V$, where $W$ is a variety.
Since ${\rm Pic}(V)=\operatorname{{\rm Cl}}(V)=\operatorname{{\rm Cl}}(Y)$ has
no $d$-torsion, we have
$\phi_{*}({\mathcal{O}}_{W})={\mathcal{O}}_{V}[T]/(T^{d}-f)$ for some
$f\in{\mathcal{O}}(V)^{\times}$ (Remark 3.2). But
${\mathcal{O}}(V)^{\times}={\mathcal{O}}(Y)^{\times}=k^{\times}$ and hence $W$
is not geometrically integral, a contradiction.

Thus, $G=H$ and we may assume that $G=\boldsymbol{\mu}_{n}$ and $\nu$ is the
defining character. Then
${\mathcal{A}}=\bigoplus_{m=0}^{n-1}{\mathcal{A}}_{m}$, where each
${\mathcal{A}}_{m}$ is a divisorial ${\mathcal{B}}$-module (Proposition 4.5).
By Theorem 6.2, the multiplication map ${\rm
mult}_{\ell,m}:{\mathcal{A}}_{\ell}\otimes_{{\mathcal{B}}}{\mathcal{A}}_{m}\to{\mathcal{A}}_{\ell+m}$
is an isomorphism on $V$ whenever $\ell+m\leq n-1$. So we may assume that
${\mathcal{A}}_{m}={\mathcal{O}}_{Y}(-mD)$ for $m=0,\ldots,n-1$, where $D$ is
a Weil divisor on $Y$. Moreover, the multiplication map
$\sigma_{1,n}:{\mathcal{A}}_{1}^{\otimes n}\to{\mathcal{B}}$ yields an
isomorphism
${\mathcal{O}}_{Y}(-nD)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{Y}(-\Delta)$
in view of Theorem 6.2 again. Equivalently, we have a section
$\sigma\in\Gamma(Y,nD)$ with divisor $\Delta$, which uniquely determines the
multiplication of the ${\mathcal{B}}$-algebra ${\mathcal{A}}$. Thus, the
$G$-variety $X$ over $Y$ is uniquely determined by the pair $(D,\sigma)$ up to
isomorphism given by linear equivalence of divisors; moreover, we have
$[\Delta]=n[D]$ in $\operatorname{{\rm Cl}}(Y)$. In turn, $X$ is uniquely
determined by $\Delta$ in view of our assumptions on $Y$. This proves the
second assertion; the final one follows from Theorem 6.2 once more. ∎

######  Remark 6.6.

Keep the asssumptions of the above proposition, and assume in addition that
$Y$ is regular. We may further assume that $G=\boldsymbol{\mu}_{n}$ and $\nu$
is the defining character. Then the sheaf ${\mathcal{L}}={\mathcal{A}}_{1}$ is
invertible; denote by

$\varphi:L={\mathbb{V}}({\mathcal{L}})={\rm Spec}\,{\rm
Sym}_{{\mathcal{B}}}({\mathcal{L}})\longrightarrow Y$

the line bundle with sheaf of local sections ${\mathcal{L}}^{\otimes-1}$. By
the above proof, we may view $X$ as the zero scheme in $L$ of the section

$\tau^{n}-\sigma\in\Gamma(L,\varphi^{*}(L^{\otimes
n}))=\bigoplus_{m=0}^{\infty}\,\Gamma(Y,{\mathcal{L}}^{\otimes m-n}),$

where
$\tau\in\Gamma(L,\varphi^{*}(L))=\bigoplus_{m=0}^{\infty}\,\Gamma(Y,{\mathcal{L}}^{\otimes
m-1})$ denotes the canonical section, corresponding to
$1\in\Gamma(Y,{\mathcal{O}}_{Y})$, and
$\sigma\in\Gamma(Y,{\mathcal{L}}^{\otimes-n})$. This gives back a classical
construction of $\boldsymbol{\mu}_{n}$-covers, see e.g. [Laz04, Prop. 4.1.6].

Also, note that for $p=2$, every $\boldsymbol{\mu}_{2}$-normal variety is
uniform. But there exist nonuniform $\boldsymbol{\mu}_{p}$-normal varieties
for any $p\geq 3$: for example, ${\mathbb{P}}^{1}$ where
$\boldsymbol{\mu}_{p}$ acts by multiplication.

######  Example 6.7.

Let $X$ be a uniform $G$-normal variety over the projective space
${\mathbb{P}}^{N}$. By Proposition 6.5, we have $G\simeq\boldsymbol{\mu}_{n}$
and $X$ is classified by its branch divisor, a reduced hypersurface
$\Delta\subset{\mathbb{P}}^{N}$ of degree $d$ divisible by $n$. Moreover, $X$
is realized as a hypersurface in the line bundle
$O_{{\mathbb{P}}^{N}}(d)={\mathbb{V}}({\mathcal{O}}_{{\mathbb{P}}^{N}}(-d))$
via the construction of Remark 6.6.

If $X$ is regular, then so is $X^{G}$ in view of [Hau20, Lem. 3.5.2]. Thus,
taking for $\Delta$ a singular hypersurface yields many examples of singular
$G$-normal varieties.

##  7 The relative dualizing sheaf

Let $G$ be a finite group scheme acting on a scheme $X$ with quotient morphism
$\pi:X\to Y$. Since $\pi$ is finite, it has a $(0)$-dualizing sheaf
$\omega_{\pi}=\pi^{!}{\mathcal{O}}_{X}$; this is the
${\mathcal{O}}_{X}$-module that corresponds to the ${\mathcal{A}}$-module
${{\mathcal{H}}}{\rm om}_{{\mathcal{B}}}({\mathcal{A}},{\mathcal{B}})$. It is
equipped with a trace map ${\rm
tr}_{\pi}:\pi_{*}(\omega_{\pi})\to{\mathcal{B}}$ via evaluation at $1$. If
$\pi$ is flat and a local complete intersection (l.c.i.), then $\omega_{\pi}$
is isomorphic to the canonical sheaf $\omega_{X/Y}$; in particular,
$\omega_{\pi}$ is invertible (see e.g. [Li10, §6.4], which will be our
reference for duality theory). In any case, $\omega_{\pi}$ is equipped with a
$G$-linearization for which the trace map is invariant.

If $G$ is linearly reductive, then the unique $G$-invariant projection
${\mathcal{A}}\to{\mathcal{B}}$ (the “Reynolds operator”) yields a global
section
$s_{\pi}\in\Gamma(X,\omega_{\pi})=\operatorname{Hom}_{{\mathcal{B}}}({\mathcal{A}},{\mathcal{B}})$.
Note that $s_{\pi}$ is $G$-invariant and satisfies ${\rm
tr}_{\pi}(s_{\pi})=1$. Also, the formation of $\omega_{\pi}$ and $s_{\pi}$
commutes with flat base change $Y^{\prime}\to Y$. If the morphism $\pi$ is
flat l. c. i, then we denote $s_{\pi}$ by $s_{X/Y}$.

Given a subgroup scheme $H\subset G$ and the corresponding factorization of
$\pi$ as
$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/H\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y=X/G$,
we may identify $\omega_{\pi}$ with
$\omega_{\varphi}\otimes_{{\mathcal{O}}_{X}}\varphi^{*}(\omega_{\psi})$ (see
[Li10, Lem. 6.4.26]). If $G$ is linearly reductive, then we have

$s_{\pi}=s_{\varphi}\otimes\varphi^{*}(s_{\psi}),$ (7.1)

where
$s_{\psi}\in\Gamma(Z,\omega_{\psi})=\operatorname{Hom}_{{\mathcal{B}}}({\mathcal{A}}^{H},{\mathcal{B}})$
is the projection ${\mathcal{A}}^{H}\to{\mathcal{B}}$. If in addition $H$ is a
normal subgroup of $G$, then $\psi$ is the quotient morphism by $G/H$, and
$s_{\psi}$ is the corresponding canonical section.

