# Galois covers of singular curves in positive characteristics

Soumyadip Das  School of Mathematics, Tata Institute of Fundamental Research,
Mumbai, Maharastra 400005<EMAIL_ADDRESS>
###### Abstract.

We study the étale fundamental groups of singular reduced connected curves
defined over an algebraically closed field of arbitrary prime characteristic.
It is shown that when the curve is projective, the étale fundamental group is
a free product of the étale fundamental group of its normalization with a free
finitely generated profinite group whose rank is well determined. As a
consequence of this result and the known results for the smooth case,
necessary conditions are given for a finite group to appear as a quotient of
the étale fundamental group. Next, we provide similar results for an affine
integral curve $U$. We provide a complete group theoretic classification on
which finite groups occur as the Galois groups for Galois étale connected
covers of $U$. In fact, when $U$ is a seminormal curve embedded in a connected
seminormal curve $X$ such that $X-U$ consists of smooth points, the tame
fundamental group $\pi_{1}^{t}(U\subset X)$ is shown to be isomorphic to a
free product of the tame fundamental group of the normalization of $U$ with a
free finitely generated profinite group whose rank is known. An analogue of
the Inertia Conjecture is also posed for certain singular curves.

###### Key words and phrases:

Étale Fundamental Group, Tame Fundamental Group, Lifting Problem for
Seminormal Curves

###### 2020 Mathematics Subject Classification:

14H30, 14G17 (Primary) 14G32, 14H20 (Secondary)

##  1\. Introduction

The main objective of this article to relate the étale fundamental group of a
reduced (singular) curve defined over an algebraically closed field of
positive characteristic with the étale fundamental group of its normalization.

Let $p$ be a prime number and $k$ be an algebraically closed field of
characteristic $p$. The étale fundamental group $\pi_{1}(X)$ of a smooth
projective connected $k$-curve is a finitely generated profinite group, but
not much is known about the structure of the group $\pi_{1}(X)$ itself. As a
finitely generated profinite group, its structure is completely determined by
its finite quotients. Due to works of Grothendieck, Shafarevich Hasse–Witt,
Nakajima, Stevenson and others (see [12, Section 7]), we know some necessary
conditions on these finite quotients. If $U\subset X$ is the affine curve
obtained by removing $r$ closed points of $X$, the étale fundamental group
$\pi_{1}(U)$ is no longer finitely generated (in contrast to what happens over
$\mathbb{C}$). In this case, we know all the finite quotients by Abhyankar’s
Conjecture on Affine Curves ([12, Conjecture 3.2], which is now a Theorem due
to Serre, Raynaud and Harbater; [12, Section 3.3]). More precisely, a finite
group $G$ occurs as a quotient of $\pi_{1}(U)$ if and only if $G/p(G)$ is
generated by at most $2g(X)+r-1$ elements (here $g(X)$ is the genus of $X$,
and $p(G)$ is the normal subgroup generated by the Sylow $p$-subgroups of
$G$). In general, this group theoretic description does not determine the
structure of the group $\pi_{1}(U)$ as we do not know how they fit into the
inverse system that defines the profinite group $\pi_{1}(U)$.

In this article, we first consider $\pi_{1}(X)$ for an arbitrary reduced
projective connected $k$-curve $X$ that is not necessarily irreducible and may
contain singular points. Let $\nu\colon X^{\nu}\longrightarrow X$ be the
normalization map. Then we have the following structure theorem for
$\pi_{1}(X)$ in terms of $\pi_{1}(X^{\nu})$ and the number of points in the
fibres of the normalization map $\nu$.

######  Theorem 1.1 (Theorem 3.3).

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $X$
be a connected projective $k$-curve having $n$ irreducible components.
Consider the normalization map $\nu\colon X^{\nu}\longrightarrow X$. Set

$\delta\coloneqq 1-n+\sum_{x\in X}\left(|\nu^{-1}(x)|-1\right).$

We have an isomorphism of finitely generated profinite groups:

$\pi_{1}(X)=\pi_{1}(X^{\nu})\ast\widehat{F_{\delta}}$

where $\widehat{F_{\delta}}$ is the profinite completion of a free group
$F_{\delta}$ on $\delta$ generators.

Note that the summation in the above definition of $\delta$ can be taken over
the singular points of $X$. To prove the result, we first reduce to the case
where $X$ is a seminormal curve (Lemma 3.2). As a consequence of [14], the
lifting problem for coves of $X$ is solvable (Proposition 2.6). It follows
that there is a complex projective connected seminormal curve $X_{\mathbb{C}}$
satisfying: (1) $X_{\mathbb{C}}$ has $n$ irreducible components; (2) The above
defined number $\delta$ for $X_{\mathbb{C}}$ is the same as $\delta$ for $X$;
(3) (see (3.1)) there is a surjective specialization map $\text{\rm
sp}\colon\pi_{1}(X_{\mathbb{C}})\twoheadrightarrow\pi_{1}(X)$. We have a
complete description of $\pi_{1}(X_{\mathbb{C}})$ by the Riemann Existence
Theorem ([24, Exp XII; Corollary 5.2]) and a topological argument. This shows
that $\pi_{1}(X)$ is a finitely generated profinite group. Further using the
surjectivity of the above map sp and the specialization map in the smooth
case, we establish that $\pi_{1}(X)$ is a quotient of
$\pi_{1}(X^{\nu})\ast\widehat{F_{\delta}}$. We then prove that there is also a
surjection
$\pi_{1}(X)\twoheadrightarrow\pi_{1}(X^{\nu})\ast\widehat{F_{\delta}}$. To see
this, we first note that (Remark 2.9) $X$ is obtained from $X^{\nu}$ by
identifying two points at a time. Then the result follows by inductively
applying Proposition 3.4 and Proposition 3.5. Roughly, these propositions
allow us to construct connected Galois covers of $X$ starting from connected
Galois covers of $X^{\nu}$. Here we suitably identify points of the induced
covers to obtain the final cover. This is managed by use of Lemma 2.10. Let us
mention that the above propositions preserve the ramification behavior of the
covers (Remark 3.6), and up to some finitely many choices involving the coset
representatives and finitely many points, this method canonically constructs
all the possible connected Galois covers of $X$ from those of $X^{\nu}$ (see
Remark 3.7).

As an application of Theorem 1.1 together with the knowledge from the smooth
case, we easily establish some important necessary conditions for a finite
group $G$ to be realized as a quotient of the group $\pi_{1}(X)$ (see
Corollary 3.8).

Next, we consider the case when $U$ is a connected affine singular $k$-curve.
For simplicity, we also assume that $U$ is irreducible. Our first main result
gives a a complete group theoretic classification for finite (continuous)
quotients of $\pi_{1}(U)$, analogous to the smooth case. Recall that for a
finite group $G$, (necessarily normal, quasi $p$-subgroup) $p(G)$ is the
subgroup generated by the Sylow $p$-subgroups of $G$.

######  Theorem 1.2 (Theorem 4.3).

Let $k$ be any algebraically closed field of arbitrary characteristic and $U$
be an integral affine $k$-curve. Let $X^{\nu}$ be the unique (up to
isomorphism) smooth projective connected $k$-curve with function field
$k(X^{\nu})=k(U)$. Let $U^{\nu}\subset X^{\nu}$ be the maximal affine open
subset together with a finite surjective morphism $\nu\colon
U^{\nu}\longrightarrow U$ that is an isomorphism over the smooth points of
$U$. Suppose that $X^{\nu}$ has genus $g$, $r=|X^{\nu}-U^{\nu}|$ and set

$\delta\coloneqq\sum_{u\in U}\left(|\nu^{-1}(u)|-1\right).$

A finite group $G$ occurs as a (continuous) quotient of the étale fundamental
group $\pi_{1}(U)$ if and only if $G/p(G)$ is generated by at most
$2g+r-1+\delta$ elements.

As in the projective case, we first reduce the problem to: $U$ is seminormal.
Additionally, we can always embed $U$ in a seminormal irreducible curve $X$ so
that $X-U$ contains $r$ smooth points. We then see (as a consequence of a
local-global principle in Proposition 4.2) that for a finite (continuous)
group quotient $\pi_{1}(U)\twoheadrightarrow G$, the prime-to-$p$ quotient
group $G/p(G)$ can be realized as the Galois group of a connected cover of
$U_{\mathbb{C}}$, where $U_{\mathbb{C}}$ is a complex connected affine curve
obtained by removing $r$ smooth points from the curve $X_{\mathbb{C}}$
described earlier. A complete (well known) realization of
$\pi_{1}(U_{\mathbb{C}})$ then produces the necessary condition on $G$.
Conversely, for any $G$ satisfying the condition, we construct a cover of $X$
that is étale over $U$. This is done using Proposition 3.4 and Proposition 3.5
together with formal patching results.

Additionally (under the notation of Theorem 1.2), using arguments already
developed for the affine and the projective cases, we see that the tame
fundamental group $\pi_{1}^{t}(U\subset X)$ for the above seminormal curves
$U\subset X$ can be related with the usual tame fundamental group
$\pi_{1}^{t}(U^{\nu})$:

  1. Theorem 4.4: $\pi_{1}^{t}(U\subset X)=\pi_{1}^{t}(U^{\nu})\ast\widehat{F_{\delta}}$.

Finally, we pose Question 4.7 similar to the generalization of the Inertia
Conjecture (see [4, Section 6]). As there is not much known about the status
of it, we do not venture into the details. Remark 4.8 shows an immediate
consequence of Theorem 3.3 and formal patching to answer a weaker version of
the general question.

The structure of the paper is as follows. Section § 2 consists of the
definitions and notation we will need; it is fairly self contained. The notion
of obtaining curves by identifying effective divisors of a normal curve is
recalled and studied briefly in Section § 2.2. The seminormal curves and their
lifting problems are the content of Section 2.3. Our main results on the étale
fundamental groups are discussed in Section § 3.1 and Section 4.1. The
consequence of these results are in Section § 3.2 and in Section § 4.2.

### Acknowledgements

I am indebted to Najmuddin Fakhruddin for discussions and insight; especially
for suggesting the statement of our main theorem in the projective case and
the use of seminormal curves.

##  2\. Preliminaries

###  2.1. Notation, Convention and Definitions

We use the following notation and convention throughout this article.

  1. (1)

We will work over an algebraically closed field $k$; our main interest is when
$\text{\rm char}(k)$ is an arbitrary prime $p$. In this case, we let $R$ be a
complete discrete valuation ring with residue field $k$ and fraction field
$K=\text{\rm Fr}(R)$.

  2. (2)

All the $k$-curves are assumed to be reduced.

  3. (3)

An $R$-curve is a reduced scheme $X_{R}\longrightarrow\text{\rm Spec}(R)$ that
is a flat, separated scheme of finite type over $\text{\rm Spec}(R)$ of
relative dimension one whose generic geometric fibres are connected. For an
$R$-curve $X_{R}$ and any ring homomorphism $R\rightarrow S$, we denote the
base change $X_{R}\times_{R}S$ over $\text{\rm Spec}(S)$ by $X_{R}$. In
particular, the special fibre $X_{R}\times_{R}k$ of $X_{R}$ is denoted by
$X_{k}$ and the generic fibre $X_{R}\times_{R}K$ by $X_{K}$.

  4. (4)

For a finite group $G$, let $p(G)$ denote the (necessarily normal) subgroup of
$G$ generated by all the Sylow $p$-subgroups of $G$. A finite group $G$ is
said to be a quasi $p$-group if $p(G)=G$.

  5. (5)

For any scheme $X$ and a point $x\in X$, the local ring at $x$ is denoted by
$\mathcal{O}_{X,x}$. We denote its completion at the maximal ideal by
$\widehat{\mathcal{O}}_{X,x}$. When $\widehat{\mathcal{O}}_{X,x}$ is a domain,
$K_{X,x}$ stands for its fraction field.

  6. (6)

For any reduced scheme $X$, we denote its normalization as $\nu\colon
X^{\nu}\longrightarrow X$.

Let $X$ be a reduced scheme. A cover of $X$ is defined to be a finite,
generically separable morphism $Y\longrightarrow X$. Let $G$ be a finite
group. A $G$-Galois cover is a cover $Y\longrightarrow X$ of reduced schemes
together with an inclusion $\rho\colon G\hookrightarrow\text{\rm Aut}_{X}(Y)$
such that $G$ acts simply transitively on each generic geometric fibre. Note
that the above inclusion is always an isomorphism if $X$ is an integral
scheme.

Let $S$ be a locally Noetherian connected reduced scheme. Let
$a\colon\text{\rm Spec}\left(\Omega\right)\longrightarrow S$ be any geometric
point. The étale fundamental group $\pi_{1}(S,a)$ of $S$ at $a$ was defined in
[24, Exp. V, Section 7, Page 140]. The category $\text{FEt}_{S}$ of finite
étale covers of $S$ is a Galois category, equipped with a fibre functor
$F\colon\text{FEt}_{S}\longrightarrow\left(\text{ finite sets}\right)$ that to
a finite étale cover $T\longrightarrow S$ assigns the set of geometric points
$\text{\rm Spec}(\Omega)\longrightarrow T$ lying over $a$. Then
$\pi_{1}(S,a)=\text{\rm Aut}(F)$, and $F$ defines an equivalence of the
category $\text{\rm FEt}_{S}$ of finite étale covers of $S$ with $\text{\rm
Finite-}\pi_{1}(S,a)\text{\rm-sets}$. For a different choice of generic point
$a^{\prime}$, there is an automorphism
$\pi_{1}(S,a)\cong\pi_{1}(S,a^{\prime})$, canonical up to inner automorphisms.
As our interest is the structure of group $\pi_{1}(S,a)$ itself, we drop the
geometric point from notation and only consider the group $\pi_{1}(S)$, up to
isomorphisms. When $S$ is an integral normal scheme, there is an ind-scheme
$\widetilde{S}$, étale over $S$, and the group $\pi_{1}(S)$ can be canonically
identified with $\text{\rm Aut}\left(\widetilde{S}/S\right)$. In general,
$\pi_{1}(S)$ is a profinite group that is a limit of its finite quotients.
Using the properties of a Galois category, we have

$\pi_{1}(S)=\underset{\begin{subarray}{c}T\longrightarrow S\\\ \text{ finite
Galois \'{e}tale cover}\end{subarray}}{\varprojlim}\text{ \rm
Aut}\left(T/S\right).$

By the general version of Riemann Existence Theorem ([24, Exp XII, Corollary
5.2, page 337]), when $k=\mathbb{C}$, the étale fundamental group $\pi_{1}(X)$
is the profinite completion of the topological fundamental group
$\pi_{1}^{\text{\rm top}}(X(\mathbb{C}))$. We refer to [12] for a survey on
the structure of $\pi_{1}(X)$ and its quotients, where $X$ is a smooth
$k$-curve.

