Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve

TL;DR

This paper computes the $S$-fundamental and Nori's fundamental group schemes of the $n$-fold symmetric product of a smooth projective curve over an algebraically closed field of characteristic p > 0.

Contribution

It provides the first explicit determination of the fundamental group schemes of symmetric products of curves in positive characteristic.

Findings

Determined the $S$-fundamental group scheme of $S^n(X)$.

Computed Nori's fundamental group scheme for $S^n(X)$.

Extended understanding of fundamental group schemes in algebraic geometry.

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.

Equations

$$\widetilde{{\psi }_{\ast }^S}:{\pi }^S(X,x{)}_{\mathrm{ab}}⟶{\pi }^S(S^n(X),nx).$$

$$\widetilde{{\psi }_{\ast }^N}:{\pi }^N(X,x{)}_{\mathrm{ab}}⟶{\pi }^N(S^n(X),nx).$$

$$\widetilde{{\psi }_{\ast }^{\text{eˊt}}}:{\pi }^{\text{eˊt}}(X,x{)}_{\mathrm{ab}}⟶{\pi }^{\text{eˊt}}(S^n(X),nx).$$

$$T_x:{\mathcal{C}}^{\mathrm{nf}}(X)⟶{\mathrm{Vect}}_k$$

$$T_x:\mathrm{EF}(X)⟶{\mathrm{Vect}}_k$$

$$n_1\ge \dots \ge n_r\ge 1.$$

$$\psi :X^n⟶S^n(X)$$

$$f_0:G^n\stackrel{{\alpha }^n}{⟶}(G_{\mathrm{ab}}{)}^n\stackrel{m}{⟶}{G}_{\mathrm{ab}},$$

$$G^n\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{F}:({\mathcal{C}}^{\mathrm{nf}}(S^n(X)),\otimes ,{\mathcal{O}}_{S^n(X)},T_{\psi (p)})\to ({\mathcal{C}}^{\mathrm{nf}}(X^n),\otimes ,{\mathcal{O}}_{X^n},T_p).$$

$${\psi }_{\ast }^S:{\pi }^S(X^n,p)⟶{\pi }^S(S^n(X),\psi (p)).$$

$${\pi }^S(X^n,(x,\dots ,x))\stackrel{\simeq }{⟶}{\pi }^S(X,x{)}^n.$$

$${\psi }_{\ast }^S:{\pi }^S(X,x{)}^n\to {\pi }^S(S^n(X),nx),$$

$$\varphi :{\pi }^S(X,x{)}^n⟶{\pi }^S(X,x{)}_{\mathrm{ab}}$$

$${\pi }^S(X,x{)}^n⟶({\pi }^S(X,x{)}_{\mathrm{ab}}{)}^n\stackrel{m}{⟶}{\pi }^S(X,x{)}_{\mathrm{ab}},$$

$$\widetilde{{\psi }_{\ast }^S}:{\pi }^S(X,x{)}_{\mathrm{ab}}⟶{\pi }^S(S^n(X),nx).$$

$${\psi }_{\ast }^S:{\pi }^S(X^n,(x,\dots ,x))⟶{\pi }^S(S^n(X),\psi (x,\dots ,x))$$

$$\tilde{\theta }:{\mathrm{Rep}}_k(G^{\prime })⟶{\mathrm{Rep}}_k(G)$$

$$\widetilde{{\psi }_{\ast }^S}:{\pi }^S(X,x{)}_{\mathrm{ab}}⟶{\pi }^S(S^n(X),nx)$$

$${\pi }^S(X^n,(x,\dots ,x))\stackrel{f_0}{⟶}{\pi }^S(X,x{)}_{\mathrm{ab}}\stackrel{{\rho }_i}{⟶}\mathrm{GL}(V_i),\forall i=1,2.$$

