$\displaystyle\operatorname{\textit{G}\mathbb{J}}(X)\times J(X)$
$\displaystyle\longrightarrow\operatorname{\textit{G}\mathbb{J}}(X)$
$\displaystyle(\mathcal{L},\mathcal{E})\hskip 15.00002pt$
$\displaystyle\longmapsto\mathcal{L}\otimes\mathcal{E}.$

Finally, we recall the definition of the geometric Abel map. Let
$\operatorname{Hilb}_{X}=\bigsqcup_{d\geq 0}\operatorname{Hilb}^{d}_{X}$ be
the Hilbert scheme of effective generalized divisor of any degree on $X$.

######  Definition 2.2.8.

Let $M$ be a line bundle of degree $e$ on $X$. The ($M$-twisted) Abel map of
degree $d$ is defined as:

$\displaystyle\mathcal{A}_{M}^{d}:\hskip
20.00003pt\operatorname{Hilb}_{X}^{d}$
$\displaystyle\longrightarrow\operatorname{\textit{G}\mathbb{J}}(X,-d+e)$
$\displaystyle D$ $\displaystyle\longmapsto\mathcal{I}_{D}\otimes M.$

The $M$-twisted Abel map in any degree is defined similary as a map:

$\mathcal{A}_{M}:\hskip
10.00002pt\operatorname{Hilb}_{X}\longrightarrow\operatorname{\textit{G}\mathbb{J}}(X)$

The restriction of $\mathcal{A}_{M}^{d}$ to
$\operatorname{{}^{\ell}Hilb}_{X}^{d}$ takes values in $J(X,-d+e)$. Taking
care of the twisting, the Abel map is equivariant with respect to the sum of
effective Cartier divisors: for any pair of line bundles $M$ and $N$ on $X$
and for any $D\in\operatorname{Hilb}_{X}$,
$E\in\operatorname{{}^{\ell}Hilb}_{X}$,

$\mathcal{A}_{M\otimes
N}(D+E)\simeq\mathcal{A}_{M}(D)\otimes\mathcal{A}_{N}(E).$

###  2.3. The direct and inverse image for generalized divisors and
generalized line bundles

Let $\pi:X\rightarrow Y$ be a finite, flat morphism of degree $n$ between
curves. In the present section, we extend the notion of direct and inverse
image from Cartier divisors to generalized divisors and generalized line
bundles.

####  2.3.1. The Fitting ideal

A fundamental tool for the definition of the direct image for generalized
divisors is the Fitting ideal of a module (and of a sheaf of modules), of
which we recall here the definition. See [Eis95, Chapter 20], [Vas04, Chapter
2.4] and [EH06, Chapter V.1.3] for a detailed treatment.

######  Definition/Lemma 2.3.1.

Let $X$ be a scheme and let $\mathcal{F}$ be a coherent sheaf on $X$. Let

$\mathcal{E}_{1}\xrightarrow{\hskip 10.00002pt\psi\hskip
10.00002pt}\mathcal{E}_{0}\longrightarrow\mathcal{F}\rightarrow 0$

be any finite presentation of $\mathcal{F}$, with $\mathcal{E}_{0}$ locally
free of rank $r$. The $0$-th Fitting ideal of $\mathcal{F}$, denoted
$\operatorname{Fitt}_{0}(\mathcal{F})$, is defined as the image of the map:

$\textstyle{\bigwedge^{r}}\mathcal{E}_{1}\otimes\left(\textstyle{\bigwedge^{r}}\mathcal{E}_{0}\right)^{-1}\xrightarrow{\hskip
10.00002pt\Psi\hskip 10.00002pt}\mathcal{O}_{X}$

induced by
$\bigwedge^{r}\psi:\bigwedge^{r}\mathcal{E}_{1}\rightarrow\bigwedge^{r}\mathcal{E}_{0}$
and is independent of the choiche of the presentation for $\mathcal{F}$. If
$\mathcal{E}_{1}$ is locally free, then $\psi$ can be locally represented by a
matrix and the $0$-th Fitting ideal is generated locally by the minors of size
$r$ of such matrix, with the convention that the determinant of the $0\times
0$ matrix is $1$.