A lot of these structure theorems in positive characteristic are obtained by
first relating to the étale fundamental group of an ‘analogous’ complex curve
which is then understood from topology. This leads us to the lifting problems.
Assume that $\text{\rm char}(k)=p>0$. Let $R$ be a complete discrete valuation
ring with residue field $k$ and its fraction field $K=\text{\rm Fr}(R)$ with
$\text{\rm char}(K)=0$. Let $X$ be a connected $k$-curve. A lift of $X$ over
$\text{\rm Spec}(R)$ is an $R$-curve $X_{R}$ with special fibre
$X_{R}\times_{R}k=X$. The usual lifting problem asks whether the curve $X$
lifts over $\text{\rm Spec}(R)$. We also have the lifting problem for finite
covers of $X$: let $f\colon Z\longrightarrow X$ be a finite cover of connected
$k$-curves. A lift of $f$ over $\text{\rm Spec}(R)$ is a finite cover
$f_{R}\colon Y_{R}\longrightarrow X_{R}$ of $R$-curves such that the special
fibre $f_{R}\times_{R}k\colon Y_{R}\times_{R}k\longrightarrow
X_{R}\times_{R}k$ of the map $f_{R}$ is $f$. Similarly, there are local
lifting problems for a complete discrete valuation ring and its extensions.
The solution to such problems are well studied when $X$ is a smooth curve. The
usual lifting problem for $X$ and the lifting problem for (Galois) covers of
$X$ can always be solved (see [13, Corollary 22.2, page 145] and [18, Theorem
1.5]). We refer to [18] and [17] for a detailed exposition on the lifting
problems for smooth curves and on the local lifting problems. In this article,
we deal with lifting of singular curves and of their étale covers. This
problem is open in general ([13, References and Further Results, page 148]).
As a consequence of [14], we will see in Proposition 2.6 that the usual and
the étale lifting problem for a seminormal curve has a solution.

###  2.2. Rosenlicht-Serre Curves

This section is devoted to the singular curves ‘defined by modulus’ in [26,
Chapter IV, Section 4, page 61–62]. Let $k$ be an algebraically closed field
of arbitrary characteristic.

Let $X$ be a (not necessarily irreducible) projective $k$-curve and
$X_{\text{Sing}}\subset X$ be its singular locus. Let $\nu\colon
X^{\nu}\longrightarrow X$ denote the normalization map. Then $X^{\nu}$ is a
smooth projective curve, and $\nu$ is a birational finite morphism that
induces an isomorphism $X^{\nu}-\nu^{-1}(X_{\text{Sing}})\cong
X-X_{\text{Sing}}$. For each closed point $x\in X$, we have an inclusion of
$k$-algebras (see [26, Chapter IV, Section 1 and 2, page 58–59])

(2.1)
$\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x}=\bigcap_{\nu(y)=x}\mathcal{O}_{X^{\nu},y}\supset
k+J_{x}\supset\mathcal{O}_{X,x}\supset k+\mathfrak{c}_{x}$

where $J_{x}$ is the Jacobson radical of the semi-local ring
$\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x}$ and $\mathfrak{c}_{x}$ is the
conductor ideal $\text{\rm
Ann}\left(\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x}/\mathcal{O}_{X,x}\right)$.
Note that $\mathfrak{c}_{x}$ is characterized by the fact that this is the
largest ideal of $\mathcal{O}_{X,x}$ that is also an ideal of
$\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x}$.

On the other hand, a curve $X$ is constructed from its normalization $X^{\nu}$
in [20] (see [26, Chapter IV, Section 3 and 4, page 60–62]). Let $X^{\nu}$ be
a smooth projective $k$-curve. Let $D_{1},\,\ldots,\,D_{l}$ be effective
(Weil) divisors on $X^{\nu}$ with pairwise disjoint support where
$D_{i}=\sum_{y\in\text{\rm Supp}\left(D_{i}\right)}n_{y}y$, $1\leq i\leq l$.
Suppose that for each $i$, $\text{\rm
deg}\left(D_{i}\right)=\sum_{y\in\text{\rm Supp}\left(D_{i}\right)}n_{y}\geq
2$. Consider the topological space

$X=\left(X^{\nu}-\bigsqcup_{1\leq i\leq l}\text{\rm
Supp}\left(D_{i}\right)\right)\sqcup\\{x_{1},\,\ldots,\,x_{l}\\}$

equipped with cofinite topology. Let $\nu\colon X^{\nu}\longrightarrow X$ be
the topological quotient map. We endow $X$ with a sheaf of rings
$\mathcal{O}_{X}$ such that

$\mathcal{O}_{X,x}=\begin{cases}\mathcal{O}_{X^{\nu},\nu^{-1}(x)}&\text{\rm if
\,}x\neq x_{i}\,\text{\rm for any }1\leq i\leq l\\\
k+\mathfrak{c}_{i}&\text{\rm if \,}x=x_{i},\,1\leq i\leq l\end{cases}$

where $\mathfrak{c}_{i}$ is the ideal of $\bigcap_{y\in\text{\rm
Supp}\left(D_{i}\right)}\mathcal{O}_{X^{\nu},y}$ formed by functions $f$ such
that $f\equiv 0\pmod{D_{i}}$, i.e. $\text{\rm v}_{y}(f)\geq n_{y}$ for all
points $y\in\text{\rm Supp}\left(D_{i}\right)$, $\text{\rm v}_{y}$ being the
valuation attached to $y$. By [26, Chapter IV, Proposition 2, page 60], the
sheaf $\mathcal{O}_{X}$ endows $X$ with a structure of an algebraic curve
having $\nu\colon X^{\nu}\longrightarrow X$ as the normalization map.
Moreover, (loc. cit. Section 4) the singular locus of $X$ is given by
$X_{\text{\rm Sing}}=\\{x_{1},\,\ldots,\,x_{l}\\}$ and $\mathfrak{c}_{i}$ is
the conductor ideal $\mathfrak{c}_{x_{i}}=\text{\rm
Ann}\left(\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x_{i}}/\mathcal{O}_{X,x_{i}}\right)$.

We have the following alternative description for $X$. For any closed point
$y\in X^{\nu}$, let $\mathfrak{m}_{y}$ denote the maximal ideal of
$\mathcal{O}_{X^{\nu},y}$. Let $Y^{\nu}\coloneqq
X^{\nu}(D_{1}+\cdots+D_{l})$111in the notation of [7, 21.7.1, page 277] be the
closed subscheme of $X$ defined by the image of
$\bigsqcup_{\begin{subarray}{c}1\leq i\leq
l\end{subarray}}\bigsqcup_{\begin{subarray}{c}y\in\text{\rm
Supp}\left(D_{i}\right)\end{subarray}}\text{Spec}\left(\mathcal{O}_{X^{\nu},y}/\mathfrak{m}_{y}^{n_{y}}\right)$
under the canonical morphism to $X^{\nu}$. The ringed space of amalgamated
union $X^{\nu}\bigsqcup_{Y^{\nu}}\\{x_{1},\,\ldots,\,x_{l}\\}$ is a scheme,
identified with $X$ via the following cocartesian (as well as cartesian; [6,
Theorem 7.1]) diagram

(*) ${Y^{\nu}}$${X^{\nu}}$${\bigsqcup_{1\leq i\leq l}\text{\rm
Spec}\left(k\right)=\\{x_{1},\ldots,x_{l}\\}}$${X}$$\scriptstyle{\nu}$

The map $\nu$ is a finite birational morphism inducing isomorphism between
$X^{\nu}-Y^{\nu}$ and $X-\\{x_{1},\ldots,x_{l}\\}$, and the natural map
$X^{\nu}\sqcup\\{x_{1},\ldots,x_{l}\\}\longrightarrow X$ is a surjective
morphism that is a quotient map of the underlying topological spaces. As we
will be dealing with such curves throughout this article, we make the
following (non-standard) definition.

######  Definition 2.1.

A curve $X$ is said to be a Rosenlicht-Serre curve (an RS curve) if $X$ is
obtained as the cocartesian diagram (* ‣ 2.2) above from a smooth projective
curve $X^{\nu}$ and a set $\\{D_{1},\,\ldots,\,D_{l}\\}$ of effective divisors
with pairwise disjoint supports. For convenience, we will denote
$X=(X^{\nu},\\{D_{i}\\}_{1\leq i\leq l})$.

Observe that $X$ is projective as $\nu$ is the normalization map and $X^{\nu}$
is projective. In general, an RS curve need not be connected. Such a curve $X$
is connected if and only if the set $\bigcup_{1\leq i\leq l}\text{\rm
Supp}\left(D_{i}\right)$ contains at least one point from each irreducible
component of $X^{\nu}$.

As every RS curve is defined using a cocartesian diagram (which are also
cartesian), we define admissible morphisms of RS curves as the morphisms of
cocartesian diagrams.

######  Definition 2.2.

Suppose that $Z=(Z^{\nu},\\{E_{j}\\}_{j\in J})$ and
$X=(X^{\nu},\\{D_{i}\\}_{i\in I})$ be two RS curves. An admissible morphism
$Z\longrightarrow X$ is defined to be a pair $(f^{\nu},f_{\rm sing})$ where
$f^{\nu}\colon Z^{\nu}\longrightarrow X^{\nu}$ is a finite morphism of smooth
projective $k$-curves, $f_{\rm sing}\colon Z_{\rm sing}=\\{z_{j}\\}_{j\in
J}\longrightarrow\\{x_{i}\\}_{i\in I}=X_{\rm sing}$ is a (necessarily finite)
surjective morphism of the corresponding singular loci satisfying the
following property.

  * •

For each $j\in J$, $f$ induces a finite morphism
$Z^{\nu}(E_{j})\longrightarrow X^{\nu}(D_{i})$ of schemes where $f_{\rm
sing}(z_{j})=x_{i}$.

Here $X^{\nu}(D_{i})$ (respectively, $Z^{\nu}(E_{j})$) is the closed subscheme
of $X^{\nu}$ (respectively, of $Z^{\nu})$ corresponding to the effective
divisor $D_{i}$ (respectively, $E_{j}$); see [7, 21.7.1, page 277].

By the universal property of a cocartesian diagram, we see that each
admissible morphism $(f^{\nu},f_{\rm sing})$ of RS curves defines a unique
finite surjective morphism $f$ of the corresponding singular curves. Moreover,
if $f^{\nu}$ is a finite cover, so is $f$. Conversely, suppose that $f\colon
Z\longrightarrow X$ be a finite surjective morphism of RS curves. Then $f$
induces a finite morphism $f^{\nu}\colon Z^{\nu}\longrightarrow X^{\nu}$.
Since the preimage of a smooth point in $X$ consists of smooth points in $Z$,
the map $f$ also induces a finite map $f_{\rm sing}\colon Z_{\rm
sing}\longrightarrow X_{\rm sing}$ between finite sets of closed points, but
this map need not be surjective. If we additionally assume that either $f$ is
an étale cover or $f$ is a cover that is étale over the singular points of
$X$, then the induced map $f_{\rm sing}$ is surjective. In this case, by the
universal property of cartesian diagrams, we obtain an admissible finite cover
$(f^{\nu},f_{\rm sing})$.

######  Remark 2.3.

Note that the association $(f^{\nu},f_{\rm sing})\mapsto f$ is a bijective
correspondence between the finite admissible covers of RS curves and the
finite covers of corresponding singular curves inducing surjection of the
singular loci.

###  2.3. Lifting of covers of Seminormal Curves

Let $k$ be an algebraically closed field of arbitrary characteristic. We start
by recalling the algebraic notion of seminormal rings (see [30] for an
exposition). For a ring $B$, let $J(B)$ denote the Jacobson radical of $B$.
For any point $x\in\text{Spec}(B)$, let $B_{x}$ denote the localization of $B$
at the prime ideal $\mathfrak{p}_{x}$ corresponding to the point $x$. We
recall the following definition of seminormality due to Traverso.

######  Definition 2.4 ([30, Definition 2.4, page 447]).

Let $A\subset B$ be an extension of rings. The seminormalization ${}^{+}_{B}A$
of $A$ in $B$ is defined to be the ring

${}^{+}_{B}A\coloneqq\\{b\in B\,|\,b_{x}\in A_{x}+J(B_{x})\text{ for all
}x\in\text{Spec}(A)\\}$

For a Mori ring $A$ (i.e., $A$ is a reduced ring $A$ and the integral closure
$\bar{A}$ of $A$ in its total ring of fractions is finite as an $A$-module),
the seminormalization $A^{+}$ of $A$ is defined to be the seminormalization
${}^{+}_{\bar{A}}A$ of $A$ in $\bar{A}$. In this case, $A$ is said to be a
seminormal ring if $A=A^{+}$.

Let $A$ be a reduced excellent ring. Let $\bar{A}$ be the integral closure of
$A$ in its total ring of fractions. Then the canonical ring extension
$A\subset\bar{A}$ factors uniquely as $A\subset A^{+}\subset\bar{A}$.
Moreover, there is no proper subintegral extension contained in
$A^{+}\subset\bar{A}$, i.e. if $A^{+}\subsetneq C\subseteq\bar{A}$ so that the
associated map
$\text{Spec}\left(C\right)\longrightarrow\text{Spec}\left(A^{+}\right)$ is a
bijection inducing isomorphisms on the residue fields, we have $C=\bar{A}$.

######  Definition 2.5.

Let $X$ be a (reduced) $k$-curve. We say that $X$ is seminormal if for each
point $x\in X$, the local ring $\mathcal{O}_{X,x}$ at $x$ is a seminormal
ring.