$$0⟶\mathcal{K}⟶{\mathcal{E}}_1⟶\mathcal{L}⟶0$$

$$\tilde{C}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to ({\psi }_{\ast }\mathcal{K}{)}^{S_n}⟶({\psi }_{\ast }{\mathcal{E}}_1{)}^{S_n}\cong E_1\stackrel{q}{⟶}({\psi }_{\ast }\mathcal{L}{)}^{S_n}=L⟶0$$

$${\mathrm{Hom}}_{S^n(X)}(E_1,E_2)⟶{\mathrm{Hom}}_{X^n}({\psi }^{\ast }{E}_1,{\psi }^{\ast }{E}_2)$$

$$f_{\ast }^S:{\pi }^S(\mathbb{P}(E),y)⟶{\pi }^S(X,f(y))$$

$$\widetilde{{\psi }_{\ast }^S}:{\pi }^S(X,x{)}_{\mathrm{ab}}⟶{\pi }^S(S^n(X),nx)$$

$$\sum _{i=1}^n{x}_i\mapsto \sum _{i=1}^n{x}_i-nx,$$

$${\eta }_{\ast }:{\pi }^S(S^n(X),nx)\stackrel{\sim }{⟶}{\pi }^S(\mathrm{Alb}(X),0).$$

$${\mathrm{alb}}_{X\ast }:{\pi }^S(X,x)\stackrel{a_0}{⟶}{\pi }^S(X,x{)}_{\mathrm{ab}}\stackrel{\widetilde{{\mathrm{alb}}_{X\ast }}}{⟶}{\pi }^S(\mathrm{Alb}(X),0),$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction

For a connected reduced complete scheme $X$ defined over a perfect field $k$ and having a $k$-rational point $x$, in [Nor76, Nor82], Nori introduced an affine $k$-group scheme $\pi^{N}(X,x)$ associated to the neutral Tannakian category of essentially finite vector bundles on $X$, known as Nori’s fundamental group scheme. This group scheme carries more informations than the étale fundamental group scheme $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)$ in positive characteristic, and is the same as $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)$ when $k=\mathbb{C}$. For a connected smooth projective curve defined over an algebraically closed field $k$, in [BPS06], Biswas, Parameswaran and Subramanian defined and studied the $S$-fundamental group scheme $\pi^{S}(X,x)$ of $X$. This is further generalized and extensively studied for higher dimensional smooth projective varieties over algebraically closed fields by Langer in [Lan11, Lan12]. It is an interesting question to find $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)$, $\pi^{N}(X,x)$ and $\pi^{S}(X,x)$ for well-known algebraic varieties.

Let $X$ be a connected smooth projective curve defined over an algebraically closed field $k$ of characteristic $p>0$. Fix an integer $n\geq 2$, and let $S_{n}$ be the permutation group of $n$ symbols. Then $S_{n}$ acts on $X^{n}$ by permutation of its factors, and the associated quotient $S^{n}(X):=X^{n}/S_{n}$ is a connected smooth projective variety over $k$. For any affine $k$-group scheme $G$ we denote by $G_{\rm ab}$ its abelianization. In this article we prove the following results.

Theorem 1** (Theorem 3.5.1).**

For any closed point $x\in X(k)$, there is an isomorphism of affine $k$-group schemes

[TABLE]

Theorem 2** (Theorem 3.5.3).**

For any closed point $x\in X(k)$, there is an isomorphism of affine $k$-group schemes

[TABLE]

As a consequence we also obtain the following result, which is already contained in [BH15], and proved using a different method. For any closed point $x\in X(k)$, there is an isomorphism of affine $k$-group schemes

[TABLE]

Note that when $n>2g-2$, where $g$ is the genus of $X$, these isomorphisms can be easily obtained from results in [Lan12, Section 7], since $S^{n}(X)$ is a projective bundle over ${\rm Alb}(X)$. We prove the above results without any restriction on $n$. Our initial strategy was to use the same method as in [PS19] under the assumption that ${\rm char}(k)>3$. However, we observed that one can avoid using the characterization of numerically flat sheaves as strongly semistable reflexive sheaves with vanishing Chern classes; proved in [Lan11]. Instead, we first show that $\widetilde{\psi_{*}^{S}}$ is faithfully flat and then use [Lan12, Section 7] to conclude that it is an isomorphism.