###### Proof.

The definition is the sheaf-theoretic reformulation of [Eis95, Corollary-
Definition 20.4]. ∎

As a consequence of the definition, the formation of Fitting ideals commutes
with restrictions and with base change [Eis95, Corollary 20.5].

######  Definition/Lemma 2.3.2.

Let $X$ be a scheme and $\mathcal{F}$ be a coherent sheaf on $X$. The $0$-th
Fitting scheme of $\mathcal{F}$ is the subscheme of $X$ defined as the zero
locus of $\operatorname{Fitt}_{0}(\mathcal{F})\subseteq\mathcal{O}_{X}$ in
$X$. The $0$-th Fitting scheme contains the support of $\mathcal{F}$, as a
closed subscheme with the same underlying topological space.

###### Proof.

The result is stated in [EH06, Definition V-10]. ∎

####  2.3.2. The direct image

Here, we define the notion of direct image for generalized divisors. First, we
start with the case of an effective generalized divisor. Let
$D\in\operatorname{GDiv}^{+}(X)$ be an effective generalized divisor on $X$,
with ideal sheaf $\mathcal{I}$. Since $\pi$ is finite and flat, the
pushforward $\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ is a coherent
$\mathcal{O}_{Y}$-module.

######  Definition/Lemma 2.3.3.

(Direct image of an effective generalized divisor) Let
$D\in\operatorname{GDiv}^{+}(X)$ be an effective generalized divisor on $X$,
with ideal sheaf $\mathcal{I}\subseteq\mathcal{O}_{X}$. The direct image of D
with respect to  $\pi$, denoted with $\pi_{*}(D)$, is the effective
generalized divisor on $Y$ defined by the $0$-th Fitting ideal of
$\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$.

###### Proof.

Let us prove that the 0-th Fitting ideal of
$\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ defines an effective generalized
divisor on $Y$. First note that
$\operatorname{Fitt}_{0}\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ is a subsheaf of
$\mathcal{O}_{Y}$, and is a coherent $\mathcal{O}_{Y}$-module; hence, it is an
effective fractional ideal. To prove that
$\operatorname{Fitt}_{0}\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ is
nondegenerate, consider a generic point $\eta\in Y$. Since the map $\pi$ is
dominant, the preimage $\pi^{-1}(\eta)=\\{\eta_{i}\\}$ is a finite set of
generic points of $X$. Then,

$\displaystyle\operatorname{Fitt}_{0}(\pi_{*}(\mathcal{O}_{X}/\mathcal{I}))_{\eta}$
$\displaystyle=\operatorname{Fitt}_{0}(\pi_{*}(\mathcal{O}_{X}/\mathcal{I})_{\eta})=$
$\displaystyle=\operatorname{Fitt}_{0}((0)_{\eta})=\mathcal{O}_{X,\eta}=\mathcal{K}_{X,\eta}.$

∎

######  Lemma 2.3.4.

(Linearity w.r.t effective Cartier divisors) Let
$D\in\operatorname{GDiv}^{+}(X)$ be an effective generalized divisor and
$E\in\operatorname{CDiv}^{+}(X)$ be an effective Cartier divisor on $X$. Then,
$\pi_{*}(E)$ is a Cartier divisor on $Y$ and
$\pi_{*}(D+E)=\pi_{*}(D)+\pi_{*}(E)$.

###### Proof.

Let $\mathcal{J}$ be the ideal sheaf of $E$ and let $\\{V_{i}\\}$ be an open
affine cover of $Y$ such that $\pi_{*}\mathcal{O}_{X}$ and
$\pi_{*}\mathcal{J}$ are trivial on each $V_{i}$ as in Lemma 2.1.2. The sheaf
$\pi_{*}\mathcal{O}_{X}/\mathcal{J}$ is locally presented by the exact
sequence:

$(\pi_{*}\mathcal{O}_{X})_{|V_{i}}\xrightarrow{\cdot
h_{i}}(\pi_{*}\mathcal{O}_{X})_{|V_{i}}\rightarrow(\pi_{*}\mathcal{O}_{X}/\mathcal{J})_{|V_{i}}\rightarrow
0$

where $h_{i}\in\Gamma(\pi^{-1}(V_{i}),\mathcal{O}_{X})$ is a local generator
for $\mathcal{J}\subset\mathcal{O}_{X}$ on $\pi^{-1}(V_{i})$. Since
$(\pi_{*}\mathcal{O}_{X})_{|V_{i}}$ is a free
$(\mathcal{O}_{Y})_{|V_{i}}$-module of rank $n$, there is a $n\times n$ matrix
$H_{i}$ with entries in $(\mathcal{O}_{Y})_{|V_{i}}$ representing the
multiplication by $h_{i}$. Then, by Definition 2.3.1, the 0-th Fitting ideal
of $\pi_{*}\mathcal{O}_{X}/\mathcal{J}$ is the principal ideal generated
locally by $\det(\cdot h_{i})$ on $V_{i}$. In particular, $\pi_{*}(E)$ is a
Cartier divisor.

Let $\mathcal{I}$ be the ideal sheaf of $D$, so that
$\mathcal{I}\cdot\mathcal{J}$ is the ideal sheaf of $D+E$. To prove the
remaining part of the thesis, we show that the equality

$\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{I}\mathcal{J})=\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{I})\cdot\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{J})$

holds locally around any point $y\in Y$. Let $V$ be an open neighborhood of
$y$ such that $(\pi_{*}\mathcal{O}_{X})_{|V}$ is a free
$\mathcal{O}_{Y}$-module, $\pi_{*}\mathcal{I}_{|V}$ is generated by sections
$s_{1},\dots,s_{r}$ of $\Gamma(V,\pi_{*}\mathcal{O}_{X})$ and
$\pi_{*}\mathcal{J}_{|V}$ is a principal ideal generated by a section $h$ of
$\Gamma(V,\pi_{*}\mathcal{O}_{X})$. In terms of exact sequences:

$\displaystyle(\pi_{*}\mathcal{O}_{X})_{|V}^{\oplus r}\xrightarrow{(\cdot
s_{1},\dots,\cdot
s_{r})}(\pi_{*}\mathcal{O}_{X})_{|V}\rightarrow(\pi_{*}\mathcal{O}_{X}/\mathcal{I})_{|V}\rightarrow
0$ $\displaystyle(\pi_{*}\mathcal{O}_{X})_{|V}\xrightarrow{\cdot
h}(\pi_{*}\mathcal{O}_{X})_{|V}\rightarrow(\pi_{*}\mathcal{O}_{X}/\mathcal{J})_{|V}\rightarrow
0$

The ideal sheaf $\mathcal{I}\cdot\mathcal{J}$ is equal to
$\mathcal{J}\cdot\mathcal{I}$, which is generated on $V$ by the sections
$hs_{1},\dots,hs_{n}$. In terms of exact sequences:

$\displaystyle(\pi_{*}\mathcal{O}_{X})_{|V}^{\oplus r}\xrightarrow{(\cdot
hs_{1},\dots,\cdot
hs_{r})}(\pi_{*}\mathcal{O}_{X})_{|V}\rightarrow(\pi_{*}\mathcal{O}_{X}/\mathcal{I}\mathcal{J})_{|V}\rightarrow
0$

Denote with $S_{i}$ and $H$ the $(\mathcal{O}_{Y})_{|V}$-matrices representing
the multiplication by $s_{i}$ and $h$ respectively. The map $(\cdot
hs_{1},\dots,\cdot hs_{r})$ in the previous exact sequence is represented by
the $n\times nr$ matrix

$M=\left[\begin{array}[]{c|c|c}HS_{1}&\dots&HS_{r}\end{array}\right]=H\left[\begin{array}[]{c|c|c}S_{1}&\dots&S_{r}\end{array}\right]$

The 0-th Fitting ideal of $\pi_{*}\mathcal{O}_{X}/\mathcal{I}\mathcal{J}$,
restricted to $V$, is the ideal of $(\pi_{*}\mathcal{O}_{X})_{|V}$ generated
by the $n\times n$ minors of the matrix $M$. Any such minor is equal to a
$n\times n$ minor of the matrix
$\left[\begin{array}[]{c|c|c}S_{1}&\dots&S_{r}\end{array}\right]$ multiplied
by $\det H$. Then, by Definition 2.3.1,