Now we see that every seminormal projective $k$-curve is an RS curve (Section
2.2). Let $X$ be any projective $k$-curve and $\nu\colon
X^{\nu}\longrightarrow X$ be the normalization map. As in Equation (2.1),
there is an inclusion of $k$-algebras

$A_{x}\coloneqq\left(\nu_{*}\mathcal{O}_{X^{\nu}}\right)_{x}\supset
k+J(A_{x})\supset\mathcal{O}_{X,x}$

for each point $x\in X$. Here
$A_{x}=\bigcap_{\nu(y)=x}\mathcal{O}_{X^{\nu},y}$ is the integral closure of
$\mathcal{O}_{X,x}$ in its total ring of fractions. By Definition 2.4, the
seminormalization of $\mathcal{O}_{X,x}$ is the ring

$\mathcal{O}_{X,x}^{+}=\\{a\in
A_{x}\,|\,a_{x}\in\mathcal{O}_{X,x}+J(A_{x})\\}.$

Now suppose that $X$ is a seminormal curve. Then for each point $x\in X$, we
have $\mathcal{O}_{X,x}=\mathcal{O}_{X,x}^{+}=k+J(A_{x})$. As the conductor
ideal $\mathfrak{c}_{x}$ is the largest common ideal of $\mathcal{O}_{X,x}$
and $A_{x}$, we have $\mathfrak{c}_{x}=J(A_{x})$. So for each point $x\in X$,
the pullback $\nu^{*}(x)$ is a reduced divisor
$\nu^{*}(x)=\nu^{-1}(x)=\sum_{\nu(y)=x}y$ on $X^{\nu}$, and we identify the
seminormal projective $k$-curve $X$ with the amalgamated union

(2.2) $X=X^{\nu}\sqcup_{\left(\bigsqcup_{x\in X_{\text{\rm
sing}}}\nu^{-1}(x)\right)}{X_{\text{\rm sing}}}.$

Now we study the lifting problems (see Section § 2.1) for seminormal curves.
As mentioned earlier, a solution to a lifting problem for an arbitrary
singular curve is not known. As a consequence of [14], we see that the étale
lifting problem for a seminormal curve has a solution.

######  Proposition 2.6.

Let $X$ be a connected projective seminormal $k$-curve. Let $R$ be a complete
discrete valuation ring with residue field $k$ and fraction field $K$. Let
$X_{\text{\rm sing}}=\\{x_{1},\cdots,x_{l}\\}$ be the singular locus of $X$,
$l\geq 1$. Consider the normalization $\nu\colon X^{\nu}\longrightarrow X$ and
a lift $X_{R}^{\nu}$ of $X^{\nu}$ to a smooth projective $R$-curve. Then there
is a connected lift $X_{R}$ of $X$ over $\text{\rm Spec}(R)$ with
geometrically connected fibres and a finite morphism $\nu_{R}\colon
X_{R}^{\nu}\longrightarrow X_{R}$ such that the special fibre
$\nu_{R}\times_{R}k$ is the map $\nu$, and the generic fibre
$\nu_{K}\coloneqq\nu_{R}\times_{R}K\colon X_{K}^{\nu}\longrightarrow X_{K}$
has the following properties for any algebraic closure $\bar{K}$ of $K$.

  1. (1)

The curve $X_{K}^{\nu}$ is a smooth projective $K$-curve; in particular, the
geometric fibre $X^{\nu}_{\bar{K}}\coloneqq X_{K}^{\nu}\times_{R}\bar{K}$ is a
smooth projective $\bar{K}$-curve. The irreducible components of
$X^{\nu}_{\bar{K}}$ are in a genus preserving bijective correspondence with
the irreducible components of $X^{\nu}$.

  2. (2)

The curve $X_{\bar{K}}\coloneqq X_{K}\times_{K}\bar{K}$ is a connected
seminormal projective $\bar{K}$-curve.

  3. (3)

The map $\nu_{\bar{K}}=\nu_{K}\times_{K}\bar{K}\colon
X^{\nu}_{\bar{K}}\longrightarrow X_{\bar{K}}$ is the normalization map.

  4. (4)

There is a bijection $\phi\colon X_{\text{\rm
sing}}\longrightarrow\left(X_{\bar{K}}\right)_{\text{\rm sing}}$ such that for
each $x\in X_{\text{\rm sing}}$,

$|\nu^{-1}(x)|=|(\nu_{\bar{K}})^{-1}(\phi(x))|.$

Moreover, any finite étale cover $f\colon Z\longrightarrow X$ of connected
$k$-curves lifts to a finite étale cover $f_{R}$ of $X_{R}$. The map $f_{R}$
is Galois with group $G$ if $f$ is so.

###### Proof.

Since $X$ is assumed to be seminormal, it is an RS curve defined by the smooth
projective curve $X^{\nu}$ and the reduced effective divisors
$\left\\{D_{x}=\nu^{-1}(x)\right\\}_{x\in X_{\text{\rm sing}}}$. As $R$ is
strictly henselian, the specialization map $X_{R}^{\nu}(R)\longrightarrow
X^{\nu}(k)$ is surjective. For each $y\in X^{\nu}$ lying over a singular point
of $X$, choose a section $s_{y}\colon\text{\rm
Spec}\left(R\right)\longrightarrow X_{R}^{\nu}$ corresponding to $y$. Let
$Y_{R}$ denote the closed subscheme of $X_{R}^{\nu}$ defined by the sections
$s_{y}$ as above. By [14, Lemma 3], we have the following cocartesian diagram

${Y_{R}}$${X_{R}^{\nu}}$${\bigsqcup_{x\in X_{\text{\rm sing}}}\text{\rm
Spec}\left(R\right)}$${X_{R}}$$\scriptstyle{\nu_{R}}$

where $X_{R}$ is a lift of $X$ over $\text{\rm Spec}(R)$. Since the special
fibre $X$ is connected, the $R$-curve $X_{R}$ is connected as well. By
flatness of $X^{\nu}_{R}$, we have (1). The closed subscheme $\nu_{R}(Y_{R})$
in $X_{R}$ is defined by a positive reduced cycle of codimension one ([7,
21.7.2, page 277]). As its closed fibre $X_{\rm sing}$ consists of finitely
many points, $\nu_{R}(Y_{R})=\sum_{x\in X_{\rm sing}}x_{R}$ where
$x_{R}\colon\text{Spec}(R)\longrightarrow X_{R}$ has special fibre $x\in
X_{\rm sing}$.

To see the other properties of the geometric generic fibre, note that we have
the cocartesian and cartesian diagram ([6, Theorem 7.1]):

${Y_{\bar{K}}}$${X_{\bar{K}}^{\nu}}$${\\{x_{\bar{K}}\\}_{x\in X_{\text{\rm
sing}}}=\bigsqcup_{x\in X_{\text{\rm sing}}}\text{\rm
Spec}\left(\bar{K}\right)}$${X_{\bar{K}}}$$\scriptstyle{\nu_{\bar{K}}}$

Here $Y_{\bar{K}}$ is the geometric generic fibre of $Y_{R}$ that is the same
as the closed subscheme of $X^{\nu}_{\bar{K}}$ defined by the generic
geometric fibres of the sections $s_{y}$, $\nu(y)\in X_{\rm sing}$, and each
point $x_{\bar{K}}$ is the generic geometric fibre of $x_{R}$. So
$X_{\bar{K}}$ has the singular locus $\left(X_{\bar{K}}\right)_{\rm
sing}=\\{x_{\bar{K}}\\}_{x\in X_{\rm sing}}$, and $X_{\bar{K}}$ is identified
as the amalgamated union
$X_{\bar{K}}^{\nu}\sqcup_{Y_{\bar{K}}}\left(X_{\bar{K}}\right)_{\rm sing}$.
Now the properties (2)–(4) are straight forward to check.

If $f\colon Z\longrightarrow X$ is a finite étale cover of connected
$k$-curves, a lift $f_{R}$ of $f$ over $\text{\rm Spec}(R)$ exists by [13,
Proposition 22.5, page 147]. Using a general argument by Grothendieck’s
Existence Theorem ([13, Theorem 21.2, page 140]) as in [18, Proof of Theorem
3.1], if $f$ is $G$-Galois, then $f_{R}$ is Galois with group $G$ as well. ∎

######  Remark 2.7.

In the above proposition, we obtained a lift $X_{R}$ for the seminormal curve
$X$. In general, a lift is not unique (when it exists). For example, for a
semistable curve $X$, there exists a lift $X^{\prime}_{R}$ with smooth generic
fibre (see [21, Theorem 4.2]). For our purpose, we want the generic geometric
fibre of the lift to have similar properties as the curve $X$ near the
singular points.

######  Remark 2.8.

Since $X$ is a proper $R$-curve, the specialization functor $\text{\rm
FEt}_{X_{R}}\longrightarrow\text{\rm FEt}_{X}$ is an equivalence of categories
by [23, Lemma 0A48].

We end this section with a construction of seminormal curves (see (2.3) below)
obtained by identifying points (extending the notion of Section § 2.2) and a
descent for covers under an identification map (Lemma 2.10). Once again, we
work over an algebraically closed field $k$ of arbitrary characteristic.

Let $Y$ be a seminormal curve and ${\sim}$ be an equivalence relation on the
closed points of $Y$ such that the set $\mathcal{C}_{{\sim}}$ of non-trivial
equivalence classes is finite and non-trivial. In the following, we obtain a
seminormal $k$-curve $Y/{\sim}$ together with a finite birational morphism

(2.3) $q\colon Y\longrightarrow Y/{\sim}$

that induces an isomorphism restricted to
$Y-\bigsqcup_{\begin{subarray}{c}y\in C\\\
C\in\mathcal{C}_{\sim}\end{subarray}}y$, and for any point $y\in C$
($C\in\mathcal{C}_{{\sim}}$), $q^{*}(q(y))$ is the reduced divisor
$\sum_{y^{\prime}\in C}y^{\prime}$ on $Y$.

Suppose that $Y_{\rm sing}=\\{y_{1},\ldots,y_{l}\\}$. Since $Y$ is seminormal,
it is an RS curve

$Y=Y^{\nu}\sqcup_{\left(\bigsqcup_{1\leq i\leq
l}\nu^{-1}(y_{i})\right)}\\{y_{1},\ldots,y_{l}\\}\hskip 20.00003pt\text{(see
\eqref{eq_sminormal_is_RS})}$

obtained from its normalization $\nu\colon Y^{\nu}\longrightarrow Y$ and the
finite set of reduced effective divisors $\\{D_{i}=\nu^{-1}(y_{i})\\}_{1\leq
i\leq l}$ on $Y^{\nu}$. For each $C\in\mathcal{C}_{{\sim}}$, consider the
reduced effective divisor $E_{C}=\sum_{y\in C}\nu^{-1}(y)$. Let $\Lambda$ be
the indexing set obtained from $\\{1,\ldots,l\\}$ by removing all $i$ such
that $y_{i}\in C$ for some $C\in\mathcal{C}_{{\sim}}$. We define $Y/{\sim}$ as
the RS curve

(2.4) $Y/{\sim}\coloneqq Y^{\nu}\sqcup_{Y^{\prime}}\left(Y/{\sim}\right)_{\rm
sing}$

where
$Y^{\prime}=\sqcup_{i\in\Lambda}\nu^{-1}(y_{i})\bigsqcup\sqcup_{\begin{subarray}{c}y\in
C\\\ C\in\mathcal{C}_{{\sim}}\end{subarray}}\nu^{-1}(y)$. Since each $E_{C}$
and $D_{j}$ are reduced effective divisors having pairwise disjoint support,
the curve $Y/{\sim}$ is a seminormal curve and $Y^{\nu}\longrightarrow
Y/{\sim}$ is the normalization map. Since we have a factorization
$Y^{\nu}\longrightarrow Y\overset{q}{\longrightarrow}Y/{\sim}$, the statements
about the map $q$ follows.

######  Remark 2.9.

Let $X$ be a connected seminormal curve over an algebraically closed field of
arbitrary characteristic. The normalization map $\nu\colon
X^{\nu}\longrightarrow X$ factors as a composition

(2.5) $\nu\colon X^{\nu}=Y_{t}\longrightarrow
Y_{t-1}\longrightarrow\cdots\longrightarrow Y_{1}\longrightarrow Y_{0}=X$

where each $Y_{i}$ is a seminormal curve, and $Y_{i}$ is obtained from
$Y_{i+1}$ by identifying two distinct closed points. In other words, there are
points $y_{1},\,y_{2}\in Y_{i+1}$ such that $Y_{i}=Y_{i+1}/{\sim}$ and the
only non-trivial equivalence class of the equivalence relation ${\sim}$ is
$\\{y_{1},y_{2}\\}$.

In the following, we see that a finite cover of curves descent under
identification maps under a natural condition. This result will be used in
ours proofs of the main results in Section § 3.1.

######  Lemma 2.10.

Let $f\colon Z\longrightarrow Y$ be a finite surjective cover of seminormal
$k$-curves that is étale over the singular points of $Y$. Let ${\sim}$ and
${\sim}^{\prime}$ be two equivalence relations on the closed points of $Y$ and
$Z$, respectively, such that the set of non-trivial equivalence classes
$\mathcal{C}_{{\sim}}$ and $\mathcal{C}_{{\sim}^{\prime}}$ are both non-empty
finite sets. Additionally, suppose that the following hold.

  1. (1)

The relations preserve $f$, i.e. $z{\sim}^{\prime}z^{\prime}$ implies
$f(z){\sim}f(z^{\prime})$.

  2. (2)

For any $C\in\mathcal{C}_{{\sim}}$, the set $f^{-1}(C)$ has a partition by
sets in $\mathcal{C}_{{\sim}^{\prime}}$ having the same size as $C$.

Then $f$ descends to a finite surjective cover $g\colon
Z/{\sim}^{\prime}\,\longrightarrow\,Y/{\sim}$ that is étale over the singular
points of $Y/{\sim}$. Moreover, if $z\in Z$ is a point, and $f$ is étale at
all points in the class containing $z$, then the cover $g$ is also étale at
the image of $z$ in $Z/{\sim}^{\prime}$ under the identification map
$Z\longrightarrow Z/{\sim}^{\prime}$.

Further assume that $f$ is $G$-Galois for a finite group $G$ and the
$G$-action on $Z$ commutes with the relation ${\sim}^{\prime}$, namely, for
any $g\in G$ and $z\in Z$,

$z{\sim}^{\prime}z^{\prime}\Rightarrow g\cdot z{\sim}^{\prime}g\cdot
z^{\prime}.$

Then the cover $g$ is also $G$-Galois.