2. Fundamental Group Schemes

Let $k$ be an algebraically closed field. Let $X$ be a reduced proper $k$-scheme, which is connected in the sense that $H^{0}(X,\mathcal{O}_{X})\cong k$.

2.1. $S$-fundamental group scheme

Let ${\rm Coh}(X)$ be the category of coherent sheaf of $\mathcal{O}_{X}$-modules on $X$. This has a full subcategory ${\rm Vect}(X)$, whose objects are locally free coherent sheaves (vector bundles) on $X$. A vector bundle $E$ on $X$ is said to be nef if $\mathcal{O}_{\mathbb{P}(E)}(1)$ is a nef line bundle on $\mathbb{P}(E)$. An object $E$ of ${\rm Coh}(X)$ is said to be numerically flat if $E$ is locally free and both $E$ and its dual $E^{\vee}$ are nef. Let $\mathscr{C}^{\rm nf}(X)$ be the full subcategory of ${\rm Coh}(X)$, whose objects are numerically flat vector bundles on $X$. It is known that, $E\in{\rm Ob}({\rm Coh}(X))$ is an object of $\mathscr{C}^{\rm nf}(X)$ if and only if $E$ is locally free and for any smooth projective curve $C$ over $k$ and any morphism $f:C\longrightarrow X$, its pullback $f^{*}E$ on $C$ is slope semistable and of degree [math] (see [Lan11, Remark 5.2]). Note that $\mathscr{C}^{\rm nf}(X)$ is closed under finite direct sum and tensor products. Choosing a closed point $x\in X(k)$, one can define a fiber functor

[TABLE]

by sending an object $E$ of $\mathscr{C}^{\rm nf}(X)$ to its fiber $E_{x}$ at $x$. The quadruple $(\mathscr{C}^{\rm nf}(X),\otimes,\mathcal{O}_{X},T_{x})$ is a neutral Tannakian category (see [Lan11, Proposition 5.5]), and the affine $k$-group scheme $\pi^{S}(X,x)$ Tannaka dual to this is known as the $S$-fundamental group scheme of $X$ with base point $x$.

Let $X$ be a connected smooth projective variety of dimension $d$ over $k$. Fix an ample divisor $H$ on $X$. Let ${\rm Vect}_{0}^{s}(X)$ be the full subcategory of ${\rm Coh}(X)$, whose objects are reflexive coherent sheaves $E$ on $X$, that are strongly $H$-semistable and ${\rm ch}_{1}(E)\cdot H^{d-1}={\rm ch}_{2}(E)\cdot H^{d-2}=0$, where ${\rm ch}_{i}(E)$ is the $i$-th Chern character of $E$. It is shown in [Lan11, Proposition 5.1] that the objects of the category ${\rm Vect}_{0}^{s}(X)$ are in fact locally free coherent sheaves on $X$ and all of their Chern classes vanishes. It follows from [Lan11, Proposition 4.5] that the category ${\rm Vect}_{0}^{s}(X)$ does not depend on choice of $H$. For $X$ smooth, the categories $\mathscr{C}^{\rm nf}(X)$ and ${\rm Vect}_{0}^{s}(X)$ are the same (see [Lan11, Proposition 5.1], [Lan12, Theorem 2.2]). We will not use this characterization here, however, this was crucial in [PS19].

It is clear from the definition of the categories ${\rm Vect}_{0}^{s}(X)$ and ${\rm EF}(X)$ that $\pi^{S}(X,x)$ carries more informations than $\pi^{N}(X,x)$. In fact, there are natural faithfully flat homomorphisms of affine $k$-group schemes $\pi^{S}(X,x)\longrightarrow\pi^{N}(X,x)\longrightarrow\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)$, (see [Lan11, Lemma 6.2]).