We will need two further observations, certainly well-known but for which we
could not locate any reference.

######  Lemma 7.1.

Let $G$ be a finite group scheme, and $\pi:X\to Y$ a $G$-torsor.

  1. (i)

The (finite flat) morphism $\pi$ is l.c.i. and satisfies
$\omega_{X/Y}\simeq{\mathcal{O}}_{X}$.

  2. (ii)

If $G$ is linearly reductive, then $s_{X/Y}$ is a trivializing section of
$\omega_{X/Y}$.

###### Proof.

(i) We use the construction at the beginning of Section 4. Choose an embedding
of $G$ in some ${\rm GL}_{n}$. Then $\pi$ is the composition

$X\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}Z={\rm
GL}_{n}\times^{G}X\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y,$

where the morphism $\iota$ is a closed immersion with image $G\times^{G}X$,
and $\psi$ is the projection ${\rm GL}_{n}\times^{G}X\to X/G=Y$. Thus, $\psi$
is a ${\rm GL}_{n}$-torsor. In particular, it is smooth and its relative
tangent sheaf is isomorphic to ${\rm Lie}({\rm
GL}_{n})\otimes_{k}{\mathcal{O}}_{Z/Y}$. As a consequence,
$\det(\Omega^{1}_{Z/Y})\simeq{\mathcal{O}}_{Z/Y}$. Also, recall that $X$ is
the fibre at the base point of the other projection $\varphi:{\rm
GL}_{n}\times^{G}X\to{\rm GL}_{n}/G$. Thus, $\iota$ is a regular immersion and
its conormal sheaf ${\mathcal{C}}_{X/Z}$ is trivial (see [Li10, Prop.
6.3.11]). This gives an isomorphism $\omega_{X/Y}\simeq{\mathcal{O}}_{X}$ in
view of loc. cit., Def. 6.4.7.

(ii) We use the commutative diagram of flat morphisms

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Spec}(k)}$$\textstyle{G\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
pr}_{X}}$$\scriptstyle{\alpha}$$\scriptstyle{{\rm
pr}_{G}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y,}$

where both squares are cartesian. Since the formation of $s_{X/Y}$ commutes
with flat base change, this yields $\alpha^{*}(s_{X/Y})=s_{G\times X/X}={\rm
pr}_{G}^{*}(s_{G})$. By fpqc descent, we are reduced to checking that
$s_{G}\in\Gamma(G,\omega_{G})=\operatorname{Hom}_{k}({\mathcal{O}}(G),k)$ is a
free generator of the ${\mathcal{O}}(G)$-module $\omega_{G}$.

In the case where $G$ is diagonalizable, the $k$-vector space
$\operatorname{Hom}_{k}({\mathcal{O}}(G),k)$ has basis
($e_{\lambda})_{\lambda\in\Lambda}$ dual to the basis $\Lambda$ of
${\mathcal{O}}(G)$, and the ${\mathcal{O}}(G)$-module structure of
$\Gamma(G,\omega_{G})$ is given by $\mu\cdot e_{\lambda}=e_{\lambda-\mu}$.
Therefore, this module is freely generated by $e_{0}=s_{G}$.

In the general case of a linearly reductive group $G$, we may assume $k$
algebraically closed by fpqc descent again. Then $G^{0}$ is diagonalizable and
$G\simeq G^{0}\rtimes\pi_{0}(G)$, where $\pi_{0}(G)$ is constant of order $N$
prime to $p$. Thus, $s_{G}:{\mathcal{O}}(G)\to k$ satisfies

$s_{G}(\varphi)=\frac{1}{N}\sum_{g\in G(k)}g\cdot\varphi_{0}$

for any $\varphi\in{\mathcal{O}}(G)$, where $\varphi_{0}$ denotes the
homogeneous component of weight $0$ relative to $G^{0}$; note that
$g\varphi_{0}=(g\cdot\varphi)_{0}$ as $g$ induces an automorphism of
$\Lambda(G^{0})$. Moreover, we have an isomorphism of algebras

${\mathcal{O}}(G)\simeq\prod_{g\in G(k)}{\mathcal{O}}(gG^{0})$

and $s_{G}(\varphi)=\frac{1}{N}\varphi_{0}$ for any $g\in G(k)$ and
$\varphi\in{\mathcal{O}}(gG^{0})$. As $gG^{0}$ is a trivial $G^{0}$-torsor,
$s_{G}$ restricts to a nonzero scalar multiple of $s_{gG^{0}}$. This implies
our assertion. ∎

######  Lemma 7.2.

Let $G$ be a finite group scheme, and $X$ a $G$-normal variety with quotient
$\pi:X\to Y$. Then the sheaf $\omega_{\pi}$ is torsion-free and satisfies
($S_{2}$).

###### Proof.

By local duality, we have an isomorphism
$\operatorname{Hom}_{{\mathcal{O}}_{X}}({\mathcal{F}},\omega_{\pi})\simeq\operatorname{Hom}_{{\mathcal{B}}}(\pi_{*}({\mathcal{F}}),{\mathcal{B}})$
for any coherent sheaf ${\mathcal{F}}$ on $X$. If ${\mathcal{F}}$ is torsion
then so is $\pi_{*}({\mathcal{F}})$; this readily implies that $\omega_{\pi}$
is torsion-free. To show that $\omega_{\pi}$ is ($S_{2}$), it suffices to
check that so is $\pi_{*}(\omega_{\pi})$ (Lemma 4.6). As
$\pi_{*}(\omega_{\pi})\simeq{{\mathcal{H}}}{\rm
om}_{{\mathcal{B}}}({\mathcal{A}},{\mathcal{B}})$, the desired assertion
follows from the normality of $Y$ and [Har80, Cor. 1.2 and Prop. 1.3]. ∎

We also record a preliminary result, to be used in Sections 8 and 9.

######  Lemma 7.3.

Consider again a finite group scheme $G$, and a $G$-normal variety $X$ with
quotient $\pi:X\to Y$. Then there exists a $G$-stable big open subset
$U\subset X$ such that the quotient $V=U/G$ is regular and the morphism
$\pi|_{U}:U\to V$ is flat l.c.i.

###### Proof.

We may replace $X$ with $\pi^{-1}(Y_{{\rm reg}})$, and hence assume that $Y$
is regular. Then $\pi$ is flat in codimension $1$, so that we may further
assume that it is flat everywhere. We now argue as in the proof of Lemma 7.1
(i). With the notation of that proof, the variety $Z$ is normal by Lemma 4.1,
and $Z_{{\rm reg}}$ is stable by ${\rm GL}_{n}$ as the latter is smooth. Thus,
we have $Z_{{\rm reg}}={\rm GL}_{n}\times^{G}U$ for a unique $G$-stable open
subset $U\subset X$. Moreover, $U$ is big in $X$, since $Z_{{\rm reg}}$ is big
in $Z$. To complete the proof, it suffices to show that $\pi|_{U}$ is l.c.i.
But $\pi|_{U}=\psi|_{Z_{{\rm reg}}}\circ\iota$, where $\iota:U\to Z_{{\rm
reg}}$ is a regular immersion and $\psi|_{Z_{{\rm reg}}}:Z_{{\rm reg}}\to Y$
is a morphism between regular varieties, and hence is l.c.i. by [Li10, Ex.
6.3.18]. So the desired assertion follows from loc. cit., Prop. 6.3.20. ∎

Next, we assume that $G$ is diagonalizable and $X$ is a $G$-normal variety on
which $G$ acts faithfully. Denote by $\iota:U\subset X$ the inclusion of the
largest strongly $G$-normal subset (Lemma 6.1). Then we have
$\omega_{\pi}=\iota_{*}(\omega_{\pi|_{U}})$ in view of Lemmas 4.6 and 7.2.
Therefore, to determine $\omega_{\pi}$, we may further assume that _$X$ is
strongly $G$-normal_.