$\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{I}\mathcal{J})_{|V}=\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{I})_{|V}\cdot\operatorname{Fitt}_{0}(\pi_{*}\mathcal{O}_{X}/\mathcal{J})_{|V}$

∎

######  Definition/Lemma 2.3.5.

(Direct image of a generalized divisor) Let $D\in\operatorname{GDiv}(X)$ be a
generalized divisor on $X$, such that $D=D^{\prime}-E$ with
$D^{\prime}\in\operatorname{GDiv}^{+}(X)$ and $E\in\operatorname{CDiv}^{+}(X)$
by Lemma 2.2.3. The direct image of D with respect to $\pi$, denoted with
$\pi_{*}(D)$, is the generalized divisor $\pi_{*}(D^{\prime})-\pi_{*}(E)$.

###### Proof.

To prove that it is well defined, let
$D=D^{\prime}-E=\widetilde{D}^{\prime}-\widetilde{E}$, with
$D^{\prime},\widetilde{D}^{\prime}$ effective and $E,\widetilde{E}$ effective
Cartier. Since $E,\widetilde{E}$ are Cartier,
$D^{\prime}+\widetilde{E}=\widetilde{D}^{\prime}+E$; then, by Lemma 2.3.4 we
have:

$\pi_{*}(D^{\prime})+\pi_{*}(\widetilde{E})=\pi_{*}(\widetilde{D}^{\prime})+\pi_{*}(E).$

Since $\pi_{*}(E)$ and $\pi_{*}(\widetilde{E})$ are also Cartier by Lemma
2.3.4, they can be subtracted from each side in order to obtain:

$\pi_{*}(D^{\prime})-\pi_{*}(E)=\pi_{*}(\widetilde{D}^{\prime})-\pi_{*}(\widetilde{E}).$

This shows that $\pi_{*}(D)$ does not depend on the choice of $D^{\prime}$ and
$E$. ∎

We study now some properties of the direct image for generalized divisors.

######  Proposition 2.3.6.

(Properties of direct image)

  1. (1)

Let $D\in\operatorname{CDiv}(X)$ be a Cartier divisor. Then, $\pi_{*}(D)$ is a
Cartier divisor and $\pi_{*}(-D)=-\pi_{*}(D)$. Moreover, $\pi_{*}(D)$
coincides with $\pi_{*}(D)$ of Definition 2.1.3.

  2. (2)

Let $D,E\in\operatorname{GDiv}(X)$ be generalized divisors, such that $E$ is
Cartier. Then, $\pi_{*}(D+E)=\pi_{*}(D)+\pi_{*}(E)$.

  3. (3)

Let $V\subset Y$ be an open subset, and denote with $\pi_{U}$ the restriction
of $\pi$ to $U=\pi^{-1}(V)\subset X$. Let $D\in\operatorname{GDiv}(X)$ be a
generalized divisor on $X$. Then,

$(\pi_{U})_{*}(D_{|U})=\pi_{*}(D)_{|V}$

  4. (4)

Let $D,D^{\prime}\in\operatorname{GDiv}(X)$ be generalized divisors such that
$D\sim D^{\prime}$. Then, $\pi_{*}(D)\sim\pi_{*}(D^{\prime})$.

###### Proof.

To prove (1), consider $D=E-F$ with $E,F$ effective and $F$ Cartier by Lemma
2.2.3. Since $D$ is Cartier and $E=D+F$, then also $E$ is Cartier. By
Definition 2.3.5,

$\pi_{*}(D)=\pi_{*}(E)-\pi_{*}(F).$

By Lemma 2.3.4, it is a difference of Cartier divisors and hence it is
Cartier. To compute $\pi_{*}(-D)$, note that since $F$ is Cartier then
$-D=F-E$, and it is a difference of effective Cartier divisors; then, apply
Definition 2.3.5 to obtain

$\displaystyle\pi_{*}(-D)$ $\displaystyle=\pi_{*}(F)-\pi_{*}(E)=$
$\displaystyle=-\pi_{*}(D).$