2.2. Nori’s fundamental group scheme

Definition 2.2.1**.**

A vector bundle $E$ on $X$ is said to be finite if there are two distinct non-zero polynomials $f$ and $g$ with positive integer coefficients such that $f(E)\cong g(E)$.

A vector bundle $E$ on $X$ is said to be essentially finite if there are finitely many finite vector bundles $E_{1},\ldots,E_{n}$ and two numerically flat vector bundles $V_{1}$ and $V_{2}$ with $V_{2}\subseteq V_{1}\subseteq\bigoplus\limits_{i=1}^{n}E_{i}$ such that $E\cong V_{1}/V_{2}$.

Let ${\rm EF}(X)$ be the full subcategory of ${\rm Vect}(X)$ whose objects are essentially finite vector bundles on $X$. Then ${\rm EF}(X)$ is an abelian rigid tensor category. Let ${\rm Vect}_{k}$ be the category of $k$-vector spaces. Fixing a closed point $x\in X(k)$, we have a fiber functor

[TABLE]

defined by sending a vector bundle $E\in{\rm Ob}({\rm EF}(X))$ to its fiber $E_{x}$ at $x$. This makes the quadruple $({\rm EF}(X),\otimes,\mathcal{O}_{X},T_{x})$ a neutral Tannakian category. The affine $k$-group scheme $\pi^{N}(X,x)$ Tannaka dual to this category is called Nori’s fundamental group scheme of $X$ with base point $x$.

3. Fundamental Group Schemes of $S^{n}(X)$

3.1. Symmetric product of curve

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be an irreducible smooth projective curve over $k$. Fix an integer $n\geq 2$, and let us denote by $S_{n}$ the permutation group of $n$ symbols. There is a natural action of $S_{n}$ on the $n$-fold product $X^{n}$, and the associated quotient $X^{n}/S_{n}$, denoted by $S^{n}(X)$, is a smooth projective variety of dimension $n$ over $k$. Note that any closed point $q\in S^{n}(X)$ can be uniquely written as $\sum\limits_{i=1}^{r}n_{i}x_{i}$, where $x_{1},\ldots,x_{r}$ are distinct closed points of $X$ and $n_{1},\ldots,n_{r}$ are integers with

[TABLE]

We call $\langle n_{1},\ldots,n_{r}\rangle$ the type of $q$. The quotient morphism

[TABLE]

is a faithfully flat finite morphism of $k$-schemes.

3.2. A group theoretic lemma

A proof of the following easy Lemma can be found in [PS19].

Lemma 3.2.1**.**

Let $G$ and $H$ be two group schemes over $k$. For an integer $n\geq 2$, we denote by $G^{n}$ the group scheme $G\times\cdots\times G$. Then $S_{n}$ acts on $G^{n}$ by permuting the factors. Let $f_{0}$ be the following composite group homomorphism

[TABLE]

where $\alpha:G\to G_{\rm ab}:=G/[G,G]$ denotes the abelianization homomorphism and $m$ denotes the multiplication homomorphism. Then a homomorphism of $k$-group schemes $f:G^{n}\longrightarrow H$ is $S_{n}$-invariant if and only if there is a homomorphism $\tilde{f}:G_{\rm ab}\longrightarrow H$ of affine $k$-group schemes such that $\tilde{f}\circ f_{0}=f$. In other words, $f$ is $S_{n}$-invariant iff there if $\tilde{f}$ which makes the following diagram commutes.

[TABLE]

3.3. Construction of homomorphism

The functor which sends $E\in\mathscr{C}^{\rm nf}(S^{n}(X))$ to $\psi^{*}E\in\mathscr{C}^{\rm nf}(X^{n})$ defines a morphism of neutral Tannakian categories (for any closed point $p\in X^{n}(k)$)

[TABLE]

Thus, we get a homomorphism

[TABLE]

It follows from [Lan12, Theorem 4.1, p. 842] that, for any closed point $x\in X(k)$, the natural homomorphism of affine $k$-group schemes

[TABLE]

is an isomorphism. By abuse of notation, denote by $\psi_{*}^{S}$ the composite of the inverse of this isomorphism and $\psi^{S}_{*}$. So now

[TABLE]

where $nx=\psi(x,\ldots,x)$.