######  Theorem 7.4.

With the above assumptions, $\pi$ is flat and l.c.i. Moreover,

${\rm div}(s_{X/Y})=\sum\big{(}|H(D)|-1\big{)}\,G\cdot D,$ (7.2)

where the sum runs over the $G$-orbits of nonfree divisors $D$.

###### Proof.

The first assertion follows from Corollary 4.7 and Theorem 6.2 (ii). With the
notation of this theorem, $X$ is covered by the $G$-stable open subsets
$U(H,\nu)$, and hence we may assume that $X=U(H,\nu)$. Then
$\pi=\psi\circ\varphi$, where $\psi:Z=X/H\to Y=X/G$ is a $G/H$-torsor. Thus,
$s_{Z/Y}$ is a trivializing section of $\omega_{Z/Y}$ (Lemma 7.1). Also,
$\omega_{X/Y}\simeq\omega_{X/Z}$ by [Li10, Thm. 6.4.9]; this identifies
$s_{X/Y}$ with $s_{X/Z}$ in view of (7.1). So it suffices to check that the
divisor of $s_{X/Z}$ equals $(|H|-1)\,\sum_{D}G\cdot D$ (sum over the
$G$-orbits of nonfree divisors).

Let $D$ be such a divisor, with generic point $x$. With the notation of
Proposition 5.4, the $A^{H}$-module $A$ is free with basis
$1,f,\ldots,f^{n-1}$, where $f$ is a local equation of $G\cdot x$ at $x$, and
$n=|H|$. Denote by $e_{0},\ldots,e_{n-1}$ the dual basis of the $A^{H}$-module
$\varphi^{!}A=\operatorname{Hom}_{A^{H}}(A,A^{H})$; then $e_{0}$ is the
projection $A\to A^{H}$. Moreover, the $A$-module structure of $\varphi^{!}A$
is given by $f\cdot e_{i}=e_{i-1}$ for $i=1,\ldots,n-1$, and $f\cdot
e_{0}=e_{n-1}$. So $\varphi^{!}A$ is freely generated by $e_{n-1}$, and
$e_{0}=f^{n-1}e_{n-1}$. In view of Proposition 5.4 again, it follows that
${\rm div}(s_{X/Y})=(n-1)G\cdot D$ in a neighborhood of $x$. ∎

######  Remark 7.5.

Still assuming $G$ diagonalizable and $X$ strongly $G$-normal, we may rewrite
(7.2) as the equality of Cartier divisors

${\rm div}(s_{X/Y})=\pi^{*}(\Delta_{Y})-G\cdot\Delta_{X},$ (7.3)

where $\Delta_{Y}=Y\setminus Y_{{\rm fr}}$ and $\Delta_{X}=X\setminus X_{{\rm
fr}}$ are viewed are reduced effective divisors, so that $\Delta_{Y}$ is the
branch divisor. Indeed,

$G\cdot\Delta_{X}=\sum G\cdot D$

(sum over the $G$-orbits of nonfree divisors); this is a $G$-stable effective
Cartier divisor on $X$. Moreover,

$\pi^{*}(\Delta_{Y})=\sum|H(D)|\,G\cdot D$

in view of Theorem 6.2 (iv).

In turn, (7.3) gives an isomorphism of $G$-linearized sheaves

$\omega_{X}(G\cdot\Delta_{X})\simeq\pi^{*}\big{(}\omega_{Y}(\Delta_{Y})\big{)},$
(7.4)

where the dualizing sheaf $\omega_{X}$ is equipped with a $G$-linearization
via the canonical isomorphism

$\omega_{X}\simeq\omega_{X/Y}\otimes\pi^{*}(\omega_{Y}).$ (7.5)

The isomorphism (7.4) will be used in the next section to determine the
“equivariant arithmetic genus” of projective $G$-normal curves. The equality
(7.3) will be generalized to linearly reductive group schemes in Section 9.

##  8 Curves

Throughout this section, $G$ denotes a finite diagonalizable group scheme, $Y$
a curve, and $X$ a $G$-normal curve over $Y$. Then $Y$ is regular, the orbit
$G\cdot x$ is an effective Cartier divisor on $X$ for any closed point $x\in
X$, and the quotient $\pi:X\to Y$ is flat l. c. i. (Lemma 7.3). Also, the
$G$-action on $X$ is generically free; thus, the nonfree locus $X\setminus
X_{{\rm fr}}$ consists of finitely many closed points. For any such point $x$,
the subgroup scheme $H=H(x)\subset G$ is cyclic (Proposition 5.1). Moreover,
we have $X=\bigcup_{H,\nu}U(H,\nu)$, where $U(H,\nu)$ is the union of $X_{{\rm
fr}}$ and those nonfree $x$ such that $H(x)=H$ and $G\cdot x$ has an equation
of weight $\nu$ in ${\mathcal{O}}_{X,G\cdot x}$ (see Theorem 6.2 for these
results). So Theorem 7.4 readily yields a version of the Hurwitz formula:

######  Corollary 8.1.

There is an isomorphism of $G$-linearized sheaves

$\omega_{X/Y}\simeq{\mathcal{O}}_{X}\left(\sum\,(|H(x)|-1)\,G\cdot x\right)$
(8.1)

(sum over the $G$-orbits of nonfree points), which identifies $s_{X/Y}$ with
the canonical section of the right-hand side.

We now assume $X$ projective. Taking degrees in (7.4) and using the equality
$\pi^{*}(\pi(x))=|H(x)|\,G\cdot x$ (which follows fromTheorem 6.2), we obtain

$2p_{a}(X)-2=|G|\,\big{(}2p_{a}(Y)-2+\sum_{y\in Y\setminus Y_{{\rm
fr}}}(1-\frac{1}{n(y)})\,{\rm deg}(y)\big{)},$ (8.2)

where $n(y)=|H(x)|$ for any $y=\pi(x)\in Y\setminus Y_{{\rm fr}}$. This
generalizes (1.1) to an arbitrary field.

We may also determine the degree of the invertible sheaves
${\mathcal{A}}_{\lambda}$ on $Y$, where $\lambda\in\Lambda$. By Remark 6.4, we
have the equality in $\operatorname{{\rm Cl}}(Y)_{{\mathbb{Q}}}$

$[{\mathcal{A}}_{\lambda}]=-\sum_{y\in Y\setminus Y_{{\rm
fr}}}\frac{m(y,\lambda)}{n(y)}y,$

where $m(y,\lambda)$ denotes the unique integer such that $0\leq
m(y,\lambda)\leq n(y)-1$ and $\lambda-m(y,\lambda)\nu(y)$ restricts trivially
to $H(y)=H(x)$. As a consequence,

${\rm deg}({\mathcal{A}}_{\lambda})=-\sum_{y\in Y\setminus Y_{{\rm
fr}}}\frac{m(y,\lambda)}{n(y)}\,{\rm deg}(y).$ (8.3)

Next, we determine the “equivariant arithmetic genus” of $X$, that is, the
structure of the $G$-module $H^{0}(X,\omega_{X})$. As $G$ is diagonalizable,
it suffices to compute the character ${\rm ch}\,H^{0}(X,\omega_{X})$, where
the character of a finite-dimensional $G$-module
$M=\bigoplus_{\lambda\in\Lambda}M_{\lambda}$ is the formal sum ${\rm
ch}(M)=\sum_{\lambda\in\Lambda}\dim(M_{\lambda})e^{\lambda}$.

######  Proposition 8.2.

With the above notation, we have

${\rm ch}\,H^{0}(X,\omega_{X})-1=\big{(}p_{a}(Y)-1\big{)}\,{\rm
ch}\,{\mathcal{O}}(G)+\sum_{y\in Y\setminus
Y_{fr}}\frac{\gamma(y)}{n(y)}\,{\rm
ch}\,{\mathcal{O}}\big{(}G/H(y)\big{)}\,{\rm deg}(y),$

where $\gamma(y)=\sum_{\ell=1}^{n(y)-1}\ell e^{-\ell\nu(y)}$.