###### Proof.

Consider the normalization maps $Y^{\nu}\longrightarrow Y$ and
$Z^{\nu}\longrightarrow Z$. Let $\\{D_{i}\\}_{i\in I}$ (respectively,
$\\{E_{j}\\}_{j\in J}$) be the set of effective reduced divisors with pairwise
disjoint support on $Y^{\nu}$ (respectively, on $Z^{\nu}$) defining $Y$
(respectively, $Z$). Then $Y/{\sim}$ is obtained from $Y^{\nu}$ as an RS curve
together with the set of reduced effective divisors
$\\{D_{i^{\prime}}\\}_{i^{\prime}\in
I^{\prime}}\sqcup\\{D_{C}\coloneqq\sum_{y\in
C}y\\}_{C\in\mathcal{C}_{{\sim}}}$ where $I^{\prime}$ is the indexing set
obtained from $I$ by removing $i\in I$ such that $D_{i}=\nu^{-1}(y)$ for some
$y\in C$, $C\in\mathcal{C}_{{\sim}}$. Then we have
$Y/{\sim}=Y^{\nu}\sqcup_{Y^{\prime}}\left(Y/{\sim}\right)_{\text{\rm sing}}$
(see (2.4)). Similarly, we have
$Z/{\sim}^{\prime}=Z^{\nu}\sqcup_{Z^{\prime}}\left(Z/{\sim}^{\prime}\right)_{\text{\rm
sing}}$ for a closed subscheme $Z^{\prime}\subset Z^{\nu}$ associated to an
effective reduced divisor on $Z^{\nu}$.

Let $f\colon Z\longrightarrow Y$ be a finite cover that is étale over the
singular points of $Y$. Then $f$ corresponds to an admissible finite cover
$(f^{\nu},f_{\text{ \rm sing}})$ as in Definition 2.2. Here $f^{\nu}\colon
Z^{\nu}\longrightarrow Y^{\nu}$ is a finite cover of smooth projective
$k$-curves and $f_{\text{ \rm sing}}$ is a surjection of the singular loci.
Under the assumptions on $f$, the map $f_{\text{ \rm sing}}$ induces a
surjection $g_{\text{ \rm
sing}}\colon\left(Z/{\sim}^{\prime}\right)_{\text{\rm
sing}}\longrightarrow\left(Y/{\sim}\right)_{\text{\rm sing}}$. Moreover, the
pair $(f^{\nu},g_{\text{ \rm sing}})$ satisfies the condition for an
admissible finite cover of RS curves between $Z/{\sim}^{\prime}$ and
$Y/{\sim}$. By the universal property of cocartesian diagrams, we obtain a
finite cover $g\colon Z/{\sim}^{\prime}\longrightarrow Y/{\sim}$.

By our construction, the pullback of $g$ via $Y\longrightarrow Y/{\sim}$ is
the cover $f$. So the covers $g$ and $f$ agree away from the points in $C$
($C\in\mathcal{C}_{{\sim}^{\prime}}$). In particular, if $f$ is étale at such
a point, then so is $g$.

Now suppose that $z\in C$ for some $C\in\mathcal{C}_{{\sim}^{\prime}}$. Set
$D\in\mathcal{C}_{{\sim}}$ to be the class containing $f(z)$. Suppose that $f$
is étale at all points in the class containing $z$. So we have

(2.6) $\mathcal{O}_{Z,u}\cong\mathcal{O}_{Y,f(u)}$

for all $u\in C$. Suppose that $z^{\prime}\in Z/{\sim}^{\prime}$ and
$y^{\prime}\in Y/{\sim}$ be the images of $z$ and $f(z)$ under the respective
identification maps. Then we have

$\displaystyle\mathcal{O}_{Z/{\sim}^{\prime},z^{\prime}}=k+J\left(\bigcap_{\begin{subarray}{c}c\in
C\\\ u\in\nu^{-1}(c)\end{subarray}}\mathcal{O}_{Z^{\nu},u}\right)\hskip
20.00003pt\text{and}$
$\displaystyle\mathcal{O}_{Y/{\sim},y^{\prime}}=k+J\left(\bigcap_{\begin{subarray}{c}d\in
D\\\ v\in\nu^{-1}(d)\end{subarray}}\mathcal{O}_{Y^{\nu},v}\right)$

As $|C|=|D|$ by our assumption, the isomorphisms in (2.6) induce an
isomorphism
$\mathcal{O}_{Z/{\sim}^{\prime},z^{\prime}}\cong\mathcal{O}_{Y/{\sim},y^{\prime}}$.
So the map $g$ is étale at $z$.

Finally, if $f$ is $G$-Galois and the $G$-action preserves the relations, the
map $g$ obtained using the universal property of cocartesian diagrams is also
$G$-Galois. ∎

##  3\. Case of Projective Curves

###  3.1. Main Results in Projective Case

Throughout this section, let $k$ be an algebraically closed field of a prime
characteristic $p$. We fix the following notation.

######  Notation 3.1.

Let $X$ be a connected projective $k$-curve (reduced, but not necessarily
irreducible). Suppose that $X_{\text{Sing}}=\\{x_{1},\ldots,x_{l}\\}$, $l\geq
1$, is the singular locus of $X$. Let $\nu\colon X^{\nu}=\sqcup_{1\leq i\leq
n}C_{i}\longrightarrow X$ be the normalization map where $C_{i}$’s are the
irreducible components of $X^{\nu}$. The finite birational morphism $\nu$
factors as

$X^{\nu}\longrightarrow X^{+}\longrightarrow X$

where $X^{+}$ is the seminormalization (Definition 2.5) of $X$.

Our goal is to relate the étale fundamental group $\pi_{1}(X)$ of the
projective curve $X$ with
$\pi_{1}(X^{\nu})=\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})$. We first
observe the following result.

######  Lemma 3.2.

Under the above notation, $X^{+}$ is a connected projective $k$-curve and we
have

$\pi_{1}(X)\cong\pi_{1}(X^{+}).$

###### Proof.

Note that $X^{+}\longrightarrow X$ is a subintegral map, i.e. for each open
affine set $U=\text{\rm Spec}(A)\subset X$ with its preimage $V=\text{\rm
Spec}(B)\subset X^{+}$, the map $V\longrightarrow U$ is a homeomorphism of the
underlying topological spaces that induces isomorphisms of residue fields
([30, Theorem 2.9(1), page 448]). In particular, the map $X^{+}\longrightarrow
X$ is a radical morphism. As $X$ is connected, so is $X^{+}$. Since the normal
curve $X^{\nu}$ is projective, $X^{+}$ is also projective.

The normalization map $\nu$ is a finite surjective morphism. So
$X^{+}\longrightarrow X$ is a surjective integral and radical morphism. By
[23, Lemma 04DF], the map $X^{+}\longrightarrow X$ is a universal
homeomorphism. Now the result follows from [7, Exp. IV, Theorem 18.1.2, page
110]. ∎

In view of the above lemma, we further assume that _$X$ is seminormal._

Now consider a complete discrete valuation ring $R$ with residue field $k$ and
fraction field $K$ of characteristic $0$. Let $\bar{K}$ be an algebraic
closure of $K$. By Proposition 2.6, there us a connected $R$-curve $X_{R}$,
the map $X_{R}\longrightarrow\text{\rm Spec}(R)$ is flat, proper with
geometrically connected fibres. We obtain the specialization map ([24, Exp.10]
or [23, Section 0BUP])

$\text{\rm sp}\colon\pi_{1}(X_{\bar{K}})\longrightarrow\pi_{1}(X)$

of Grothendieck that is a continuous homomorphism. Since $X$ is a reduced
curve, the map sp is surjective by [23, Lemma 0C0P]. Without lose of
generality, we may assume (see the argument in [18, Remark 1.7]) that
$\bar{K}=\mathbb{C}$. Then the above surjection becomes

(3.1) $\text{\rm
sp}\colon\pi_{1}(X_{\mathbb{C}})\twoheadrightarrow\pi_{1}(X).$

Further, by Proposition 2.6, $X_{\mathbb{C}}$ is obtained from its
normalization $X_{\mathbb{C}}^{\nu}=\sqcup_{1\leq i\leq n}D_{i}$ via
identifying points the same way $X$ is obtained from $X^{\nu}$ (here $D_{i}$’s
are the irreducible components of $X^{\nu}$), and $g(C_{i})=g(D_{i})$ for each
$i$. By the general version of the Riemann Existence Theorem ([24, Exp XII;
Corollary 5.2]), $\pi_{1}(X_{\mathbb{C}})$ is the profinite completion of the
topological fundamental group $\pi_{1}^{\text{\rm
top}}(X_{\mathbb{C}}(\mathbb{C}))$.

We have a complete description of the group $\pi_{1}(X_{\mathbb{C}})$ as
follows. By Remark 2.9, the normalization map $\nu_{\mathbb{C}}\colon
X_{\mathbb{C}}^{\nu}\longrightarrow X_{\mathbb{C}}$ factors as a composition

$X_{\mathbb{C}}^{\nu}=Y_{t}\longrightarrow
Y_{t-1}\longrightarrow\cdots\longrightarrow Y_{1}\longrightarrow
Y_{0}=X_{\mathbb{C}}$

where each $Y_{i}$ is a seminormal projective $\mathbb{C}$-curve and $Y_{i}$
is obtained from $Y_{i+1}$ by identifying two distinct closed points
$y_{i+1,1}$ and $y_{i+1,2}$. The underlying topological space
$Y_{i}(\mathbb{C})$ is a deformation retract of the space $U_{i+1}$ obtained
from $Y_{i+1}(\mathbb{C})$ by joining $y_{i+1,1}$ to $y_{i+1,2}$ via a
contractible arc (i.e. a ‘handle’). Then $\pi_{1}^{\text{\rm
top}}(Y_{i})=\pi_{1}^{\text{\rm top}}(U_{i+1})$. Using Van Kampen’s Theorem,
$\pi_{1}^{\text{\rm top}}(U_{i+1})=\pi_{1}^{\text{\rm
top}}(Y_{i+1}(\mathbb{C}))\ast\mathbb{Z}$ if the points $y_{i+1,1}$ and
$y_{i+1,2}$ lie in a connected component, and $\pi_{1}^{\text{\rm
top}}(U_{i+1})=\pi_{1}^{\text{\rm top}}(Y_{i+1}(\mathbb{C}))$ otherwise. We
conclude that $\pi_{1}(X_{\mathbb{C}})$ is a free product of finitely
generated profinite groups

$\pi_{1}(X_{\mathbb{C}})=\pi_{1}(D_{1})\ast\cdots\ast\pi_{1}(D_{n})\ast\widehat{F_{\delta}}=\pi_{1}(X^{\nu}_{\mathbb{C}})\ast\widehat{F_{\delta}},$

and $F_{\delta}$ is a free finitely generated group on $\delta\coloneqq
1-n+\sum_{1\leq i\leq l}\left(|\nu^{-1}(x_{i})|-1\right)$ generators,
recursively defined as follows. Set $F_{t}$ to be the trivial group. If the
two points in $Y_{i+1}$ getting identified under $Y_{i+1}\longrightarrow
Y_{i}$ lie in the same connected component of $Y_{i+1}$, set $F_{i}\coloneqq
F_{i+1}\ast\mathbb{Z}$. Otherwise, set $F_{i}\coloneqq F_{i+1}$. If
$g_{i}=g(D_{i})$ denote the genus of the compact Riemann surface $D_{i}$,
$1\leq i\leq n$, it is well known that the étale fundamental group
$\pi_{1}(D_{i})$ is a the profinite completion of a finitely generated group
on $2g_{i}$ generators $a_{1},b_{1},\ldots,a_{g_{i}},b_{g_{i}}$ subject to the
only relation $\prod_{1\leq j\leq g_{i}}\left[a_{j},b_{j}\right]=1$.

As a consequence of the surjective specialization homomorphism sp in (3.1) and
the description of the group $\pi_{1}(X_{\mathbb{C}})$ as above, $\pi_{1}(X)$
is a finitely generated group of rank at most $2\sum_{1\leq i\leq
n}g_{i}+\delta$ (recall that $g(C_{i})=g_{i}=g(D_{i})$), and

$\delta=1-n+\sum_{1\leq i\leq l}\left(|\nu^{-1}(x_{i})|-1\right).$

In particular, $\pi_{1}(X)$ is completely determined by its finite group
quotients (as a topological group; [16, Theorem 1.1]).

We further see that there is a surjective homomorphism

$\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}\twoheadrightarrow\pi_{1}(X)$

of finitely generated profinite groups. Recall that we obtained a surjective
specialization map $\text{\rm
sp}\colon\pi_{1}(X_{\mathbb{C}})\twoheadrightarrow\pi_{1}(X)$ in (3.1). In
terms of covers, the existence of the specialization map is equivalent to: the
functor $\text{\rm FEt}_{X_{R}}\longrightarrow\text{\rm FEt}_{X}$ taking
covers to their special fibres is an equivalence of categories (with the quasi
inverse $\theta$, say). Then we obtain a functor

$\lambda\colon\text{\rm
FEt}_{X}\overset{\theta}{\underset{\sim}{\longrightarrow}}\text{\rm
FEt}_{X_{R}}\longrightarrow\text{\rm FEt}_{X_{\mathbb{C}}}.$

The surjectivity of sp is equivalent to: $\lambda$ maps connected finite étale
covers of $X$ to connected finite étale covers of $X_{\mathbb{C}}$.

Now let $f\colon Z\longrightarrow X$ be a $G$-Galois étale cover of connected
$k$-curves. Using the surjection sp in (3.1), $G$ is generated by its
subgroups $G_{1},\ldots,G_{n}$ and $H$ with $F_{\delta}\twoheadrightarrow H$
and $\pi_{1}(D_{i})\twoheadrightarrow G_{i}$ (as before, $D_{i}$’s are the
irreducible components of $X_{\mathbb{C}}^{\nu}$). So the $G$-Galois cover
$\nu^{*}(\lambda(f))$ is a disjoint union of $G_{i}$-Galois étale connected
covers of $D_{i}$’s. We have the following $2$-commutative diagram of Galois
categories.