The natural $S_{n}$-action on $X^{n}$ gives rise to automorphisms $\sigma_{*}$ of the affine $k$-group scheme $\pi^{S}(X^{n},(x,\ldots,x))\cong\pi^{S}(X,x)^{n}$, for all $\sigma\in S_{n}$. Now one can check that $\psi_{*}^{S}\circ\sigma_{*}=\psi_{*}^{S}$, where $\psi_{*}^{S}$ is the homomorphism defined in (3.3.2) with $p=(x,\ldots,x)$. Consider the natural homomorphism of affine $k$-group schemes

[TABLE]

defined as the following composite homomorphism

[TABLE]

where the first homomorphism is given by abelianization at each factor, and the second homomorphism is the multiplication. Then it follows from Lemma 3.2.1 that the homomorphism $\psi_{*}^{S}$ in (3.3.2) factors through a homomorphism

[TABLE]

We record the above discussion in the following proposition.

Proposition 3.3.4**.**

The map

[TABLE]

factors to give a homomorphism $\widetilde{\psi_{*}^{S}}:\pi^{S}(X,x)_{\rm ab}\longrightarrow\pi^{S}(S^{n}(X),nx)$.

A vector bundle $E$ on $X^{n}$ is said to be $S_{n}$-invariant if $\sigma^{*}E\cong E$, for all $\sigma\in S_{n}$.

Proposition 3.3.5**.**

Let $E$ be a vector bundle in $\mathscr{C}^{\rm nf}(X^{n})$ associated to a representation $\rho:\pi^{S}(X^{n},(x,\ldots,x))\cong\pi^{S}(X,x)^{n}\to\operatorname{\textnormal{GL}}(V)$. If $\rho$ factors through $\pi^{S}(X,x)_{\rm ab}$, as in Lemma 3.2.1, then $E$ is $S_{n}$-invariant.

Proof.

From the hypothesis it follows that $\rho\circ\sigma_{*}=\rho$. The proposition follows. ∎

3.4. Faithfully flat

In this subsection we use [DMOS82, Proposition 2.21] to show that the homomorphism $\widetilde{\psi_{*}^{S}}$ in (3.3.3) is faithfully flat. We begin by recalling this result for the convenience of the reader. Let $\theta:G\longrightarrow G^{\prime}$ be a homomorphism of affine group schemes over $k$, and let

[TABLE]

be the functor given by sending $\rho^{\prime}:G^{\prime}\to\operatorname{\textnormal{GL}}(V)$ to $\rho^{\prime}\circ\theta:G\to\operatorname{\textnormal{GL}}(V)$. An object $\rho:G\to\operatorname{\textnormal{GL}}(V)$ in $\operatorname{\textnormal{Rep}}_{k}(G)$ is said to be a subquotient of an object $\eta:G\to\operatorname{\textnormal{GL}}(W)$ in $\operatorname{\textnormal{Rep}}_{k}(G)$ if there are two $G$-submodules $V_{1}\subset V_{2}$ of $W$ such that $V\cong V_{2}/V_{1}$ as $G$-modules.

Proposition 3.4.2** (Proposition 2.21, [DMOS82]).**

Let $\theta:G\longrightarrow G^{\prime}$ be a homomorphism of affine algebraic groups over $k$. Then

1. (a)

$\theta$* is faithfully flat if and only if the functor $\widetilde{\theta}$ in (3.4.1) is fully faithful and given any subobject $W\subset\widetilde{\theta}(V^{\prime})$, with $V^{\prime}\in\operatorname{\textnormal{Rep}}_{k}(G^{\prime})$, there is a subobject $W^{\prime}\subset V^{\prime}$ in $\operatorname{\textnormal{Rep}}_{k}(G^{\prime})$ such that $\widetilde{\theta}(W^{\prime})\cong W$ in $\operatorname{\textnormal{Rep}}_{k}(G)$.* 2. (b)

$f$* is a closed immersion if and only if every object of $\operatorname{\textnormal{Rep}}_{k}(G)$ is isomorphic to a subquotient of an object of the form $\widetilde{\theta}(V^{\prime})$, for some $V^{\prime}\in\operatorname{\textnormal{Rep}}_{k}(G^{\prime})$.*

Proposition 3.4.3**.**

The homomorphism

[TABLE]

defined in (3.3.3) is faithfully flat.