###### Proof.

For any $\lambda\in\Lambda$, we set
$h^{0}(\omega_{X})_{\lambda}=\dim\big{(}H^{0}(X,\omega_{X})_{\lambda}\big{)}$.
If $G$ acts freely on $X$, then $\omega_{X}=\pi^{*}(\omega_{Y})$ by (7.4), or
directly by Lemma 7.1. As a consequence, we obtain isomorphisms of $G$-modules

$H^{0}(X,\omega_{X})\simeq H^{0}(Y,\omega_{Y}\otimes{\mathcal{A}}),\quad
H^{1}(X,\omega_{X})\simeq H^{1}(Y,\omega_{Y}\otimes{\mathcal{A}})$

by using the projection formula. Moreover, $H^{1}(X,\omega_{X})$ is the
trivial $G$-module $k$ by Serre duality. This yields
$h^{0}(\omega_{X})_{0}=h^{0}(\omega_{Y})=g(Y)$ and
$h^{0}(\omega_{X})_{\lambda}=\chi(Y,\omega_{Y}\otimes{\mathcal{A}}_{\lambda})$
for any nonzero $\lambda$. Since ${\rm deg}({\mathcal{A}}_{\lambda})=0$ (as
follows e.g. from Remark 3.2), we obtain $h^{0}(\omega_{X})_{\lambda}=g(Y)-1$
by using the Riemann–Roch theorem. Thus, ${\rm
ch}\,H^{0}(X,\omega_{X})-1=\big{(}p_{a}(Y)-1\big{)}\,{\rm
ch}\,{\mathcal{O}}(G)$ as desired.

So we may assume that $G$ does not act freely on $X$, i.e., $G\cdot\Delta_{X}$
and $\Delta_{Y}$ are nonzero. Denoting by $\iota$ the closed immersion
$G\cdot\Delta_{X}\to X$, we have a short exact sequence of $G$-linearized
sheaves

$0\longrightarrow\omega_{X}\longrightarrow\omega_{X}(G\cdot\Delta_{X})\longrightarrow\iota_{*}\iota^{*}\omega_{X}(G\cdot\Delta_{X})\longrightarrow
0$

and hence a long exact sequence of $G$-modules

$0\longrightarrow H^{0}(X,\omega_{X})\longrightarrow
H^{0}\big{(}X,\omega_{X}(G\cdot\Delta_{X})\big{)}\longrightarrow\bigoplus
H^{0}\big{(}G\cdot x,\omega_{X}(G\cdot\Delta_{X})\big{)}$ $\longrightarrow
H^{1}(X,\omega_{X})\longrightarrow
H^{1}\big{(}X,\omega_{X}(G\cdot\Delta_{X})\big{)}\longrightarrow 0,$

where the sum runs over the $G$-orbits of nonfree points. Moreover,

$H^{1}\big{(}X,\omega_{X}(G\cdot\Delta_{X})\big{)}=H^{0}\big{(}X,{\mathcal{O}}_{X}(-G\cdot\Delta_{X})\big{)}^{*}=0.$

In addition, we obtain isomorphisms

$H^{0}\big{(}X,\omega_{X}(G\cdot\Delta_{X})\big{)}_{\lambda}\simeq
H^{0}\big{(}Y,\omega_{Y}(\Delta_{Y})\otimes{\mathcal{A}}_{\lambda}\big{)}$

for all $\lambda$, by arguing as above. Thus,

$h^{0}\big{(}\omega_{X}(G\cdot\Delta_{X})\big{)}_{\lambda}=p_{a}(Y)-1+\sum_{y\in
Y\setminus Y_{{\rm fr}}}\big{(}1-\frac{m(y,\lambda)}{n(y)}\big{)}\,{\rm
deg}(y)$

in view of (8.3) and the Riemann-Roch theorem again.

We also obtain isomorphisms of $G$-modules $H^{0}\big{(}G\cdot
x,\omega_{X}(G\cdot\Delta_{X})\big{)}\simeq{\mathcal{O}}(G\cdot x)$ by using
(7.4) and the fact that $G\cdot x$ is contained in the fibre of $\pi$ at
$y=\pi(x)$. Moreover,

$\dim{\mathcal{O}}(G\cdot x)_{\lambda}=\begin{cases}{\rm deg}(y)&\text{if
$m(y,\lambda)=0$,}\\\ 0&\text{else},\end{cases}$

as follows from Proposition 5.1. In particular,
$h^{0}\big{(}\omega_{X}(G\cdot\Delta_{X})\big{)}_{0}=p_{a}(Y)-1+{\rm
deg}(\Delta_{Y})$ and $\sum\dim{\mathcal{O}}(G\cdot x)_{0}={\rm
deg}(\Delta_{Y})$. In view of our long exact sequence of $G$-modules, this
yields the equality

$h^{0}(\omega_{X})_{0}=p_{a}(Y).$

For $\lambda\neq 0$, we obtain similarly

$h^{0}(\omega_{X})_{\lambda}=p_{a}(Y)-1+\sum\big{(}1-\frac{m(y,\lambda)}{n(y)}\big{)}\,{\rm
deg}(y),$

where the sum runs over the points $y\in Y\setminus Y_{{\rm fr}}$ such that
$m(y,\lambda)\neq 0$. Translating the two latter equalities in terms of
characters yields the assertion. ∎

The above proposition expresses the equivariant arithmetic genus as the sum of
a “main term” associated with a free action, and of “correcting terms”
associated with the nonfree points. Of course, it gives back the Hurwitz
formula (8.2) by specializing characters to dimensions. But the latter formula
has a much more direct proof.

Next, we obtain detailed information on the local structure of the $G$-normal
curve $X$ in the case where $G$ is infinitesimal; equivalently, we have
$G\simeq\boldsymbol{\mu}_{p^{r}}$ by [Br24, Lem. 5.3]. (The general case
reduces to this in view of the factorization (2.3)
$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/G^{0}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y$,
where $\psi$ is a finite abelian cover). We identify $G$ with
$\boldsymbol{\mu}_{p^{r}}$, and hence the character group $\Lambda$ with
${\mathbb{Z}}/p^{r}{\mathbb{Z}}$. So the subgroup schemes of $G$ are exactly
the $\boldsymbol{\mu}_{p^{s}}$ for $s=0,\ldots,r$.

Choose a nonfree point $x\in X$ and let $A={\mathcal{O}}_{X,x}$, $y=\pi(x)$
and $B={\mathcal{O}}_{Y,y}$ as in Section 5; then
$H(x)=\boldsymbol{\mu}_{p^{s}}$ for some positive integer $s=s(x)\leq r$. We
may now state a refinement of Proposition 5.4:

######  Proposition 8.3.

With the above notation, we have an isomorphism of
${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded $B$-algebras

$A\simeq B[T_{1},T_{2}]/(T_{1}^{p^{r-s}}-u,T_{2}^{p^{s}}-t\,T_{1}^{\nu})$

where $T_{1}$ (resp. $T_{2}$) is homogeneous of weight $p^{s}$ (resp. $\nu$
prime to $p$), $u\in B$ is a unit, and $t\in B$ generates the maximal ideal.