${\lambda^{\nu}\colon\text{\rm FEt}_{X^{\nu}}}$${\text{\rm
FEt}_{X_{R}^{\nu}}}$${\text{\rm
FEt}_{X^{\nu}_{\mathbb{C}}}}$${\lambda\colon\text{\rm FEt}_{X}}$${\text{\rm
FEt}_{X_{R}}}$${\text{\rm
FEt}_{X_{\mathbb{C}}}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\nu^{*}}$$\scriptstyle{\nu^{*}}$$\scriptstyle{\nu^{*}}$

As $\lambda^{\nu}(\nu^{*}(f))=\nu^{*}(\lambda(f))$, the $G$-Galois cover
$\nu^{*}(f)$ is also a disjoint union of $G_{i}$-Galois étale connected covers
of $C_{i}$’s. This shows that each $G_{i}$ is in fact a finite quotient of
$\pi_{1}(C_{i})$. As the profinite groups in question are finitely generated,
we obtain a surjective homomorphism

(3.2)
$\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}\twoheadrightarrow\pi_{1}(X).$

We have the following structure theorem for the fundamental group $\pi_{1}(X)$
in terms of its normalizer and singular points.

######  Theorem 3.3.

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $X$
be a connected projective $k$-curve. Let $X_{\rm
sing}=\\{x_{1},\ldots,x_{l}\\}$. Let $\nu\colon X^{\nu}=\sqcup_{1\leq i\leq
n}C_{i}\longrightarrow X$ be the normalization map where $C_{i}$’s are the
irreducible components of $X^{\nu}$. Then the étale fundamental group
$\pi_{1}(X)$ is the following profinite group.

$\pi_{1}(X)=\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}$

where $\widehat{F_{\delta}}$ is the profinite completion of a free group
$F_{\delta}$ on

$\delta\coloneqq 1-n+\sum_{1\leq i\leq l}\left(|\nu^{-1}(x_{i})|-1\right)$

generators. In particular, $\pi_{1}(X)$ is a finitely generated profinite
group on $\sum_{1\leq i\leq n}r_{i}+\delta$ generators, where $r_{i}$ is the
rank of $\pi_{1}(C_{i})$ ($r_{i}\leq 2g(C_{i})$).

###### Proof.

We assume that $\text{\rm char}(k)>p$ and $X_{\text{\rm sing}}\neq\emptyset$
since the statement is known otherwise. In view of Lemma 3.2, we may assume
that $X$ is a seminormal curve. By our previous discussion, $\pi_{1}(X)$ is a
finitely generated group of rank at most $2\sum_{1\leq i\leq
n}g(C_{i})+\delta$, and hence it is completely determined as a topological
group by its finite quotients.

The normalization map $\nu$ admits a factorization (see (2.5))

$\nu\colon X^{\nu}=Y_{t}\longrightarrow
Y_{t-1}\longrightarrow\cdots\longrightarrow Y_{1}\longrightarrow Y_{0}=X$

where each $Y_{i}$ is a seminormal curve, and $Y_{i}$ is obtained from
$Y_{i+1}$ by identifying two distinct closed points. By inductively applying
the next two propositions, we show that any finite quotient of
$\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}$ is also a
finite quotient of $\pi_{1}(X)$, and hence determine a surjection

$\pi_{1}(X)\twoheadrightarrow\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}.$

We already have a surjective homomorphism
$\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}\twoheadrightarrow\pi_{1}(X)$
in (3.2), and this concludes the proof of the theorem. ∎

######  Proposition 3.4.

Let $Y$ be a seminormal projective $k$-curve, $y_{1},\,y_{2}\in Y$ be two
closed points. Suppose that $X$ is the seminormal curve obtained from $Y$ by
identifying $y_{1}$ with $y_{2}$ (see (2.3)). Assume that $Y$ is connected.
Let $\Gamma$ be a finite group generated by a subgroup $G$ and an element
$\gamma\in\Gamma$. Let $f\colon Z\longrightarrow Y$ be a $G$-Galois cover of
connected projective $k$-curves, étale over $Y_{\text{\rm
sing}}\cup\\{y_{1},y_{2}\\}$. Then there exists a $\Gamma$-Galois cover
$g\colon W\longrightarrow X$ of connected projective $k$-curves, étale over
$X_{\text{\rm sing}}$.

######  Proposition 3.5.

Let $Y$ be a seminormal projective $k$-curve, $y_{1},\,y_{2}\in Y$ be two
closed points. Suppose that $X$ is the seminormal curve obtained from $Y$ by
identifying $y_{1}$ with $y_{2}$ (see (2.3)). Assume that $Y$ is a union of
two distinct connected components $Y_{1}$ and $Y_{2}$ containing the points
$y_{1}$ and $y_{2}$, respectively. Let $G$ be a finite group generated by two
of its subgroups $G_{1}$ and $G_{2}$. Suppose that for $a=1,\,2$, $f_{a}\colon
Z_{a}\longrightarrow Y_{a}$ be a $G_{i}$-Galois cover of connected projective
$k$-curves, étale over $\left(Y_{a}\right)_{\text{\rm sing}}\cup\\{y_{a}\\}$.
Then there exists a $G$-Galois cover $g\colon W\longrightarrow X$ of connected
projective $k$-curves, étale over $X_{\text{\rm sing}}$.

Before proving the above propositions, we make some observation on the Galois
action on the induced covers. Let $X$ be any projective connected $k$-curve,
and let $f\colon Z\longrightarrow X$ be a $G$-Galois cover of connected
$k$-curves that is étale over the singular locus $X_{\text{\rm sing}}$ of $X$.
This implies that the image of the singular points of $Z$ are mapped to
singular points of $X$, and the fibre of $f$ over any singular point of $X$
consists only of singular points of $Z$. Fix a closed point $x\in X$ in the
étale locus of $f$. Then $G$ acts simply transitively on the finite set of
points $f^{-1}(x)$ in $Z$. In other words, the closed subscheme $f^{-1}(x)$ of
$Z$ is a principal homogeneous space for the finite constant $k$-group scheme
$G$ over $\text{\rm Spec}(k)$.

We have the following group theoretic observation. For any finite group $G$
and a finite set $S$, there is a bijective correspondence

(3.3) $\begin{Bmatrix}\text{ A simply transitive }G\text{-action on }S\\\
\text{ together with an element }s_{0}\in
S\end{Bmatrix}\overset{\sim}{\longrightarrow}\begin{Bmatrix}\text{ A bijection
}\alpha\colon S\longrightarrow G\end{Bmatrix}$

defined by $(\Lambda\colon G\times S\longrightarrow S)\,\mapsto\,\alpha(s)=g$
where $g\in G$ is the unique element $\Lambda(g,s)=s_{0}$ obtained by the
transitivity of $\Lambda$ (Here $\alpha^{-1}(g)=\Lambda(g,s_{0})$). The
inverse map of the above correspondence is given by
$\alpha\mapsto\left(\Lambda,\alpha^{-1}(e)\right)$, $e$ is the identity
element of $G$, $\Lambda(g,s)=\alpha^{-1}\left(g\cdot\alpha(s)\right)$ where
$g\cdot\alpha(s)$ is obtained by the regular action of $G$ on itself.

Now suppose that $\Gamma$ is a finite group generated by its subgroups $G$ and
$H$. Consider the induced (disconnected) $\Gamma$-Galois cover $f^{\text{ \rm
ind}}\colon\text{\rm Ind}_{G}^{\Gamma}Z\longrightarrow X$. Then $f^{\text{ \rm
ind}}$ is also étale over $X_{\text{\rm sing}}$. Since $f$ is étale over the
point $x\in X$, so is the cover $f^{\text{ \rm ind}}$. We study the
$\Gamma$-action on the fibre $\left(f^{\textbf{ \rm ind}}\right)^{-1}(x)$ in
detail. Fix a point $z_{0}\in f^{-1}(x)$.

Fix a set of coset representatives
$C(\Gamma/G)=\\{e=h_{0},h_{1},\ldots,h_{N-1}\\}$ of the left coset $\Gamma/G$
where $N=[\Gamma\colon G]$ and $h_{i}\in H$. Then $\text{\rm
Ind}_{G}^{\Gamma}Z=\sqcup_{0\leq i\leq N-1}Z_{i}$ is a disjoint union where
each $Z_{i}$ is a copy of $Z$ indexed by $h_{i}\in C(\Gamma/G)$. Let
$z_{0}^{(i)}\in Z_{i}$ be the unique point over $z_{0}\in Z$ under the above
identification $Z_{i}\cong Z$. As $f$ is étale over $x$, for each $0\leq i\leq
N-1$, the simply transitive $G$-action on $f^{-1}(x)$ and the point
$z_{0}^{(i)}$ determines a unique bijection by (3.3):

$\alpha_{i}\colon\left(f^{\text{ \rm ind}}\right)^{-1}(x)\cap
Z_{i}\overset{{\sim}}{\longrightarrow}G.$

These define a bijection

(3.4) $\alpha\colon\left(f^{\text{ \rm
ind}}\right)^{-1}(x)\longrightarrow\Gamma$

as follows. For $z\in\left(f^{\text{ \rm ind}}\right)^{-1}(x)$, there is a
unique $0\leq i\leq N-1$ such that $z\in\left(f^{\text{ \rm
ind}}\right)^{-1}(x)\cap Z_{i}$. Set $\alpha(z)\coloneqq h_{i}\alpha_{i}(z)$.
It has the inverse map given by: $\gamma\in\Gamma$ can be written uniquely as
$\gamma=h_{i}g$ for $i$ and $g\in G$; set
$\alpha^{-1}(\gamma)\coloneqq\alpha_{i}^{-1}(g)=g\cdot z_{0}^{(i)}$. By (3.3),
this bijection corresponds to the point $\alpha^{-1}(e)=z_{0}^{(0)}\in Z_{0}$
and the $\Gamma$-action $\Lambda$ on $\left(f^{\text{ \rm
ind}}\right)^{-1}(x)$ given by

$\Lambda(\gamma,z)=\alpha^{-1}\left(\gamma\cdot\alpha(z)\right)$

where $\gamma\cdot\alpha(z)$ is obtained from the regular $G$-action on
itself. More precisely, let $\gamma=h_{i}g,\,z\in Z_{j}$. Then
$h_{i}g\alpha_{j}(z)=h_{i^{\prime}}g^{\prime}$ for some uniquely determined
$i^{\prime}$ and $g^{\prime}\in G$. We have $\Lambda(\gamma,z)=g^{\prime}\cdot
z_{0}^{(i^{\prime})}$ determined by the $G$-action on $f^{-1}(x)$.

######  Proof of Proposition 3.4.

Let $x$ be the image of $y_{1}$ (and of $y_{2}$) under the identification map
$Y\longrightarrow X$. By our hypothesis, $f$ is possibly ramified only over a
finitely many smooth points of $X$. For any closed point $z\in Z$ over which
$f$ is étale, we have $\mathcal{O}_{Z,z}\cong\mathcal{O}_{Y,f(z)}$. If $f$ is
ramified at a point $z\in Z$, then $f(z)$ is a smooth point of $X$; so the
local ring extension $\mathcal{O}_{X,x}\subset\mathcal{O}_{Z,z}$ is
necessarily an extension of discrete valuation domains. Thus $Z$ is also a
seminormal curve (see Definition 2.5).

Let $N=[\Gamma\colon G]$. Then
$C(\Gamma/G)=\\{e=\gamma^{0},\gamma,\ldots,\gamma^{N-1}\\}$ is a set of coset
representatives for the left coset $\Gamma/G$. Consider the (possibly
disconnected) $\Gamma$-Galois étale cover

$f^{\text{ \rm ind}}\colon\text{Ind}_{G}^{\Gamma}Z=\sqcup_{0\leq i\leq
N-1}Z_{i}\longrightarrow Y$

where each $Z_{i}$ is a copy of $Z$ indexed by $\gamma^{i}$. Let
$a\in\\{1,2\\}$. Fix a point $z_{a}\in f^{-1}(y_{a})$, and let $z_{a}^{(i)}\in
Z_{i}$ denote its image under the isomorphism $Z_{i}\cong Z$. We have the
corresponding bijections ($0\leq i\leq N-1$)

$\alpha_{a,i}\colon\left(f^{\text{ \rm ind}}\right)^{-1}(y_{a})\cap
Z_{i}\overset{{\sim}}{\longrightarrow}G.$

Recall from (3.4) that the above bijections define a bijection
$\alpha_{a}\colon\left(f^{\text{ \rm
ind}}\right)^{-1}(y_{a})\longrightarrow\Gamma$ that determine the
$\Gamma$-action on $\left(f^{\text{ \rm ind}}\right)^{-1}(y_{a})$.

We define an equivalence relation ${\sim}$ on $\text{\rm Ind}_{G}^{\Gamma}Z$
whose non-trivial classes are obtained using the following rule.

  * •

A point $z\in\left(f^{\text{ \rm ind}}\right)^{-1}(y_{1})\cap Z_{i}$ is
identified with a point $z^{\prime}\in\left(f^{\text{ \rm
ind}}\right)^{-1}(y_{2})\cap Z_{i+1\pmod{N}}$ if and only if
$\alpha_{1,i}(z)=\alpha_{2,i+1\pmod{N}}(z^{\prime})$.