To compare $\pi_{*}(D)$ with Definition 2.1.3, let $\mathcal{I}$ and
$\mathcal{J}$ be the ideal sheaves of $E$ and $F$ and let $\\{V_{i}\\}_{i\in
I}$ be an open cover of $Y$ such that $\pi_{*}\mathcal{I}_{|V_{i}}$ and
$\pi_{*}\mathcal{J}_{|V_{i}}$ are non-degenerate principal ideals of
$(\pi_{*}\mathcal{O}_{X})_{|V_{i}}$-modules generated by the regular sections
$f_{i}$ and $g_{i}$ of $\Gamma(V,\pi_{*}\mathcal{O}_{X})$, respectively on
each $i\in I$. The fractional ideal of $D$ is then generated on each $V_{i}$
by the meromorphic regular section $h_{i}=f_{i}/g_{i}$ of
$\Gamma(V_{i},\pi_{*}\mathcal{K}_{X}^{*})$. By the proof of Lemma 2.3.4, the
ideal sheaves of $\pi_{*}(E)$ and $\pi_{*}(F)$ are generated on each $V_{i}$
by $\det(\cdot f_{i})$ and $\det(\cdot g_{i})$ , so the fractional ideal of
$\pi_{*}(D)$ is generated on each $V_{i}$ by the meromorphic regular section
$\det(\cdot f_{i})/\det(\cdot g_{i})$ of $\Gamma(V,\mathcal{K}_{Y}^{*})$.
Using the same cover in Definition 2.1.3 and applying Definition 2.1.1, the
sheaf $\pi_{*}(D)$ defined above and the sheaf $\pi_{*}(D)$ defined in Section
2.1 have the same local generators, hence they are equal.

To prove (2), consider $D=D_{1}-D_{2}$ and $E=E_{1}-E_{2}$, with
$D_{1},D_{2},E_{1},E_{2}$ effective and $D_{2},E_{2}$ Cartier. Note that
$D+E=(D_{1}+E_{1})-(D_{2}+E_{2})$, and it is a difference of effective
divisors with $D_{2}+E_{2}$ Cartier. Then, applying Definition 2.3.5 and Lemma
2.3.4, we obtain:

$\displaystyle\pi_{*}(D+E)$
$\displaystyle=\pi_{*}((D_{1}+E_{1})-(D_{2}+E_{2}))=$
$\displaystyle=\pi_{*}(D_{1}+E_{1})-\pi_{*}(D_{2}+E_{2})=$
$\displaystyle=\pi_{*}(D_{1})+\pi_{*}(E_{1})-\pi_{*}(D_{2})-\pi_{*}(E_{2})=$
$\displaystyle=(\pi_{*}(D_{1})-\pi_{*}(D_{2}))-(\pi_{*}(E_{1})-\pi_{*}(E_{2}))=$
$\displaystyle=\pi_{*}(D)+\pi_{*}(E).$

To prove (3), if $D$ is effective, the result follows from Definition 2.3.1.
Then, observe that the operations of product and inverse of fractional ideals
commute with restrictions.

To prove (4), let $f\in\Gamma(X,\mathcal{K}_{X})$ be a global section that
generates a principal divisor $E=(f)\in\operatorname{Prin}(X)$. By linearity,
it is sufficient to prove that $\pi_{*}(E)$ is again principal. Let
$\\{V_{i}\\}$ be an affine open cover of $Y$, such that locally

$f_{|\pi^{-1}(V_{i})}=g_{i}/h_{i},\hskip
30.00005ptg_{i},h_{i}\in\Gamma(\pi^{-1}(V_{i}),\mathcal{O}_{X})$

In terms of divisors, this means that
$(f)_{|\pi^{-1}(V_{i})}=(g_{i})-(h_{i})$, and it is a difference of effective
Cartier divisors on $\pi^{-1}(V_{i})$. Then, applying Part (3) and reasoning
simiarly to Part (1), we obtain:

$\displaystyle\pi_{*}((f))_{|V}$
$\displaystyle=(\pi_{i})_{*}((g_{i}))-(\pi_{i})_{*}((h_{i}))=$
$\displaystyle=(\det[\cdot g_{i}])-(\det[\cdot h_{i}])=$
$\displaystyle=(\det[\cdot g_{i}]/\det[\cdot h_{i}]).$

Since taking determinants is multiplicative, the local sections $\det[\cdot
g_{i}]/\det[\cdot h_{i}]$ glue together to give a global section
$\widetilde{f}$ of $\mathcal{K}_{Y}$, such that $\pi_{*}(E)=(\widetilde{f})$.
∎

We are now ready to define the direct image for generalized line bundles.