###### Proof.

By Corollary 5.3, the morphism $X/H\to Y$ is a torsor under
$G/H=\boldsymbol{\mu}_{p^{r-s}}$ in a neighborhood of $y$. Together with
Remark 3.2, this yields an isomorphism of
${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded $B$-algebras

$A^{H}\simeq B[T_{1}]/(T_{1}^{p^{r-s}}-u),$

where $T_{1}$ is homogeneous of weight $p^{s}$ and $u\in B^{\times}$. We also
have an isomorphism of ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded $B$-algebras

$A\simeq A^{H}[T_{2}]/(T_{2}^{p^{s}}-g),$

where $T_{2}$ is homogeneous of weight $\nu$ prime to $p$, and $g\in A^{H}$ is
homogeneous of weight $p^{s}\nu$; moreover, $gA=\mathfrak{n}A$ (Proposition
5.4). Thus, $g=t\,T_{1}^{\nu}$, where $t\in B$ generates $\mathfrak{n}$.
Combining both displayed isomorphisms yields the statement. ∎

######  Remark 8.4.

The above proposition takes a much simpler form in the case where $r=s$, i.e.,
$x$ is a $G$-fixed point: then $A\simeq B[T]/(T^{p^{r}}-t)$ for a uniformizer
$t\in B$. As a consequence, the homomorphism $B/Bt\to A/Af$ is bijective,
where $f$ denotes the image of $T$ in $A$. Equivalently, $X$ is regular at $x$
and $\kappa(x)=B/Bt=\kappa(y)$. This yields a refinement of [Br24, Prop. 5.5].

On the other hand, if $r>s$ then we have with the notation of Section 5:

$\bar{A}=A/tA\simeq\kappa(y)[\bar{T_{1}},\bar{T_{2}}]/(\bar{T_{1}}^{p^{r-s}}-\bar{u},\bar{T_{2}}^{p^{s}}),$

where $\bar{u}\in\kappa(y)^{\times}$ denotes the image of $u\in A^{\times}$.
If in addition $k$ is perfect, then so is $\kappa(y)$ (as the extension
$\kappa(y)/k$ is finite), and hence we obtain

$\bar{A}\simeq\kappa(y)[U_{1},U_{2}]/(U_{1}^{p^{r-s}},U_{2}^{p^{s}}).$

Thus, $U_{1},U_{2}$ form a minimal system of generators of the ideal
$\bar{\mathfrak{m}}$. So $X$ is not regular at $x$, and $\kappa(x)=\kappa(y)$.
But if $\kappa(y)$ is imperfect, then we may choose $u\in A^{\times}$ such
that $\bar{u}\notin\kappa(y)^{p}$, and hence
$A/Af\simeq\kappa(y)[T]/(T^{p^{r-s}}-\bar{u})$ is a field, where $f$ denotes
the image of $T_{2}$ in $A$. In that case, $X$ is regular but not smooth at
$x$, and $\kappa(y)\subsetneq\kappa(x)$; the morphism ${\rm
Spec}\big{(}\kappa(x)\big{)}\to{\rm Spec}\big{(}\kappa(y)\big{)}$ is the
torsor under $(\boldsymbol{\mu}_{p^{r-s}})_{\kappa(y)}$ considered in
Proposition 5.1.

Next, let $s$, $\nu$ be integers such that $1\leq s\leq r$ and $0\leq\nu\leq
p^{r}-1$, and let $U=U(s,\nu)$ be the open subset of $X$ consisting of the
free locus $X_{{\rm fr}}$ together with the closed points $x\in X$ such that
$H(x)=\boldsymbol{\mu}_{p^{s}}$ and the largest
${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded ideal of ${\mathcal{O}}_{X,x}$ is
generated by a homogeneous element of weight $\nu$. Then $V=V(s,\nu)=U/G$ is
an open subset of $Y$ containing $Y_{{\rm fr}}$, and
$\Delta=\Delta(s,\nu)=V\setminus V_{{\rm fr}}$ is a reduced effective divisor
on $V$. With the notation of the beginning of Section 6, we have $U=U(H,\nu)$
and $\Delta=\Delta(H,\nu)$, where $H=\boldsymbol{\mu}_{p^{s}}$. So the
multiplication in ${\mathcal{A}}|_{V}$ yields isomorphisms of
${\mathcal{B}}|_{V}$-modules

$\sigma=\sigma_{\nu,p^{s}}:{\mathcal{A}}_{\nu}^{\otimes
p^{s}}|_{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}_{\nu
p^{s}}|_{V}(-\Delta),\quad\tau=\sigma_{p^{s}\nu,p^{r-s}}:{\mathcal{A}}_{p^{s}}^{\otimes
p^{r-s}}|_{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{B}}|_{V},$

since ${\mathcal{A}}_{p^{s}}\subset{\mathcal{A}}^{H}$ and the morphism
$U/H={\rm Spec}_{{\mathcal{B}}|_{V}}({\mathcal{A}}^{H})\to V$ is a
$G/H$-torsor (Theorem 6.2). We may now state:

######  Proposition 8.5.

There is an isomorphism of ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded
${\mathcal{B}}|_{V}$-algebras

${\mathcal{A}}|_{V}\simeq{\rm
Sym}_{{\mathcal{B}}|_{V}}({\mathcal{A}}_{\nu}\oplus{\mathcal{A}}_{p^{s}})/{\mathcal{I}},$

where ${\mathcal{I}}$ denotes the ideal generated by the
$a^{p^{s}}-\sigma(a^{\otimes p^{s}})$ where $a\in{\mathcal{A}}_{\nu}$, and the
$b^{p^{r-s}}-\tau(b^{\otimes p^{r-s}})$ where $b\in{\mathcal{A}}_{p^{s}}$.

###### Proof.

Using the fact that ${\rm Spec}_{{\mathcal{B}}|_{V}}({\mathcal{A}}^{H})\to V$
is a $G/H$-torsor together with Remark 3.2, we obtain an isomorphism

${\mathcal{A}}^{H}|_{V}\simeq{\rm
Sym}_{{\mathcal{B}}|_{V}}({\mathcal{A}}_{p^{s}})/{\mathcal{J}},$

where ${\mathcal{J}}$ denotes the ideal generated by the
$b^{p^{r-s}}-\tau(b^{\otimes p^{r-s}})$ for $b\in{\mathcal{A}}_{p^{s}}$. The
assertion follows by combining this isomorphism with that of Theorem 6.2 (ii).
∎

The above proposition takes again a much simpler form in the case where $r=s$:
then we just have an isomorphism of ${\mathcal{B}}|_{V}$-modules
$\sigma:{\mathcal{A}}_{\nu}^{\otimes
p^{r}}|_{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{V}(-\Delta)$,
and the algebra ${\mathcal{A}}|_{V}$ is the quotient of ${\rm
Sym}_{{\mathcal{B}}|_{V}}({\mathcal{A}}_{\nu})$ by the ideal generated by the
$a^{p^{r}}-\sigma(a^{\otimes p^{r}})$. Moreover, $U$ is regular along $U^{H}$
by Remark 8.4.

Finally, we consider the tangent sheaf ${\mathcal{T}}_{X}$ consisting of the
$k$-linear derivations of the structure sheaf ${\mathcal{O}}_{X}$, and its
relative version ${\mathcal{T}}_{X/Y}$ consisting of the
${\mathcal{O}}_{Y}$-linear derivations. Recall that
${\mathcal{T}}_{X}={{\mathcal{H}}}{\rm
om}_{{\mathcal{O}}_{X}}(\Omega^{1}_{X},{\mathcal{O}}_{X})$ is equipped with a
$G$-linearization, since so is $\Omega^{1}_{X}$; likewise,
${\mathcal{T}}_{X/Y}$ is $G$-linearized as well. As a consequence, for any
$x\in X$, the fibre ${\mathcal{T}}_{X}(x)$ is a linear representation of
$H(x)$. By [Br24, Prop. 5.4], the sheaf ${\mathcal{T}}_{X}$ is invertible; in
particular, its fibre at $x$ is a vector space of dimension $1$ over the
residue field $\kappa(x)$, for any $x$ as above. We will recover the latter
result in a more explicit way:

######  Corollary 8.6.

With the above notation, we have ${\mathcal{T}}_{U}={\mathcal{T}}_{U/V}$.
Moreover, ${\mathcal{T}}_{U}(x)$ is a $\kappa(x)$-vector space of dimension
$1$ on which $H(x)$ acts with weight $-\nu$, for any nonfree point $x\in U$.