Proof.

We will apply [DMOS82, Proposition 2.21 (a)]. Let $E_{1}$ be an object in the category $\mathscr{C}^{\rm nf}(S^{n}(X))$. Clearly $\psi^{*}E_{1}$ has the same rank as that of $E_{1}$. If $\mathcal{E}_{2}\subset\mathcal{E}_{1}:=\psi^{*}E_{1}$ is a subbundle corresponding to a representation of $\pi^{S}(X,x)_{\rm ab}$, we need to show that there is a subbundle $E_{2}\subset E_{1}$ such that $\psi^{*}E_{2}=\mathcal{E}_{2}$. We will prove this by induction on rank of $E_{1}$. If ${\rm rank}(E_{1})=1$, there is nothing to prove. Assume that ${\rm rank}(E_{1})\geq 2$.

The vector bundles $\mathcal{E}_{i}$ corresponds to a representation

[TABLE]

It follows from Proposition 3.3.5 that $\mathcal{E}_{2}$ is a $S_{n}$-invariant numerically flat vector bundle on $X^{n}$. Since $\pi^{S}(X,x)_{\rm ab}$ is an abelian $k$-group scheme, it follows from [Wat79, Theorem 9.4, p. 70], that we can find a surjective $\pi^{S}(X,x)_{\rm ab}$-module homomorphism $V_{1}\to L_{1}$, where $L_{1}$ is one dimensional and $V_{2}$ is a $\pi^{S}(X,x)_{\rm ab}$-submodule of the kernel of this homomorphism. Let $\mathcal{L}$ be the line bundle on $X^{n}$ corresponding to the representation $L_{1}$. Then it is clear that $\mathcal{L}$ is $S_{n}$-invariant (see Proposition 3.3.5) and there is an $S_{n}$-equivariant exact sequence of vector bundles

[TABLE]

on $X^{n}$ such that $\mathcal{E}_{2}\subset\mathcal{K}$.

Every $S_{n}$-invariant line bundle on $X^{n}$ is the pullback of a line bundle from $S^{n}(X)$ (see [Fog77, Proposition 3.6], also [PS19, Proposition 5.1.1]). Therefore, it follows that $L:=(\psi_{*}\mathcal{L})^{S_{n}}$ is a line bundle on all of $S^{n}(X)$. We now show that $L$ is numerically flat on $S^{n}(X)$. Given a morphism $C\longrightarrow S^{n}(X)$ from a smooth projective curve $C$ into $S^{n}(X)$, we can find a curve $\widetilde{C}$ and a morphism $\widetilde{C}\longrightarrow C$ making the following diagram commutative.

[TABLE]

Since $\mathcal{L}$ is numerically flat on $X^{n}$ and $\mathcal{L}\cong\psi^{*}L$, it follows that $L$ is numerically flat.

We claim that

[TABLE]

is exact. The sequence (3.4.4) can fail to be exact only on the right. Since both $E_{1}$ and $L$ are numerically flat and $L$ is a line bundle, $q$ is surjective since it is nonzero. This proves the exactness of (3.4.4). It follows that $K:=(\psi_{*}\mathcal{K})^{S_{n}}$ is locally free and numerically flat on $S^{n}(X)$. It is clear that $\psi^{*}K=\mathcal{K}$ on $X^{n}$. Since $\mathcal{E}_{2}\subset\mathcal{K}$ the assertion that there is $E_{2}\subset E_{1}$ such that $\mathcal{E}_{2}=\psi^{*}E_{2}$ on $X^{n}$ follows by induction on rank.