######  Definition/Lemma 2.3.7.

(Direct image for generalized line bundles) The direct image for generalized
divisors induce a direct image map between the sets of generalized line
bundles, defined as:

$\displaystyle[\pi_{*}]:\hskip 20.00003pt\operatorname{GPic}(X)$
$\displaystyle\longrightarrow\operatorname{GPic}(Y)$ $\displaystyle[D]$
$\displaystyle\mapsto[\pi_{*}(D)].$

###### Proof.

Recall that for any curve $X$, the set $\operatorname{GPic}(X)$ can be seen
equivalently as the set of generalized line bundles or as the set of
generalized divisors modulo linear equivalence. Here, $[\pi_{*}]$ is defined
in terms of generalized divisors modulo linear equivalence. By Proposition
2.3.6, the direct images of linearly equivalent divisors are linearly
equivalent, hence $[\pi_{*}]$ is well defined. ∎

In the remaining part of this subsection, we study an alternative formula for
$[\pi_{*}]$ in terms of generalized line bundles. First, we need a technical
lemma.

######  Lemma 2.3.8.

Let $\mathcal{F}$ be a coherent sheaf on a curve $X$ which is locally free of
rank 1 at any generic point and let $\omega=\omega_{X}$ be the canonical, or
dualizing sheaf of $X$. Let $T(\mathcal{F})$ be the torsion subsheaf of
$\mathcal{F}$ and
$\mathcal{F}^{\omega\omega}=\operatorname{\mathcal{H}\textit{om}}(\operatorname{\mathcal{H}\textit{om}}(\mathcal{F},\omega),\omega)$
be the double $\omega$-dual. There, there is a canonical isomorphism

$\mathcal{F}^{tf}=\mathcal{F}/T(\mathcal{F})\xrightarrow{\hskip
10.00002pt\sim\hskip 10.00002pt}\mathcal{F}^{\omega\omega}.$

###### Proof.

The sheaf $\mathcal{F}^{tf}$ is endowed with a quotient map
$q:\mathcal{F}\twoheadrightarrow\mathcal{F}^{tf}$, which is universal among
all arrows from $\mathcal{F}$ to torsion-free sheaves (i.e. pure of dimension
1). Note that, since $\mathcal{F}$ is locally free of rank 1 at any generic
point of $X$, then $q$ is generically an isomorphism. Let
$\alpha(\mathcal{F}):\mathcal{F}\rightarrow\mathcal{F}^{\omega\omega}$ be the
canonical map from $\mathcal{F}$ to its double $\omega$-dual. Since taking
double $\omega$-duals is functorial, there is a commutative diagram of
homomorphism of sheaves:

${\mathcal{F}}$${\mathcal{F}^{tf}}$${\mathcal{F}^{\omega\omega}}$${(\mathcal{F}^{tf})^{\omega\omega}.}$$\scriptstyle{q}$$\scriptstyle{\alpha(\mathcal{F})}$$\scriptstyle{\alpha(\mathcal{F}^{tf})}$$\scriptstyle{q^{\omega\omega}}$

Using [Har07, Proposition 1.5] and recalling that $S_{1}=S_{2}$ for sheaves on
curves, we note that a sheaf on $X$ is $\omega$-reflexive if and only if it is
$S_{1}$. In particular, $\mathcal{F}^{\omega\omega}$ is $\omega$-reflexive by
[Har07, Proposition 1.6] and hence it is $S_{1}$. Then, by the universal
property of $q$, there is a unique map
$\psi:\mathcal{F}^{tf}\rightarrow\mathcal{F}^{\omega\omega}$ such that
$\psi\circ q=\alpha(\mathcal{F})$. Note that

$q^{\omega\omega}\circ\psi\circ
q=q^{\omega\omega}\circ\alpha(\mathcal{F})=\alpha(\mathcal{F}^{tf})\circ q,$

so by surjectivity of $q$ we conclude that
$q^{\omega\omega}\circ\psi=\alpha(\mathcal{F}^{tf})$.