###### Proof.

Since $U$ is affine, the fraction field of ${\mathcal{O}}(U)$ is the function
field $k(X)$. Moreover, the invariant subfield $k(X)^{G}$ is generated by $k$
and $k(X)^{p^{r}}$ (see e.g. [Br24, Prop. 5.1]). Thus, every $k$-linear
derivation of ${\mathcal{O}}(U)$ vanishes on
${\mathcal{O}}(U)^{G}={\mathcal{O}}(V)$. This shows the first assertion.

For the second assertion, we use the structure of the
${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-graded $B$-algebra $A$ (Proposition 8.5),
which gives an isomorphism of graded $A$-modules

$\Omega^{1}_{A/B}\simeq(A\,dT_{1}\oplus
A\,dT_{2})/(A\,t\,T_{1}^{\nu-1}dT_{1})$

with the notation of this proposition. Taking $A$-duals, we obtain an
isomorphism of graded $A$-modules ${\rm Der}_{B}(A)\simeq A\,\partial_{2}$,
where $\partial_{2}$ arises from the derivation $\partial/\partial T_{2}$ of
$B[T_{1},T_{2}]$. As $T_{2}$ has weight $\nu$, this yields the result. ∎

For instance, if $x\in X(k)$ is $G$-fixed, then it is a smooth point of $X$
(e.g. by Remark 8.4) and hence the Zariski tangent space $T_{x}(X)$ is a line
on which $G$ acts with weight $-\nu$.

######  Remark 8.7.

The degree of the tangent sheaf satisfies

${\rm deg}({\mathcal{T}}_{X})=|G|\sum_{y\in Y\setminus Y_{{\rm fr}}}\frac{{\rm
deg}(y)}{n(y)}=\sum_{y\in Y\setminus Y_{{\rm fr}}}p^{r-s(y)}\,{\rm deg}(y).$

Indeed, ${\mathcal{T}}_{X}$ is equipped with a $G$-invariant global section:
the generator $\xi$ of the Lie algebra ${\rm Lie}(G)={\rm
Lie}(\boldsymbol{\mu}_{p^{r}})={\rm Lie}({\mathbb{G}}_{m})=k$. Clearly, the
divisor of $\xi$ is supported at the nonfree points. Given such a point $x$,
we have $\xi(T_{1})=0$ and $\xi(T_{2})=\nu T_{2}$ with the notation of
Proposition 8.3; thus, $\xi=\nu\partial_{2}$. As a consequence, the ideal of
zeroes of $\xi$ in $A$ equals $AT_{2}$. Since

$A/AT_{2}=B[T_{1}]/(T_{1}^{p^{r-s}}-u,tT_{1}^{\nu})=B[T_{1}]/(T_{1}^{p^{r-s}}-u,t)=\kappa(y)[T_{1}]/(T_{1}^{p^{r-s}}-u(y)),$

we obtain $\dim_{\kappa(y)}(A/AT_{2})=p^{r-s}{\rm deg}(y)$. This implies our
formula.

In particular, we have ${\rm deg}({\mathcal{T}}_{X})\geq 0$, and equality
holds if and only if $X$ is a $G$-torsor over $Y$. Also, note that the fixed
point subscheme $X^{G}$ is reduced and satisfies ${\rm
deg}({\mathcal{T}}_{X})\equiv{\rm deg}(X^{G})$ (mod $p$).

##  9 Linearly reductive group schemes

Throughout this section, we denote by $G$ a finite linearly reductive group
scheme, and by $X$ a $G$-normal variety on which $G$ acts faithfully with
quotient $\pi:X\to Y$. Our aim is to determine the relative dualizing sheaf
$\omega_{\pi}$ as a $G$-linearized sheaf. Using Lemmas 7.2 and 7.3, we may
assume that $Y$ is regular and $\pi$ is flat l. c. i. Then
$\omega_{\pi}=\omega_{X/Y}$ is invertible and has a canonical $G$-invariant
section $s_{X/Y}$, which trivializes it over the free locus and whose
formation commutes with flat base change on $Y$.

######  Theorem 9.1.

With the above assumptions, we have ${\rm
div}(s_{X/Y})=\pi^{*}(\Delta_{Y})-G\cdot\Delta_{X}$.

###### Proof.

We start with a reduction to the case where _$G^{0}$ is diagonalizable and
$\pi_{0}(G)$ is constant_. As in Remark 4.4, we may choose a finite Galois
extension $K/k$ such that $G^{0}_{K}$ is diagonalizable and $\pi_{0}(G)_{K}$
is constant. Then the $G_{K}$-variety $X_{K}$ satisfies our assumptions;
moreover, $\Delta_{Y_{K}}=(\Delta_{Y})_{K}$ as the base change to $K$
preserves the free locus and reducedness, and
$(G\cdot\Delta_{X})_{K}=G_{K}\cdot(\Delta_{X})_{K}$ as the formation of the
schematic image commutes with flat base change. By Galois descent, it thus
suffices to prove the theorem for the $G_{K}$-variety $X_{K}$.

If $G=G^{0}$ then the desired assertion follows from Remark 7.5. We now prove
this assertion when $G=\pi_{0}(G)$, i.e., _$G$ is constant_. Since $G$ is
linearly reductive, its order is prime to $p$; also, recall that $X$ is
normal.

We will need a folklore result for which we could not find any reference.
Consider a point $x\in X$ with centralizer $C_{G}(x)$. Then the quotient
morphism $\pi:X\to Y$ factors as

$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/C_{G}(x)\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y=X/G,\quad
x\longmapsto z\longmapsto y.$

In turn, $\psi$ factors as

$Z\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}W=X/N_{G}(x)\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}Y,\quad
z\longmapsto w\longmapsto y,$

where the normalizer $N_{G}(x)$ is the largest subgroup of $G$ that stabilizes
$x$, and $\tau$ is the quotient by $N_{G}(x)/C_{G}(x)$.

######  Lemma 9.2.

With the above notation, we have:

  1. (i)

$\kappa(z)=\kappa(x)$.

  2. (ii)

$\tau$ is a torsor under $N_{G}(x)/C_{G}(x)$ in a neighborhood of $z$.

  3. (iii)

$\eta$ induces an isomorphism of completed local rings
$\widehat{{\mathcal{O}}}_{Y,y}\simeq\widehat{{\mathcal{O}}}_{W,w}$.

  4. (iv)

$\psi$ is étale at $z$ and $s_{\psi}$ is a trivializing section of
$\omega_{\psi}$ at that point.

  5. (v)

If $x$ is the generic point of a prime divisor, then $C_{G}(x)$ is cyclic.

###### Proof.

(i) This follows from the isomorphism of local rings
${\mathcal{O}}_{Z,z}\simeq{\mathcal{O}}_{X,x}^{C_{G}(x)}$ in view of the
linear reductivity of $C_{G}(x)$.

(ii) Likewise, the linear reductivity of $N_{G}(x)$ and the isomorphism
${\mathcal{O}}_{W,w}\simeq{\mathcal{O}}_{X,x}^{N_{G}(x)}$ imply that
$\kappa(w)=\kappa(x)^{N_{G}(x)}=\kappa(x)^{N_{G}(x)/C_{G}(x)}$. As
$N_{G}(x)/C_{G}(x)$ acts faithfully on $x$, the desired assertion follows from
Galois theory.