To complete the proof of the Proposition, we need to show that if $E_{1}$ and $E_{2}$ are numerically flat vector bundles on $S^{n}(X)$ then the natural map

[TABLE]

is bijective. It is clear that this natural map is injective (faithful). Therefore, it suffices to show that given any numerically flat vector bundle $E$ on $S^{n}(X)$, any nonzero homomorphism $\phi:\mathcal{O}_{X^{n}}\longrightarrow\psi^{*}E$ comes from a nonzero homomorphism $\widetilde{\phi}:\mathcal{O}_{S^{n}(X)}\longrightarrow E$. Since the homomorphism $\pi^{S}(X^{n},x)\longrightarrow\pi^{S}(X,x)_{\rm ab}$ is faithfully flat, and $\psi^{*}E$ corresponds to a representation of $\pi^{S}(X,x)_{\rm ab}$, it follows that $\phi$ is a morphism between two representations of $\pi^{S}(X,x)_{\rm ab}$. This shows that $\phi$ is $S_{n}$-equivariant on $X^{n}$. Now from the preceding discussion it follows that $\phi$ arises from a morphism $\mathcal{O}_{S^{n}(X)}\longrightarrow E$. ∎

3.5. Proofs of Theorems

Let $X$ be a connected smooth projective variety over $k$ and $f:\mathbb{P}(E)\longrightarrow X$ be a projective bundle over $X$. It is easy to see, using $\pi^{S}(\mathbb{P}^{n},s)=\{1\}$ and [Har77, Corollary 12.9, Chapter III], that for a numerically flat sheaf $F$ on $\mathbb{P}(E)$, the sheaf $f_{*}F$ is locally free and the natural map $f^{*}f_{*}F\to F$ is an isomorphism. From this it easily follows that the homomorphism of $S$-fundamental group schemes

[TABLE]

is an isomorphism, for all $y\in\mathbb{P}(E)$.

Theorem 3.5.1**.**

The homomorphism of affine $k$-group schemes

[TABLE]

is an isomorphism, for all $x\in X(k)$.

Proof.

Let ${\rm Alb}(X)$ be the Albanese variety of $X$. Let $g$ be the genus of the curve $X$. Fix a closed point $x\in X(k)$. If $n\geq 2g-1$, then the morphism $\eta:S^{n}(X)\to{\rm Alb}(X)$ given by

[TABLE]

makes $S^{n}(X)$ into a projective bundle over ${\rm Alb}(X)$. It follows that the induced homomorphism of affine $k$-group schemes is an isomorphism,

[TABLE]

From [Lan12, Section 7] it follows that the Albanese morphism ${\rm alb}_{X}:X\longrightarrow{\rm Alb}(X)$ given by $t\mapsto t-x$ induces maps

[TABLE]

where $\widetilde{{\rm alb}_{X*}}$ is an isomorphism. Consider the commutative diagram

[TABLE]

where $a(t)=(t,x,x,\ldots,x)$. At the level of $S$-fundamental group schemes we get the commutative diagram

[TABLE]

We have $\eta\circ\psi\circ a={\rm alb}_{X}$. It is easy to check that $(\psi\circ a)_{*}=\widetilde{\psi^{S}_{*}}\circ a_{0}$. Since $a_{0}$ is faithfully flat and

[TABLE]

it follows that $\widetilde{\psi^{S}_{*}}=\eta_{*}^{-1}\circ\widetilde{{\rm alb}_{X*}}$ and so the theorem is true is $n\geq 2g-1$.