Let $W$ be the seminormal curve $\left(\text{\rm
Ind}_{G}^{\Gamma}Z\right)/{\sim}$ (see (2.3)); the finite birational morphism
$\beta\colon\text{\rm Ind}_{G}^{\Gamma}Z\longrightarrow W$ is an isomorphism
away from the set of points $S\coloneqq\left(f^{\text{ \rm
ind}}\right)^{-1}\left(\\{y_{1},y_{2}\\}\right)$. By our construction, the
equivalence relations satisfy the hypothesis of (1) and (2) of Lemma 2.10.
Moreover, $f$ is étale over both $y_{1}$ and $y_{2}$. As $\alpha_{a,i}(g\cdot
z_{a}^{(i)})=g$ for any $g\in G,\,0\leq i\leq N-1$, we have

$z_{1}^{(i)}{\sim}z_{2}^{(i+1\pmod{N})}\text{ and for any }g\in G,\,g\cdot
z_{1}^{(i)}{\sim}z_{2}^{(i+1\pmod{N})}.$

Thus $f$ induces a $\Gamma$-Galois cover $g\colon W\longrightarrow X$ that is
étale over $X_{\text{\rm sing}}$. Since for each $0\leq i\leq N-1$, the set
$\\{z\in\text{\rm Ind}_{G}^{\Gamma}Z\,|\,z\text{ lies in a non-trivial
equivalence class of }{\sim}\\}\cap Z_{i}$ is non-empty and $Z$ is connected,
$W$ is a connected seminormal curve. ∎

######  Proof of Proposition 3.5.

Let $x\in X$ be the image of $y_{1}$ ( and of $y_{2}$) under the
identification map $Y\longrightarrow X$. It also follows that $Z_{1}$ and
$Z_{2}$ are seminormal curves. Let $a\in\\{1,2\\}$. Since
$G=\langle\,G_{1},G_{2}\,\rangle$, we choose a set of coset representatives
$C(G/G_{a})=\\{e=g_{3-a,0},g_{3-a,1},\ldots,g_{3-a,n_{a}-1}\\}$ for the left
coset $G/G_{a}$ where $n_{a}=[G\colon G_{a}]$ and each $g_{3-a,i_{a}}\in
G_{3-a}$. Consider the induced (possibly disconnected) $\Gamma$-Galois cover

$f_{a}^{\text{ \rm ind}}\colon\text{\rm Ind}_{G_{a}}^{G}Z_{a}=\sqcup_{0\leq
i_{a}\leq n_{a}-1}Z_{a,i_{a}}\longrightarrow Y_{a},$

étale over $\left(Y_{a}\right)_{\text{\rm sing}}\cup\\{y_{a}\\}$; each
$Z_{a,i_{a}}$ is a copy of $Z_{a}$ indexed by $g_{3-a,i_{a}}\in C(G/G_{a})$.
Fix a point $z_{a}\in f_{a}^{-1}(y_{a})$. We obtain bijections (see (3.4))

$\alpha_{a}\colon\left(f_{a}^{\text{ \rm
ind}}\right)^{-1}(y_{a})\overset{{\sim}}{\longrightarrow}G$

that determines the $G$-action on the set $\left(f_{a}^{\text{ \rm
ind}}\right)^{-1}(y_{a})$.

We have the following (possibly disconnected) $G$-Galois cover

$\phi\colon f_{1}^{\text{ \rm ind}}\sqcup f_{2}^{\text{ \rm ind}}\colon
Z^{\prime}\coloneqq\text{\rm Ind}_{G_{1}}^{G}Z_{1}\sqcup\text{\rm
Ind}_{G_{2}}^{G}Z_{2}\longrightarrow Y_{1}\sqcup Y_{2}$

that is étale over $Y_{\text{\rm sing}}\cup\\{y_{1},y_{2}\\}$. Consider the
equivalence relation ${\sim}$ on the closed points of $Z^{\prime}$ where the
non-trivial classes are given as follows.

  * •

A point $z_{1}\in\left(f_{1}^{\text{ \rm ind}}\right)^{-1}(y_{1})$ is
identified with a point $z_{2}\in\left(f_{2}^{\text{ \rm
ind}}\right)^{-1}(y_{2})$ if and only if
$\alpha_{1}(z_{1})=\alpha_{2}(z_{2})$.

In other words, if $g\in G$, then $g$ can be uniquely written as
$g=g_{1,i}g_{2}=g_{2,j}g_{1},\,g_{1}\in G_{1},\,g_{2}\in G_{2},\,0\leq i\leq
n_{2},\,0\leq j\leq n_{1}$. We identify the point $\alpha_{1,j}^{-1}(g_{1})$
on $Z_{1,j}$ with the point $\alpha_{2,i}^{-1}(g_{2})$ on $Z_{2,i}$. Let
$\beta\colon Z^{\prime}\longrightarrow W\coloneqq Z^{\prime}/{\sim}$ be the
identification map. It is easy to check that the conditions of Lemma 2.10 are
satisfied, and we obtain a $G$-Galois cover $W\longrightarrow X$, étale over
$X_{\text{\rm sing}}$.

Now we show that $W$ is connected. It is enough to show that for any $0\leq
i\leq n_{1}-1$ and any $0\leq j\leq n_{2}-1$, there exist points $z_{1}\in
Z_{1,j},\,z_{2}\in Z_{2,i}$ that are identified via ${\sim}$. Consider any
point $z_{1}=\alpha_{1,j}^{-1}(g_{1})$ on $Z_{1,j}$. Then $g\coloneqq
g_{2,j}g_{1}$ can be written uniquely as $g=g_{1,i}g_{2}$, and $z_{1}$ is
identified with $z_{2}\coloneqq\alpha_{2,i}^{-1}(g_{2})$ on $Z_{2,i}$. ∎

######  Remark 3.6.

The above proofs also show the following. Let $y\in Y$ be any smooth point,
$y\neq y_{1},\,y_{2}$. Let $x$ denote the smooth point in $X$ obtained as the
image of $y$ under the identification map $Y\longrightarrow
X=Y/{y_{1}{\sim}y_{2}}$. If $I_{y}$ is an inertia group over $y$ for the cover
$f$ in Proposition 3.4 (or for either of the covers $f_{1}$ and $f_{2}$ in
Proposition 3.5), then $I_{y}$ is also an inertia group above $x$ for the
cover $W\longrightarrow X$ (in either of the propositions). So the above
constructions do not change the ramification behavior above any point.

######  Remark 3.7.

In Theorem 3.3, we have

$\pi_{1}(X)=\pi_{1}(C_{1})\ast\cdots\ast\pi_{1}(C_{n})\ast\widehat{F_{\delta}}=\pi_{1}(X^{\nu})\ast\widehat{F_{\delta}}.$

In particular, we obtain the injective homomorphism
$\pi_{1}(X^{\nu})\hookrightarrow\pi_{1}(X)$. The corresponding exact functor
of the Galois categories is the pullback of finite étale covers of $X$ under
the normalization map. Similarly, each surjection
$\pi_{1}(X)\longrightarrow\pi_{i}(C_{i})$ corresponds to the exact fully
faithful ([23, Lemma 0BN6]) functor $\text{\rm
FEt}_{C_{i}}\longrightarrow\text{\rm FEt}_{X}$. These are precisely the maps
obtained (for connected objects in the categories) in Proposition 3.4 and
Proposition 3.5. So the above propositions describe a canonical way (up to the
choices of coset representatives and of one point in each fibre of covers over
the points getting identified) to obtain all the Galois covers of any singular
curve $X$ starting from the Galois covers of its normalization.

###  3.2. Necessary Conditions on Galois groups

We retain the notation from our previous section. As application of our main
theorem 3.3, we have the following necessary conditions for any finite
quotient of $\pi_{1}(X)$. These are known when $X$ is smooth (see [28] or [12,
Section 7]).

######  Corollary 3.8.

Let $X$ be a connected singular curve over an algebraically closed field $k$
of characteristic $p>0$. Let $\delta$ and $g_{i}=g(C_{i})$ ($1\leq i\leq n$)
be as in Theorem 3.3. Let $X^{+}$ be the seminormalization of $X$.

  1. (1)

The specialization map $\text{\rm
sp}\colon\pi_{1}(X^{+}_{\mathbb{C}})\twoheadrightarrow\pi_{1}(X)$ obtained by
composing the surjection (3.1) with the isomorphism
$\pi_{1}(X^{+})\cong\pi_{1}(X)$ of Lemma 3.2 induces an isomorphism of the
maximal $p^{\prime}$-quotients

$\pi_{1}(X^{+}_{\mathbb{C}})^{(p^{\prime})}\twoheadrightarrow\pi_{1}(X)^{(p^{\prime})}.$

  2. (2)

The maximal pro-$p$ quotient $\pi_{1}(X)^{(p)}$ is a free profinite group on
$\sum_{1\leq i\leq n}s_{i}+\delta$ generators; here $s_{i}$ denote the
$p$-rank of $C_{i}$.

Further, suppose that a finite group $G$ occurs as the Galois group of a
Galois étale connected cover of $X$. Let $I_{G}$ denote the augmented ideal

$\left\\{\sum_{g\in G}a_{g}g\,|\,\sum_{g}a_{g}=0\right\\}$

of the group ring $k[G]$. Let $t_{G}$ be the minimal number of generators of
$I_{G}$. Let $\sigma(G)$ be the $p$-rank of the abelianization $G_{\text{\rm
ab}}$ of $G$. Then $G$ satisfies the following properties.

  1. (3)

(Nakajima’s Condition for singular curves)222For integral semistable curves,
this can be proved using the original ideas of [15]; in fact, Theorem 3 and
Theorem 4 of loc. cit. holds in this set up. This insight is missing for
general curves as the Riemann Roch Theorem and related notions are known when
the curve is Gorenstein. $t_{G}\leq\sum_{1\leq i\leq n}g_{i}+\delta$.

  2. (4)

(Hasse-Witt for singular curves) $\sigma(G)\leq\sum_{1\leq i\leq
n}g_{i}+\delta$.

###### Proof.

We first note that if $\Gamma$ is a free product of finitely generated
profinite groups $\Gamma_{1}$ and $\Gamma_{2}$, then there is an isomorphism
$\Gamma^{(p^{\prime})}\cong\left(\Gamma_{1}^{(p^{\prime})}\ast\Gamma_{2}^{(p^{\prime})}\right)^{(p^{\prime})}$
of maximal $p^{\prime}$-quotients and an isomorphism
$\Gamma^{(p)}\cong\left(\Gamma_{1}^{(p)}\ast\Gamma_{2}^{(p)}\right)^{(p)}$ of
maximal pro-$p$ quotients. If $G$ is a finite quotient of
$\Gamma_{1}^{(p^{\prime})}\ast\Gamma_{2}^{(p^{\prime})}$ of order prime-
to-$p$, then $G$ is also a quotient of $\Gamma$ by the normal subgroup
generated by the kernels of the maps $\Gamma\twoheadrightarrow\Gamma_{i}$.
Conversely, suppose that $\Gamma^{(p^{\prime})}\twoheadrightarrow G$ be a
finite quotient. Then $G$ admits generators which are images of generators of
$\Gamma$ and satisfies the relations in $\Gamma$. Consider the subgroups
$G_{1}$ and $G_{2}$ of $G$ that are generated by those elements which are
images of generators of $\Gamma_{1}$ and $\Gamma_{2}$, respectively. These
generators necessarily satisfy the relations in respective $\Gamma_{i}$’s. So
$G_{i}$ is a finite quotient of $\Gamma_{i}$ and is of order prime-to-$p$.
Since $\Gamma_{i}$’s are finitely generated, we see the first isomorphism of
maximal $p^{\prime}$-quotients. The other isomorphism follows similarly by
considering finite quotients which are $p$-groups.

In view of the above, (1) follows from the isomorphism induced by the
specialization map in smooth case ([24, Exp XIII, Corollary 2.12, page 392]).
Similarly, since $\pi_{1}(Y)^{(p)}$ for a smooth projective $k$-curve $Y$ is a
free group on $s_{Y}$ ($=p\text{\rm-rank of }Y$) generators ([22]), (2)
follows.

Now we prove (2). By Theorem 3.3, $G$ is generated by its subgroups
$H_{1},\ldots,H_{n}$ and $H$ (which may be trivial) such that $H_{i}$ a finite
quotient of $\pi_{1}(C_{i})$ and $H$ is a finite group on $\leq\delta$
generators. Then $t_{G}\leq\sum_{i}t_{G_{i}}+t_{H}$. By [28, Corollary 1.4],
we have $t_{H_{i}}\leq g_{i}$. We also have $t_{H}\leq\delta$ by [28,
Proposition 2.1(i)]. Thus $t_{G}\leq\sum_{i}g_{i}+\delta$.

For (4), suppose that $f\colon Z\longrightarrow X$ be a $G$-Galois étale cover
of connected $k$-curves. Let $A$ denote the maximal elementary abelian
$p$-quotient of $G$. Then $f$ factors through a connected $A$-Galois cover
$Y\longrightarrow X$. Then by (2), the number of generators for $A$
($=\sigma(G)$ by definition) is bounded above by $\sum_{1\leq i\leq
n}s_{i}+\delta\leq\sum_{1\leq i\leq n}g_{i}+\delta$. ∎

##  4\. Case of Affine Curves

###  4.1. Main Results for affine case

In this section, we study the Galois covers of an affine $k$-curve $U$. As
before, $k$ is an algebraically closed field of characteristic $p>0$. We also
assume that $U$ is irreducible.

Even when $U$ is smooth, the étale fundamental group $\pi_{1}(U)$ is not
finitely generated; so understanding its finite quotients does not fully
determine the structure of the profinite group $\pi_{1}(U)$. Nevertheless, it
is important to understand the finite continuous quotients. Abhyankar’s
Conjecture on Affine Curves (now a theorem) states that for a smooth
irreducible affine $k$-curve $U$ with its smooth projective completion
$U\subset X$ and $r=|X-U|$, a group $G$ is a finite continuous quotient of
$\pi_{1}(U)$ if and only if $G/p(G)$ has a generating set of at most
$2g(X)+r-1$ elements. We seek a similar statement when $U$ is a singular
curve.

Embed $U$ in a connected projective curve $X$ such that $B\coloneqq X-U$
consists of smooth points. Set $r\coloneqq|B|$. Consider the normalization map
$\nu\colon X^{\nu}\longrightarrow X$. Then $X^{\nu}$ is the unique (up to
isomorphism) smooth projective connected $k$-curve with function field
$k(X)=k(U)$. In general, the pullback divisor $\nu^{*}(x)$ on $X^{\nu}$ for a
closed point $x\in X$ need not be reduced. Let $X^{+}$ be the seminormal curve
obtained from $X^{\nu}$ together with the set of effective reduced divisors
$\\{\nu^{*}(x)_{\text{ \rm red}}\coloneqq\nu^{-1}(x)\\}_{x\in U_{\rm sing}}$
(see (2.2)). Since $U$ is connected, $X^{+}$ is also connected. The map $\nu$
factors as a composition

$\nu\colon
X^{\nu}\overset{\nu^{+}}{\longrightarrow}X^{+}\overset{\eta}{\longrightarrow}X.$

Then $B$, $B^{+}\coloneqq\eta^{-1}(B)$ and $\nu^{-1}(B)$ are all isomorphic
sets of $r$-many closed smooth points in the respective curves. We make the
following useful note.