(iii) This is a special case of [SGA1, Exp. V, Prop. 2.2]) and can be proved
directly as follows: $\pi$ induces an isomorphism
$\widehat{{\mathcal{O}}}_{Y,y}\simeq\widehat{{\mathcal{O}}}_{X,G\cdot x}^{G}$.
Moreover, $G\cdot x$ is the disjoint union of the $g\cdot x$, where $g\in
G/N_{G}(x)$. Thus, we have a $G$-equivariant isomorphism of semi-local rings
$\widehat{{\mathcal{O}}}_{X,G\cdot x}\simeq\prod_{g\in
G/N_{G}(x)}\widehat{{\mathcal{O}}}_{X,g\cdot x}$. Taking $G$-invariants, we
obtain
$\widehat{{\mathcal{O}}}_{Y,y}\simeq\widehat{{\mathcal{O}}}_{X,x}^{N_{G}(x)}$.
Moreover, ${\mathcal{O}}_{X,x}^{N_{G}(x)}\simeq{\mathcal{O}}_{W,w}$; this
yields the assertion.

(iv)The morphism $\tau$ is étale at $z$ by (ii), and hence $\omega_{\tau}$ is
invertible at that point; moreover, $s_{\tau}$ is a trivializing section in
view of Lemma 7.1. Also, $\eta$ is étale at $w$ by (iii), so that
$\omega_{\eta}$ is invertible at that point. Moreover, $s_{\eta}$ is a
trivializing section, since its formation commutes with flat base change. This
implies the assertions in view of the isomorphism
$\omega_{\psi}\simeq\omega_{\tau}\otimes\tau^{*}(\omega_{\eta})$ identifying
$s_{\psi}$ with $s_{\tau}\otimes\tau^{*}(s_{\eta})$.

(v) The group $C_{G}(x)$ acts faithfully on the local ring
${\mathcal{O}}_{X,x}$ and stabilizes the powers $\mathfrak{m}^{n}_{x}$ of the
maximal ideal. Since $\bigcap_{n\geq 1}\,\mathfrak{m}^{n}_{x}=0$, the induced
action on some quotient ${\mathcal{O}}_{X,x}/\mathfrak{m}^{n}_{x}$ is faithful
as well. By linear reductivity, it follows that the induced action on some
subquotient $\mathfrak{m}_{x}^{m}/\mathfrak{m}_{x}^{m+1}$ is faithful. As
$\mathfrak{m}_{x}^{m}/\mathfrak{m}_{x}^{m+1}$ is a vector space of dimension
$1$ over $\kappa(x)$, we obtain an injective homomorphism
$C_{G}(x)\to\kappa(x)^{\times}$. ∎

We may now prove Theorem 9.1 in the case where $G$ constant. Let $D$ be a
prime divisor in $X$, with generic point $x$. By (7.1) and Lemma 9.2 (iv), the
multiplicity of $D$ in $s_{X/Y}$ equals that in $s_{X/Z}$. The latter
multiplicity equals $|C_{G}(x)|-1$ in view of Theorem 7.4 together with Lemma
9.2 (i), (v). On the other hand, denoting by $E$ (resp. $F$) the image of $D$
in $Y$ (resp. $Z$), we see that $F$ occurs in $\psi^{*}(E)$ with multiplicity
$1$ (as $\psi$ is étale at $z$), and $D$ occurs in $\varphi^{*}(F)$ with
multiplicity $|C_{G}(x)|$ (by Theorem 6.2). So $D$ occurs in
$\pi^{*}(\Delta_{Y})-G\cdot\Delta_{X}$ with multiplicity $|C_{G}(x)|-1$ as
desired.

Finally, we handle the general case (where $G^{0}$ is diagonalizable and
$\pi_{0}(G)$ is constant of order prime to $p$). Recall the factorization of
$\pi$ as

$X\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Z=X/G^{0}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Y=X/G=Z/\pi_{0}(G).$
(9.1)

Moreover, $X$ is a $G^{0}$-normal variety on which $G^{0}$ acts generically
freely, and $Z$ is a normal variety on which $\pi_{0}(G)$ acts generically
freely (Lemmas 2.1 and 3.4). Thus, $\psi$ is a tamely ramified Galois cover
with group $\pi_{0}(G)(k)$. We may further assume that $X$ is strongly
reductive relative to $G^{0}$.

Let again $D$ be a prime divisor in $X$ with generic point $x$ and set
$E=\pi(D)$, $F=\varphi(D)$ with generic points $y$, $z$. Then $F$ occurs in
$\psi^{*}(E)$ with multiplicity $|C_{\pi_{0}(G)}(z)|$ by the above step.
Moreover, we have

$\varphi^{*}(F)=|H(x)|\,G^{0}\cdot D$

by Theorem 6.2. Thus, $\pi^{*}(\Delta_{Y})-G\cdot\Delta_{X}$ equals

$\big{(}|H(x)|\,|C_{\pi_{0}(G)}(z)|-1\big{)}\,G^{0}\cdot D$

in a neighborhood of $x$. On the other hand, we have

${\rm div}(s_{X/Y})={\rm div}(s_{X/Z})+\varphi^{*}{\rm div}(s_{Y/Z})$

by (7.1), and

${\rm div}(s_{X/Z})=\big{(}|H(x)|-1\big{)}G^{0}\cdot D$

in a neighborhood of $x$ (Theorem 7.4), whereas

${\rm div}(s_{Y/Z})=\big{(}|C_{\pi_{0}(G)}(z)|-1\big{)}\,F$

in a neighborhood of $z$ (by the above step again). As a consequence,
$G^{0}\cdot D$ occurs in ${\rm div}(s_{X/Y})$ with multiplicity

$|H(x)|-1+|H(x)|\,\big{(}|C_{\pi_{0}(G)}(z)|-1\big{)}=|H(x)|\,|C_{\pi_{0}(G)}(z)|-1.$

So ${\rm div}(s_{X/Y})$ and $\pi^{*}(\Delta_{Y})-G\cdot\Delta_{X}$ have the
same multiplicity along $G^{0}\cdot D$.

To complete the proof, it suffices to show that $G^{0}\cdot D$ coincides with
$G\cdot D$ in a neighborhood of $x$. Since $\pi_{0}(G)$ is constant, there
exists a finite purely inseparable extension $K/k$ such that
$G_{K}=G^{0}_{K}\rtimes\pi_{0}(G)_{K}=\coprod_{g\in G(K)}G^{0}_{K}\,g$.
Therefore, we have $G_{K}\cdot D_{K}=\bigcup_{g\in G(K)}G^{0}_{K}\cdot g\cdot
D_{K}$. Moreover, $D_{K}$ is irreducible (possibly nonreduced). Denoting by
$x^{\prime}$ its generic point, the generic points of $G_{K}\cdot D_{K}$ are
exactly the $g\cdot x^{\prime}$, where $g\in G(K)$. As a consequence,
$G^{0}_{K}\cdot D_{K}$ coincides with $G_{K}\cdot D_{K}$ in a neighborhood of
$x^{\prime}$. Taking the schematic image under the projection ${\rm
pr}:X_{K}\to X$, $D_{K}\to D$, $x^{\prime}\to x$ yields the desired assertion.
∎

Next, we assume that $k$ is algebraically closed and $X$ is a curve. Then
$Y=X/G$ is a smooth curve, and $\pi:X\to Y$ is flat l. c. i. by Lemma 7.3. We
now extend parts of Theorems 1 and 2 to this setting:

######  Proposition 9.3.

  1. (i)

The group $\operatorname{Stab}_{G}(x)$ is cyclic for any $x\in X(k)$.

  2. (ii)

We have

${\rm div}(s_{X/Y})=\sum\big{(}|\operatorname{Stab}_{G}(x)|\,-1\big{)}\,G\cdot
x$

(sum over the $G$-orbits of $k$-rational points).

###### Proof.

(i) Consider again the factorization (9.1) of $\pi$, where $Z=X/G^{0}$ is a
smooth curve. Also, note that $G^{0}\simeq\boldsymbol{\mu}_{p^{r}}$ for some
$r$.