We show moreover that $\psi$ is an isomorphism. By construction
$\mathcal{F}^{tf}$ is pure, hence it is $\omega$-reflexive and
$\alpha(\mathcal{F}^{tf})$ is an isomorphism. Moreover, $q^{\omega\omega}$ is
surjective and generically an isomorphism since $q$ is surjective and
generically an isomorphism. Then, the kernel $K$ of $q^{\omega\omega}$ is a
subsheaf of $\mathcal{F}^{\omega\omega}$ which is generically zero. Since
$\mathcal{F}^{\omega\omega}$ is pure, we conclude that $K$ is everywhere zero
and then $q^{\omega\omega}$ is an isomorphism. This shows that
$\psi=(q^{\omega\omega})^{-1}\circ\alpha(\mathcal{F}^{tf})$ is an isomorphism.
∎

In order to study the formula for $[\pi_{*}]$, we study a preliminary formula
for $\pi_{*}$ in the case of effective generalized divisors.

######  Lemma 2.3.9.

Let $D\in\operatorname{GDiv}^{+}(X)$ be an effective generalized divisor with
ideal sheaf $\mathcal{I}\subset\mathcal{O}_{X}$. Then, there is an injection:

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}\hookrightarrow\det(\pi_{*}\mathcal{O}_{X})\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}\xrightarrow{{}_{\sim}}\mathcal{O}_{Y}$

whose image in $\mathcal{O}_{Y}$ is the $0$-th Fitting ideal of
$\pi_{*}\mathcal{O}_{X}/\mathcal{I}$.

###### Proof.

Consider the short exact sequence:

$0\rightarrow\mathcal{I}\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{O}_{D}\rightarrow
0.$

Since $\pi$ is finite and flat, the pushforward induces a a short exact
sequence:

$0\rightarrow\pi_{*}\mathcal{I}\xrightarrow{\varphi}\pi_{*}\mathcal{O}_{X}\rightarrow\pi_{*}(\mathcal{O}_{X}/\mathcal{I})\rightarrow
0.$

In particular, the last exact sequence is a finite presentation of
$\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ whose middle term is locally free.
Hence, by Definition 2.3.1, the $0$-th Fitting ideal of
$\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$ is equal to the image of the morphism:

$\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}\big{)}\otimes\det\big{(}\pi_{*}\mathcal{O}_{X}\big{)}^{-1}\xrightarrow{\hskip
5.0pt\det\varphi\otimes 1\hskip
5.0pt}\det(\pi_{*}\mathcal{O}_{X})\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}\xrightarrow{{}_{\sim}}\mathcal{O}_{Y}.$

Consider now the determinant map $\det\varphi$. Since
$\det(\pi_{*}\mathcal{O}_{X})$ is also a pure sheaf, applying the universal
property of the torsion-free quotient together with Lemma 2.3.8 we obtain the
following commutative diagram:

${\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}}$${\det(\pi_{*}\mathcal{O}_{X})}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}}$$\scriptstyle{\det\varphi}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

The canonical map $\alpha$ is surjective by Lemma 2.3.8. Since $\mathcal{I}$
is locally free of rank 1 at the generic points of $X$, both $\det\phi$ and
$\alpha$ are generically isomorphisms; hence, the map $\beta$ is generically
an isomorphism. Since its domain
$\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}$ is
pure, we conclude that its kernel is zero, hence $\beta$ is injective. We have
then factorized $\det\varphi$ as the composition of a surjective map $\alpha$
followed by an injective map $\beta$.

Tensoring by $\det(\pi_{*}\mathcal{O}_{X})^{-1}$, we obtain then the following
commutative diagram:

${\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}}$${\det(\pi_{*}\mathcal{O}_{X})\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}}$${\mathcal{O}_{Y}}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}}$$\scriptstyle{\det\varphi\otimes
1}$$\scriptstyle{\alpha\otimes
1}$$\scriptstyle{\sim}$$\scriptstyle{\eta}$$\scriptstyle{\beta\otimes 1}$

The map $\alpha\otimes 1$ is surjective, so the composition
$\eta\circ(\beta\otimes 1)$ is an injective map whose image in
$\mathcal{O}_{Y}$ is equal to the image of the map
$\eta\circ(\det\varphi\otimes 1)$. By the previous remark, such image
coincides with the $0$-th Fitting ideal of
$\pi_{*}(\mathcal{O}_{X}/\mathcal{I})$, proving the lemma. ∎

We are now ready to give a sheaf-theoretic formula for $[\pi_{*}]$. Recall
that, for any curve $X$, the set $\operatorname{GPic}(X)$ can be seen as the
set of isomorphism classes of generalized line bundles on on $X$.