Assume that $n<2g-1$. Consider the maps

[TABLE]

where $a(t)=(t,x,x,\ldots,x)$ and $c(\sum_{i=1}^{n}x_{i})=\sum_{i=1}^{n}x_{i}+(2g-1-n)x$. It is clear that the composite morphism is ${\rm alb}_{X}$. As above, one easily checks that

[TABLE]

Thus, we get homomorphisms of affine $k$-group schemes

[TABLE]

such that the composite homomorphism is an isomorphism. This forces that the first map is a closed immersion. Since we know from Proposition 3.4.3 that the first map is faithfully flat, the theorem follows. ∎

Remark 3.5.2**.**

That $\widetilde{\psi_{*}^{S}}$ is a closed immersion could have been proved using the same method in [PS19] under the assumption that ${\rm char}(k)>3$.

Let $X$ be a reduced proper $k$-scheme with $H^{0}(X,\mathcal{O}_{X})=k$. Let $E$ be an essentially finite vector bundle on $X$. Then there is a finite $k$-group scheme $G$, a principal $G$-bundle $p:P\to X$ and a finite dimensional $k$-linear representation $\rho:G\to\operatorname{\textnormal{GL}}(W)$ such that $E\cong P\times^{\rho}W$, the vector bundle associated to the representation $\rho$. It follows from the proof of [Nor76, Proposition 3.8] that there is a finite vector bundle $\mathcal{V}$ on $X$ such that $E$ is a subbundle of $\mathcal{V}$.

As before, let $X$ be a connected smooth projective curve over $k$ and $S^{n}(X)$ the $n$-fold symmetric product of $X$. It is clear that $\psi^{*}$ takes a finite vector bundle to a finite vector bundle. Thus, $\psi^{*}E\subset\psi^{*}\mathcal{V}$, which shows that $\mathscr{F}$ takes essentially finite vector bundles to essentially finite vector bundles. Note that there is a commutative diagram of homomorphisms of affine $k$-group schemes

[TABLE]

where the vertical arrows are faithfully flat by [Lan11, Lemma 6.2]. It follows that $\widetilde{\psi_{*}^{N}}$ is faithfully flat.

Now let $\mathcal{E}$ be an essentially finite $S_{n}$-invariant vector bundle on $X^{n}$. It is easy to find a finite and $S_{n}$-invariant bundle $\mathcal{V}$ on $X^{n}$ and an $S_{n}$-equivariant inclusion $\mathcal{E}\subset\mathcal{V}$. Define $E=(\psi_{*}\mathcal{E})^{S_{n}}$ and $V:=(\psi_{*}\mathcal{V})^{S_{n}}$. It is clear that $V$ is a finite vector bundle and $E\subset V$. So $E$ is essentially finite and $\mathscr{F}(E)=\mathcal{E}$. This shows that $\widetilde{f}^{N}$ is a closed immersion. Thus, we have the following.

Theorem 3.5.3**.**

There is a natural isomorphism of affine $k$-group schemes

[TABLE]

3.6. Étale Fundamental Group Scheme of $S^{n}(X)$

In this subsection we sketch how to deduce from Theorem 3.5.3 the same assertion for étale fundamental group schemes. This result is a special case of [BH15, Theorem 1.2]. Note that there is a commutative diagram

[TABLE]

From this it follows that $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)_{\rm ab}\longrightarrow\pi^{\operatorname{\textnormal{\'{e}t}}}(S^{n}(X),nx)$ is faithfully flat. Consider a homomorphism $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)_{\rm ab}\to\operatorname{\textnormal{GL}}(V)$. It follows using [Nor76, Proposition 3.10] that this homomorphism factors through a finite and reduced $k$-group scheme $G$. Now consider the diagram

[TABLE]

The right vertical arrow is the unique map which makes the square commute. It factors through $d$ since $G$ is finite and reduced. Now it follows from [DMOS82, Proposition 2.21 (b)] that $\pi^{\operatorname{\textnormal{\'{e}t}}}(X,x)_{\rm ab}\longrightarrow\pi^{\operatorname{\textnormal{\'{e}t}}}(S^{n}(X),nx)$ is a closed immersion. This proves the following.

Theorem 3.6.1**.**

For any closed point $x\in X(k)$, there is an isomorphism of affine $k$-group schemes

[TABLE]