######  Remark 4.1.

Set $U^{\nu}=\nu^{-1}(U)$. The genus $g$ of the curve $X^{\nu}$, the number
$r=|\nu^{-1}(B)|=|X^{\nu}-U^{\nu}|$ and the number

$\delta\coloneqq\sum_{u\in U_{\text{\rm sing}}}\left(|\nu^{-1}(u)|-1\right)$

are all uniquely determined by the curve $U$.

By construction, $U^{+}$ is a seminormal affine $k$-curve, and $\eta$
restricts to a finite surjective morphism $\eta|_{U^{+}}\colon
U^{+}\longrightarrow U$ which is also subintegral (for each point $u\in U$,
there is a unique point in $\eta^{-1}(u)$ having the same residue field). In
particular, $\eta|_{U^{+}}$ is a radical morphism. By [23, Lemma 04DF], it is
a universal homeomorphism. By [7, Theorem 18.1.2, page 110], we have the
following isomorphism of étale fundamental groups.

(4.1) $\pi_{1}(U)\cong\pi_{1}(U^{+}).$

_In whatever follows, we assume that $U$ is seminormal._

Now suppose that $\pi_{1}(U)\twoheadrightarrow G$ be a finite continuous
quotient. This uniquely corresponds to a $G$-Galois cover $f\colon
Y\longrightarrow X$ of connected projective $k$-curves that is étale over $U$.
Since $X$ is seminormal and the branch locus of $f$ is contained in the smooth
locus of $X$, the curve $Z$ is necessarily seminormal. We need a local-global
principle (see [18, Theorem 3.1] for a smooth curve) as follows.

######  Proposition 4.2.

Let $R$ be a complete discrete valuation ring with residue field $k$. Under
the above notation, let $X_{R}$ be a connected projective flat $R$-curve with
special fibre $X$ obtained in Proposition 2.6. Let $B_{R}\subset X_{R}$ be the
set of $r$ sections in $X_{R}(\text{\rm Spec}(R))$ with pairwise disjoint
supports corresponding to the set $B$. Suppose that for each point $x\in B$, a
subgroup $I_{x}\subset G$ occurs as an inertia group over $x$. Also assume
that for each point $x\in B$, a point $z\in f^{-1}(x)$, the local
$I_{x}$-Galois extension
$\widehat{\mathcal{O}}_{Z,z}/\widehat{\mathcal{O}}_{X,x}$ of complete local
rings lifts over $\text{\rm Spec}(R)$. Then the cover $f$ lifts to a
$G$-Galois cover $f_{R}\colon Z_{R}\longrightarrow X_{R}$ of connected
$R$-curves that is étale away from $B_{R}$.

######  Sketch of Proof.

The proof is essentially the same as the proof of [18, Theorem 3.1] that uses
formal patching. Set $U_{R}=X_{R}-B_{R}$. Since the cover $f$ is étale over
$U$, by [13, Proposition 22.5, page 147], there is a $G$-Galois étale cover
$g_{R}\colon V_{R}\longrightarrow U_{R}$ of connected $R$-curves such that
$g_{R}\times_{R}k=f|_{U}$. As each point $x\in B$ is a smooth point,
$\widehat{\mathcal{O}}_{X,x}$ is a complete discrete valuation ring. By our
assumption, there are $I_{x}$-Galois covers

$h_{x}\colon S_{x}\longrightarrow\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},x}\right)$

with respective special fibres $\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{Z,z}\right)\longrightarrow\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)$. Consider the induced
$G$-Galois covers

$g_{x}\colon\text{\rm Ind}_{I_{x}}^{G}S_{x}\longrightarrow\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},x}\right).$

Since the covers $h_{x}$ are generically étale, and $f_{R}$ is étale over
$\bigcup_{x\in X}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},\tilde{x}}\right)$ (here $\tilde{x}$
is the deleted formal neighborhood $\text{\rm Spec}\left(K_{X,x}\right)$ and
$K_{X,x}=\text{\rm Fr}(\widehat{\mathcal{O}}_{X,x})$), there is an
$G$-equivariant isomorphism

$V_{R}\times_{U_{R}}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},\tilde{x}}\right)\cong\text{\rm
Ind}_{I_{x}}^{G}S_{x}\times_{\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},x}\right)}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X_{R},\tilde{x}}\right)$

of Galois étale covers. We obtain our required $G$-Galois cover $f_{R}$ by
applying [10, Theorem 3.2.8]. ∎

We come back to our set up where $f\colon Y\longrightarrow X$ is a $G$-Galois
cover of connected projective seminormal $k$-curves. Recall from Section § 2.1
that the (necessarily normal) subgroup of $G$ generated by the Sylow
$p$-subgroups is denoted by $p(G)$. The cover $f$ factors as as composition

$f\colon Y\xrightarrow[]{p(G)\text{\rm-Galois}}Y^{\prime}\coloneqq
Y/p(G)\xrightarrow[]{G/p(G)\text{\rm-Galois}}X$

of Galois covers. Moreover, the cover $Y^{\prime}\longrightarrow X$ is étale
over $U$, of order prime-to-$p$. In particular, the inertia groups above
points in $x\in B$ are cyclic groups of order prime-to-$p$. Then the
extensions of complete local rings near each point $x\in B$ lifts over
$\text{\rm Spec}(R)$ ([18, Theorem 1.5, discussion preceding Theorem 3.1]). By
Proposition 4.2, there is a $G/p(G)$-Galois cover
$Y^{\prime}_{R}\longrightarrow X_{R}$ of connected $R$-curves that is étale
over $U_{R}$. Further assume that $K=\text{\rm Fr}(R)$ has characteristic $0$.
Without lose of generality, we may assume that $\mathbb{C}$ is an algebraic
closure of $K$. The generic fibre $Y^{\prime}_{R}\times_{R}K\longrightarrow
X_{R}\times_{R}K$ is again $G/p(G)$-Galois, étale over $U_{R}\times_{R}K$. Its
geometric fibre is a $G/p(G)$-Galois cover
$Y^{\prime}_{\mathbb{C}}\longrightarrow X_{\mathbb{C}}$ of connected
projective $\mathbb{C}$-curves that is étale over $U_{\mathbb{C}}\coloneqq
U_{R}\times_{R}\mathbb{C}$. As $U$ is irreducible, so is $U_{\mathbb{C}}$.
Moreover, $|X_{\mathbb{C}}-U_{\mathbb{C}}|=|B|=r$. Thus we can realize
$G/p(G)$ as a (continuous) finite quotient

(4.2) $\pi_{1}(U_{\mathbb{C}})\twoheadrightarrow G/p(G).$

######  Theorem 4.3.

Let $k$ be any algebraically closed field of arbitrary characteristic and $U$
be an integral affine $k$-curve. Let $X^{\nu}$ be the unique (up to
isomorphism) smooth projective connected $k$-curve with function field
$k(X^{\nu})=k(U)$. Let $U^{\nu}\subset X^{\nu}$ be the maximal affine open
subset together with a finite surjective morphism $\nu\colon
U^{\nu}\longrightarrow U$ that is an isomorphism over the smooth locus
$U_{\text{\rm smooth}}=U-U_{\text{\rm sing}}$. Suppose that $X^{\nu}$ has
genus $g$, $r=|X^{\nu}-U^{\nu}|$ and set

$\delta\coloneqq\sum_{u\in U_{\text{\rm sing}}}\left(|\nu^{-1}(u)|-1\right).$

A finite group $G$ occurs as a (continuous) quotient of the étale fundamental
group $\pi_{1}(U)$ if and only if $G/p(G)$ is generated by at most
$2g+r-1+\delta$ elements.

###### Proof.

We may assume that $\text{\rm char}(k)=p>0$ and $U$ is a singular curve
(otherwise, the theorem is well known). Observe that the triple $(g,r,\delta)$
in the statement is the same as the triple of numbers in Remark 4.1. By our
previous discussion and the isomorphism (4.1), we may assume that $U$ is a
seminormal curve embedded in an irreducible projective seminormal curve $X$
such that the finite set $B=X-U$ of $r$ points are all smooth points of $X$.
The map $\nu$ in the statement is the restriction of the normalization map
$\nu\colon X^{\nu}\longrightarrow X$.

Suppose that $G$ occurs as a continuous finite quotient of $\pi_{1}(U)$. Then
we have a surjection $\pi_{1}(U_{\mathbb{C}})\twoheadrightarrow G/p(G)$ as in
(4.2) for a similar complex affine curve $U_{\mathbb{C}}$. In particular,
$U_{\mathbb{C}}$ is an irreducible seminormal curve which have the same
associated triple $(g,r,\delta)$. By the general form of the Riemann Existence
Theorem ([24, Exp XII; Corollary 5.2]), $\pi_{1}(U_{\mathbb{C}})$ is the
profinite completion of the topological fundamental group
$\pi_{1}^{\text{top}}(U_{\mathbb{C}}(\mathbb{C}))$. As the projective curve
$X_{\mathbb{C}}$ is obtained from its normalization via identifying two points
at a time (see (2.5)), we conclude that the topological fundamental group
$\pi^{\text{top}}_{1}(U_{\mathbb{C}})$ is a finitely generate free group of
the form $\pi^{\text{top}}_{1}(\nu^{-1}\left(U_{\mathbb{C}}\right))\ast
F_{\delta}$ where $F_{\delta}$ is a finitely generated free group on
$\delta$-many generators and we again denote the normalization map
$X^{\nu}_{\mathbb{C}}\longrightarrow X_{\mathbb{C}}$ by $\nu$. Since
$\pi^{\text{top}}_{1}(\nu^{-1}\left(U_{\mathbb{C}}\right))$ is the free group
on $2g+r-1$ generators, the forward direction follows.

For the converse, suppose that $G$ is a finite group and $G/p(G)$ has a
generating set of size $m$, $m\leq 2g+r-1+\delta$. Let $P$ be a Sylow
$p$-subgroup of $G$, thus necessarily contained in $p(G)$. Let $N$ be the
normalizer subgroup $N=N_{G}(P)$. Consider the surjective homomorphism
$\pi\colon G\twoheadrightarrow G/p(G)$. By [8, Lemma 5.3], $N$ contains a
subgroup $H$ of order prime-to-$p$ such that $\pi(H)=G/p(G)$. In particular,
$G$ is generated by $H$ and $p(G)$. Let $a_{1},\,\ldots,\,a_{m}$ be the
generators of $G/p(G)$. Choose elements $h_{1},\,\ldots,\,h_{m}\in H$ with
$\pi(h_{i})=a_{i}$. Let $m^{\prime}\coloneqq\text{\rm min}\\{m,2g+r-1\\}$. Set
$H_{1}$ to be the subgroup of $H$ generated by
$h_{1},\,\ldots,\,h_{m^{\prime}}$. As $m^{\prime}\leq 2g+r-1$ and $H_{1}$ is a
group of order prime-to-$p$, by [24, Exp. XIII, Corollary 2.12], there is a
$H_{1}$-Galois cover $V_{1}\longrightarrow X^{\nu}$ of smooth connected
$k$-curves, étale away from the $r$ points in $\nu^{-1}(B)$. We apply
Proposition 3.4 inductively to obtain an $H$-Galois cover
$V_{2}\longrightarrow X$ of connected projective $k$-curves that is étale away
from the set $B$ of $r$ smooth points. Since $H$ has order prime-to-$p$, the
cover $V_{2}\longrightarrow X$ is at most tamely ramified. Fix a point $x\in
B$, and let $\langle c\rangle\cong\mathbb{Z}/s$ be an inertia group above $x$,
$(s,p)=1$. Then $c$ normalizes $P$. Consider the semi-direct product
$P\rtimes\langle c^{-1}\rangle$. Let
$g_{1}\colon\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ be any
$P\rtimes\langle c^{-1}\rangle$-Galois (Harbater-Katz-Gabber) cover, étale
away from $\\{0,\infty\\}$, totally ramified over $\infty$ and $\langle
c^{-1}\rangle$ occurs as an inertia group above $0$. By [11, Theorem 2.1], we
obtain a $\langle\,p(G),P\rtimes\langle\,c^{-1}\,\rangle\,\rangle$-Galois
cover $W\longrightarrow\mathbb{P}^{1}$, étale away from $\\{0,\infty\\}$ such
that $P\rtimes\langle\,c^{-1}\,\rangle$ occurs as an inertia group above
$\infty$ and $\langle\,c^{-1}\,\rangle$ occurs as an inertia group above $0$.
Finally we apply [9, Corollary to Patching Theorem]333The same proof works in
our set up as the branch point $x$ is assumed to be a smooth point. to this
cover and the $H$-Galois cover $V_{2}\longrightarrow X$ to obtain a $G$-Galois
cover $Z\longrightarrow X$, étale over $U$ and
$P\rtimes\langle\,c^{-1}\,\rangle$ occurs as an inertia group above $x$. This
uniquely corresponds to a (continuous) surjection
$\pi_{1}(U)\twoheadrightarrow G$. ∎

###  4.2. Tame Fundamental Groups and Inertia Conjecture

We retain the notation of Section § 4.1. As in the smooth case, we would also
like to get information about the tame fundamental group $\pi_{1}^{t}(U\subset
X)$ when $X$ is seminormal and $B\coloneqq X-U$ consists of $r$ smooth points.
Similar groups are studied in [21] and [31] when $X$ is a semistable curve and
$X-U$ consists of singular points.

The finite covers of $X$ that are étale over $U$ and tamely ramified over $B$
form a Galois Category $\text{\rm FT}_{U\subset X}$ with the obvious fibre
functor, $\pi_{1}^{t}(U\subset X)$ is the automorphism group of the fibre
functor, and we have an equivalence of categories $\text{\rm FT}_{U\subset
X}\overset{\sim}{\longrightarrow}\text{\rm Finite-}\pi_{1}^{t}(U\subset
X)\text{\rm-sets}$. For analogous complex curves $U_{\mathbb{C}}\subset
X_{\mathbb{C}}$, we also consider the covers of $X_{\mathbb{C}}$ that are
étale over $U_{\mathbb{C}}$, and the inertia groups of order prime-to-$p$; we
have the corresponding category $\text{\rm FT}_{U_{\mathbb{C}}\subset
X_{\mathbb{C}}}$ and the group $\pi_{1}^{p\text{\rm-
tame}}(U_{\mathbb{C}}\subset X_{\mathbb{C}})$. The results below are almost
similar to what we have obtained so far. To avoid repetition, we give a brief
sketch without going into the details.