Let $x\in X(k)$ and set $z=\varphi(x)$, $y=\psi(z)=\pi(x)$. Since $k$ is
algebraically closed, we have $G=G^{0}\rtimes\pi_{0}(G)$ and
$\operatorname{Stab}_{G}(x)=\operatorname{Stab}_{G^{0}}(x)\rtimes\operatorname{Stab}_{\pi_{0}(G)}(x)$.
Moreover, $\operatorname{Stab}_{G^{0}}(x)\simeq\boldsymbol{\mu}_{p^{s}}$ for
some $s$, and
$\operatorname{Stab}_{\pi_{0}(G)}(x)=\operatorname{Stab}_{\pi_{0}(G)}(z)$ as
$\varphi$ is bijective and $\pi_{0}(G)$-equivariant. Also,
$\operatorname{Stab}_{\pi_{0}(G)}(z)$ is cyclic by Lemma 9.2 (v).

To complete the proof, it suffices to check that
$\operatorname{Stab}_{\pi_{0}(G)}$ centralizes
$\operatorname{Stab}_{G^{0}}(x)$, since both groups are cyclic of coprime
orders. Consider the subgroup scheme
$H=G^{0}\rtimes\operatorname{Stab}_{\pi_{0}(G)}(x)$ of $G$, and the
corresponding factorization $X\to X/H\to Y$ of $\pi$. Note that $X$ is an
$H$-normal curve (since $H\supset G^{0}$); in particular, $X/H$ is a smooth
curve. Also, $\operatorname{Stab}_{G}(x)\subset H$. Thus, we may replace $G$
with $H$, and assume that $\pi_{0}(G)$ _fixes_ $x$.

We now use some constructions and results from Section 5. Clearly, we have
$G\cdot x=G^{0}\cdot x$, and $G$ fixes $z$. Thus, $A={\mathcal{O}}_{X,G\cdot
x}={\mathcal{O}}_{X,x}$ is a local algebra equipped with a $G$-action, as well
as its quotient algebra $\bar{A}={\mathcal{O}}(X_{z})$. The ideal $\bar{I}$ of
$G\cdot x$ in $\bar{A}$ is $G$-stable and generated by a $G^{0}$-eigenvector
$\bar{f}$ of weight $\nu\in\Lambda(G^{0})$. The action of $\pi_{0}(G)$ on
$G^{0}$ by conjugation yields an action on $\Lambda(G^{0})$ by group
automorphisms, which stabilizes the subgroup
$\Lambda\big{(}G^{0}/\operatorname{Stab}_{G^{0}}(x)\big{)}$ (as $\pi_{0}(G)$
normalizes $\operatorname{Stab}_{G^{0}}(x)$). For any
$g\in\pi_{0}(G)(k)=G(k)$, we have $g\cdot\bar{f}=u_{g}\bar{f}$ for some
$u_{g}\in\bar{A}^{\times}$. Moreover, $g\cdot\bar{f}$ is a $G^{0}$-eigenvector
of weight $g\cdot\nu$, and hence $u_{g}$ is a $G^{0}$-eigenvector of weight
$g\cdot\nu-\nu$. By Proposition 5.1, we have
$g\nu-\nu\in\Lambda\big{(}G^{0}/\operatorname{Stab}_{G^{0}}(x)\big{)}$ and the
character group of $\operatorname{Stab}_{G^{0}}(x)$ is generated by the
restriction of $\nu$. As a consequence, $\pi_{0}(G)$ acts trivially on this
character group. Equivalently, $\pi_{0}(G)$ centralizes
$\operatorname{Stab}_{G^{0}}(x)$ as desired.

(ii) By the proof of Theorem 9.1, the orbit $G\cdot x$ (viewed as a Cartier
divisor) occurs in ${\rm div}(s_{X/Y})$ with multiplicity
$|\operatorname{Stab}_{G^{0}}(x)|\,|\operatorname{Stab}_{\pi_{0}(G)}(z)|-1$.
Moreover,
$|\operatorname{Stab}_{G^{0}}(x)|\,|\operatorname{Stab}_{\pi_{0}(G)}(z)|=|\operatorname{Stab}_{G^{0}}(x)|\,|\operatorname{Stab}_{\pi_{0}(G)}(x)|=|\operatorname{Stab}_{G}(x)|$.
∎

######  Remark 9.4.

Still assuming $k$ algebraically closed, we discuss the structure of finite
linearly reductive group schemes $G$ acting faithfully on a curve. Recall that
we have $G\simeq\boldsymbol{\mu}_{p^{r}}\rtimes H$, where $H$ is constant of
order prime to $p$. As the automorphism group of $\boldsymbol{\mu}_{p^{r}}$ is
isomorphic to
$({\mathbb{Z}}/p^{r}{\mathbb{Z}})^{\times}\simeq({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\times{\mathbb{Z}}/p^{r-1}{\mathbb{Z}}$,
it follows that the conjugation action of $G$ on $\boldsymbol{\mu}_{p^{r}}$ is
given by a character

$\chi:G\longrightarrow({\mathbb{Z}}/p{\mathbb{Z}})^{\times},$

which may be identified with the weight of the $G$-action on its Lie algebra.
In particular, we have an exact sequence of finite group schemes

$1\longrightarrow\boldsymbol{\mu}_{p^{r}}\times F\longrightarrow
G\stackrel{{\scriptstyle\chi}}{{\longrightarrow}}({\mathbb{Z}}/p{\mathbb{Z}})^{\times},$
(9.2)

where $F$ (the centralizer of $\boldsymbol{\mu}_{p^{r}}$ in $H$) is constant
of order prime to $p$.

Conversely, given such an exact sequence, does $G$ act faithfully on some
curve (which may then be taken projective and $G$-normal)? We only know a
partial answer to this question, in the case where
$G=\boldsymbol{\mu}_{p^{r}}\times F$ with $F$ as above. Then by the main
result of [MV83], there exists a smooth projective curve $X$ such that $F={\rm
Aut}(X)$. We first construct a faithful _rational_ action of
$\boldsymbol{\mu}_{p^{r}}$ on $X$ that commutes with the $F$-action.

Let $K$ be the function field of $X$, and $L=K^{F}$. Choose $t\in L\setminus
L^{p}$; then $t\in K\setminus K^{p}$ and hence we have a decomposition

$K=\bigoplus_{m=0}^{p-1}K^{p}\,t^{m},$

which is a ${\mathbb{Z}}/p{\mathbb{Z}}$-grading of the $K^{p}$-algebra $K$ by
$F$-stable subspaces. In view of [Br24, Cor. 3.4], this yields a faithful
rational action of $\boldsymbol{\mu}_{p}\times F$ on $X$, which extends the
given $F$-action. More generally, we obtain a decomposition

$K=\bigoplus_{m=0}^{p^{r}-1}K^{p^{r}}\,t^{m}$

by induction on $r$, and hence the $F$-action on $X$ extends to a faithful
rational action of $\boldsymbol{\mu}_{p^{r}}\times F$. Moreover, these actions
are compatible with the inclusions
$\boldsymbol{\mu}_{p^{r}}\subset\boldsymbol{\mu}_{p^{r+1}}$.

By loc. cit., Cor. 4.4, these rational actions can be “regularized”, i.e., for
any $r$, there exists a $\boldsymbol{\mu}_{p^{r}}\times F$-normal projective
curve $X_{r}$ which is equivariantly birationally isomorphic to $X$; then we
still have $F={\rm Aut}(X_{r})$. If $g(X)\geq 2$ then $X_{r}\neq X$, as the
automorphism group scheme ${\rm Aut}_{X}$ is étale. Under this assumption,
there exists no projective curve $Y$ that is $\boldsymbol{\mu}_{p^{r}}$-normal
for infinitely many values of $r$, and birational to $X$: otherwise, the
connected automorphism group scheme ${\rm Aut}^{0}_{Y}$ is not finite, and
hence $Y$ must have infinitely many automorphisms, a contradiction. In
particular, we obtain infinitely many pairwise nonisomorphic projective models
$X_{r}$.

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