Suppose that $f\colon Z\longrightarrow X$ is a $G$-Galois cover of connected
curves that is étale over $U$ and is tamely ramified over $B$. Using the same
notation and argument for the Galois cover $f$ in the beginning of this
section, we see that $f$ lifts over $\text{\rm Spec}(R)$ where $R$ is any
complete discrete valuation ring with residue field $k$, the fraction field
$K$ of characteristic $0$, and $\mathbb{C}$ is an algebraic closure of $K$. In
particular, we obtain a $G$-Galois cover $f_{\mathbb{C}}\colon
Z_{\mathbb{C}}\longrightarrow X_{\mathbb{C}}$ that is étale over
$U_{\mathbb{C}}$ and tamely ramified over $B_{\mathbb{C}}$. Further, the
ramification indices are also the same for $f$ and $f_{\mathbb{C}}$. The
construction of the cover also shows that the specialization functor
$\text{\rm FT}_{U_{R}\subset X_{R}}\longrightarrow\text{\rm FT}_{U\subset X}$
is an equivalence of categories, and the composite functor $\text{\rm
FT}_{U\subset X}\overset{\sim}{\longrightarrow}\text{\rm FT}_{U_{R}\subset
X_{R}}\longrightarrow\text{\rm FT}_{U_{\mathbb{C}}\subset X_{\mathbb{C}}}$
sends connected objects to connected objects. Thus we also have a surjective
specialization map

(4.3) $\text{\rm sp}^{t}\colon\pi_{1}^{p\text{\rm-tame}}(U_{\mathbb{C}}\subset
X_{\mathbb{C}})\twoheadrightarrow\pi_{1}^{t}(U\subset X).$

From the usual topological argument, we again see that $\pi_{1}^{p\text{\rm-
tame}}(U_{\mathbb{C}}\subset X_{\mathbb{C}})$ is the free product
$\pi_{1}^{p\text{\rm-tame}}(U^{\nu}_{\mathbb{C}})\ast\widehat{F_{\delta}}$.
Thus $\pi_{1}^{p\text{\rm-tame}}(U_{\mathbb{C}}\subset X_{\mathbb{C}})$ is a
free finitely generated profinite group, and the same is true for the group
$\pi_{1}^{t}(U\subset X)$. In particular, $\pi_{1}^{t}(U\subset X)$ is
completely determined by its finite quotients, and if the group has rank $a$,
then every finite group on $\leq a$ generators is a finite quotient of it.

Further, we can use the same argument as in the projective case (replacing FEt
by FT in (3.1)) to obtain a surjective homomorphism

$\pi_{1}^{t}(U^{\nu})\ast\widehat{F_{\delta}}\twoheadrightarrow\pi_{1}^{t}(U\subset
X).$

Finally, using the similar argument for the projective case—Theorem 3.3, we
see there is also a surjection in the other direction. As a consequence, we
obtain the following structure theorem for the tame fundamental group.

######  Theorem 4.4.

Let $X$ be an irreducible seminormal $k$-curve. Let $U\subset X$ be an affine
curve obtained by removing $r$ many closed points from $X$. Let $\nu\colon
X^{\nu}\longrightarrow X$ be the normalization map. Set $U^{\nu}=\nu^{-1}(U)$.
Then the tame fundamental group $\pi_{1}^{t}(U\subset X)$ is given as a free
product of free finitely generated profinite groups as follows.

$\pi_{1}^{t}(U\subset X)=\pi_{1}^{t}(U^{\nu})\ast\widehat{F_{\delta}}.$

The same proof of Corollary 3.8 1 then shows the following.

######  Corollary 4.5.

Under the above notation, the tame specialization map $\text{\rm sp}^{t}$ in
(4.3) induces an isomorphism of the maximal $p^{\prime}$-quotients.

######  Remark 4.6.

We remark that anabelian type conjectures for seminormal $k$-curves can be
posed; namely, what information do we get about the curve from its étale
fundamental group and subsequent quotients? The results from [29] for smooth
curves then translates to seminormal singular curves once we know how the
points in its normalizer are identified.

To understand the Galois covers of $\mathbb{P}^{1}_{k}$, Abhyankar posed the
Inertia Conjecture in [2, Section 16]. The status of the conjecture remain
open at this moment (see [12], [4] and [5] for the progress towards this
conjecture and for the generalizations). Since we now know the Galois groups
for covers of singular curves, such questions can be posed for singular curves
as well. In the following, we treat a particular case. We see that that whole
set up for [4, Section 6] comes into picture, even in this simple case.

Suppose that $Y=\mathbb{P}^{1}$, and let $X=Y/\left(0{\sim}1\right)$ be the
seminormal curve obtained from $Y$ by identifying the points $0$ and $1$. By
Theorem 4.3, a finite group $G$ occurs as a Galois group for some Galois cover
$Z\longrightarrow X$ that is étale away from $\infty$ if and only if $G$ is an
extension of the normal quasi $p$-subgroup $p(G)$ by a cyclic group. We want
to understand the possible inertia groups that occur over $\infty$ for all
such covers. We proceed as in [4, Section 6]. By [25, IV, Corollaries 3 and
4], an inertia group $I$ is necessarily of the form $I=P\rtimes\mathbb{Z}/m$
for a $p$-group $P$ and $(m,p)=1$. Suppose that $f\colon Z\longrightarrow X$
be a $G$-Galois cover of connected $k$-curves that is étale away from
$\infty$. Let $I=P\rtimes\mathbb{Z}/m$ be an inertia group above $\infty$.
Consider the (necessarily normal) subgroup
$H_{1}\coloneqq\langle\,P^{G}\rangle$, generated by the conjugates of $P$ in
$G$. Then $f$ factors as a composition
$Z\overset{h}{\longrightarrow}Z^{\prime}=Z/H_{1}\overset{g_{1}}{\longrightarrow}X$.
Since $P\subset H_{1}$, the cover $g_{1}$ is at most tamely ramified over
$\infty$, and étale away from $\infty$. If the cover $g_{1}$ has ramification
index $\geq 2$ over $\infty$, the same remains true for the normalized
pullback of $g_{1}$ over $\mathbb{P}^{1}$. Since $\pi_{1}^{t}(\mathbb{A}^{1})$
is trivial, this is not possible. So $g_{1}$ is étale everywhere. In
particular, $I\subset H_{1}$. Applying the Riemann Hurwitz theorem for étale
covers of Gorenstein curves, it also follows that $Z^{\prime}\cong X$.
Further, we can consider the normal subgroup
$H_{2}=\langle\,P^{H_{1}}\,\rangle$ of $H_{1}$ and factorize the cover $h$ as
a composition of an $H_{2}$-Galois cover followed by an étale $H/H_{1}$-Galois
cover $X\longrightarrow X$. This way, we obtain a factorization of $f$ as a
tower

$Z\longrightarrow
X\overset{g_{l}}{\longrightarrow}X\overset{g_{l-1}}{\longrightarrow}\cdots\overset{g_{1}}{\longrightarrow}X$

where $g_{j}$ is an $H_{j-1}/H_{j}$-Galois étale cover, and $H_{j}$’s are
inductively defined as follows.

$H_{0}=G\text{ \rm and }H_{j+1}=\langle\,P^{H_{j}}\,\rangle.$

We ask whether the converse holds.

######  Question 4.7 ([4, Question 6.2] for smooth case).

Let $X$, $G$, $I$ and $H_{j}$’s be as above.

  1. (1)

(Stronger Version) Given a tower

$g\colon
X\overset{g_{l}}{\longrightarrow}X\overset{g_{l-1}}{\longrightarrow}\cdots\overset{g_{1}}{\longrightarrow}X$

of $H_{j-1}/H_{j}$-Galois étale covers, does there exit a $G$-Galois cover
$Z\longrightarrow X$ dominating $g$ that is étale away from $\infty$ and $I$
occurs as as an inertia group above $\infty$?

  2. (2)

(Weaker Version) Suppose that $p(G)$ is generated by the conjugates of $P$ in
$p(G)$. does there exit a $G$-Galois cover $Z\longrightarrow X$, étale away
from $\infty$, and $I$ occurs as as an inertia group above $\infty$?

We end with a final remark.

######  Remark 4.8.

In the above notation, if there is a connected $p(G)$-Galois cover $f\colon
W\longrightarrow\mathbb{P}^{1}$ that is étale away from $\infty$ and $I$
occurs as an inertia group above $\infty$, the Weaker Version of Question 4.7
(2) has an affirmative answer. This is because $G$ is generated by $p(G)$ and
an element $g\in G$ of prime-to-$p$ order (see [8, Lemma 5.3]), and we can
apply the formal patching result [9, Corollary to Patching Theorem] to the
above cover $f$ and any $\langle g\rangle$ étale connected cover of $X$ (as
mentioned earlier, the proof of the loc. cit. holds in our situation of
singular curves; the later $\langle g\rangle$-Galois cover exists by Theorem
3.3). In particular, if the Inertia Conjecture holds for the group $p(G)$, the
weaker question has an affirmative answer. This observation applies when
$p(G)$ is an Alternating group of $d\geq p$ with $I=P$ a $p$-group ([3,
Theorem 1.7] and [5, Theorem 1.4] for $p$ odd; [5, Theorem 1.6] when $4\nmid
d$ and $p=2$).

## References

  * [1] Abhyankar S. S., Coverings of algebraic curves, Amer. J. Math., 79, 825–856, 1957.
  * [2] Abhyankar S. S., Resolution of singularities and modular Galois theory, Bull. Amer. Math. Soc. (N.S.), 38(2), 131–169, 2001.
  * [3] Das S., Kumar M., On the Inertia Conjecture for alternating group covers, J. Pure Appl. Algebra 224, no. 9, 106363, 14 pp., 2020.
  * [4] Das S., On the Inertia Conjecture and Its Generalizations, to appear in Israel J. Math. arXiv:2002.04934 [math.AG].
  * [5] Das S., Towards the Purely Wild Inertia Conjecture, submitted, arXiv:2111.15495 [math.AG].
  * [6] Ferrand, D., Conducteur, descente et pincement, Bull. Soc. Math. France, 131, 4, 553–585, 2003.
  * [7] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math., 32, pg 361, 1967.
  * [8] Harbater, D., Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117, no. 1, 1–25, 1994.
  * [9] Harbater, D., Fundamental groups and embedding problems in characteristic $p$. In Shreeram Shankar Abhyankar, editor, Recent developments in the inverse Galois problem (Seattle, WA, 1993), volume 186 of Contemp. Math., pages 353–369. American Mathematical Society, Providence, RI, 1995.
  * [10] Harbater, D., Patching and Galois theory, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41, 313–424, Cambridge Univ. Press, Cambridge, 2003.
  * [11] Harbater D., Abhyankar’s conjecture and embedding problems, J. Reine Angew. Math., 559, 1–24, 2003.
  * [12] Harbater D., Obus A., Pries R., Stevenson, K., Abhyankar’s Conjectures in Galois Theory: Current Status and Future Directions, Bull. Amer. Math. Soc. (N.S.) 55, no. 2, 239–-287, 2018.
  * [13] Hartshorne, R., Deformation theory, Graduate Texts in Mathematics, 257, Springer, New York, viii+234, 2010.
  * [14] Kaya, C., On lifting singular curves, Indian J. Pure Appl. Math., 43, 5, 423–428, 2012.
  * [15] Nakajima, S., On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math., vol. 75, no. 1, 1–8, 1984.
  * [16] Nikolov, N., Segal, D., On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2), vol. 165, no. 1, 171–238, 2007.
  * [17] Obus A., The (local) lifting problem for curves, Galois-Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math., vol. 63, Math. Soc. Japan, Tokyo, 359–412, 2012.
  * [18] Obus A., Lifting of curves with automorphisms, Open problems in arithmetic algebraic geometry, Adv. Lect. Math. (ALM), 46, 9–59, Int. Press, Somerville, MA, 2019.
  * [19] Raynaud M., Revêtements de la droite affine en caractéristique  $p>0$ et conjecture d’Abhyankar, Invent. Math., 116(1-3), 425–462, 1994.
  * [20] Rosenlicht M., Equivalence relations on algebraic curves, Ann. of Math. (2), 169–191, 56, 1952.
  * [21] Saïdi, Mohamed, Revêtements modérés et groupe fondamental de graphe de groupes, Compositio Math., 107, 3, 319–338, 1997.
  * [22] Shafarevitch, I., On $p$-extensions, Rec. Math. [Mat. Sbornik] N.S., vol. 20(62), 351–363, 1947.
  * [23] The Stacks project authors, The Stacks project, https://stacks.math.columbia.edu, 2022.
  * [24] Revêtements étales et groupe fondamental (French), Séminaire de Géométrie Algébrique du Bois Marie 1960-1961 (SGA 1). Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de Michéle Raynaud. Lecture Notes in Mathematics, Vol. 224. Springer-Verlag, Berlin-New York, 1971.
  * [25] Serre J. P., Local fields, vol. 67, Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg.
  * [26] Serre, J. P., Algebraic groups and class fields, Graduate Texts in Mathematics, 117, Translated from the French, Springer-Verlag, New York, x+207, 1988.
  * [27] Serre J. P., Construction de revêtements étales de la droite affine en caractêristique $p$., C. R. Acad. Sci. Paris Sêr. I Math. 311 (1990), no. 6, 341-–346.
  * [28] Stevenson K. F., Conditions related to $\pi_{1}$ of projective curves, J. Number Theory, vol. 69, no. 1, 62–79, 1998.
  * [29] Tamagawa, A., On the fundamental groups of curves over algebraically closed fields of characteristic $>0$, Internat. Math. Res. Notices, no. 16, 853–873, 1999.
  * [30] Vitulli, M. A., Weak normality and seminormality, Commutative algebra–Noetherian and non-Noetherian perspectives, 441–480, Springer, New York, 2011.
  * [31] Wewers, S., Deformation of tame admissible covers of curves, Aspects of Galois theory (Gainesville, FL, 1996), London Math. Soc. Lecture Note Ser., vol. 256, 239–282, Cambridge Univ. Press, Cambridge, 1999.