######  Proposition 2.3.10.

(Formula for the direct image of generalized line bundles)  Let $\mathcal{L}$
be a generalized line bundle on $X$. Then,

$[\pi_{*}](\mathcal{L})\simeq\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

###### Proof.

By the surjectivity of
$\operatorname{GDiv}(X)\twoheadrightarrow\operatorname{GPic}(X)$, we can pick
a generalized divisor $D\in\operatorname{GDiv}(X)$ with fractional ideal
$\mathcal{I}$ isomorphic to $\mathcal{L}$; then,

$[\pi_{*}](\mathcal{L})=[\pi_{*}]([D])=[\pi_{*}(D)]$

by Definition 2.3.7. By Lemma 2.2.3, there are effective generalized divisors
$E,F$ on $X$ such that $D=E-F$ and $F$ Cartier. Denote with
$\mathcal{I}^{\prime}$ the ideal sheaf of $E$ and with $\mathcal{J}$ the ideal
sheaf of $F$. Since $F$ is Cartier, the condition $D=E-F$ can be rewritten as
$E=D+F$, or in terms of sheaves:

$\mathcal{I}^{\prime}=\mathcal{I}\cdot\mathcal{J}.$

Consider the direct images of $E$ and $F$. By Lemma 2.3.9, the ideal sheaf of
$\pi_{*}(E)$ is isomorphic to

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1},$

while the ideal sheaf of $\pi_{*}(F)$ is isomorphic to

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}\simeq\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

By Definition 2.3.5, $\pi_{*}(D)=\pi_{*}(E)-\pi_{*}(F)$. Then, the fractional
ideal of $\pi_{*}(D)$ is isomorphic to:

$\displaystyle\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}\otimes\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right)^{-1}\otimes\det(\pi_{*}\mathcal{O}_{X}),$

which in turn is isomorphic to:

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right)^{-1}.$

Then, we are left to prove that:

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right)^{-1}\simeq\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1},$

or, equivalently, that:

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})\simeq\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right),$

under the assumptions $\mathcal{I}^{\prime}=\mathcal{I}\cdot\mathcal{J}$ and
$\mathcal{J}$ locally principal. Consider an open cover $\\{V_{i}\\}_{i\in I}$
of $Y$ such that $\mathcal{J}$ is trivial on each $U_{i}=\pi^{-1}(V_{i})$,
i.e. there is an isomorphism
$\lambda_{i}:\mathcal{J}_{|U_{i}}=(f_{i})\xrightarrow{{}_{\sim}}\mathcal{O}_{X|U_{i}}$
for each $i\in I$. On the intersections $U_{i}\cap U_{j}$, the collection
$\\{\lambda_{i}\circ\lambda_{j}^{-1}\\}$ of automorphisms of
${\mathcal{O}_{X}}_{|U_{i}\cap U_{j}}$ is a cochain of elements of
${\mathcal{O}_{X}^{*}}_{|U_{i}\cap U_{j}}$ that measures the obstruction for
the $\lambda_{i}$’s to glue to a global isomorphism. We define now an
isomorphism

$\alpha:\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})\longrightarrow\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{J}\big{)}\right)$

by glueing a collection of isomorphisms $\alpha_{i}$ defined on each $V_{i}$.
To do so, we define each $\alpha_{i}$ as the following composition of arrows:

${\left(\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}^{\prime}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})\right)_{|V_{i}}}$${\left(\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}\big{)}\right)^{\omega\omega}\otimes\bigwedge^{n}\big{(}\pi_{*}\mathcal{J}\big{)}\right)_{|V_{i}}}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}^{\prime}_{|U_{i}}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X|U_{i}})}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}_{|U_{i}}\big{)}\right)^{\omega\omega}\otimes\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{J}_{|U_{i}}\big{)}\right)^{\omega\omega}}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}_{|U_{i}}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X|U_{i}})}$${\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{I}_{|U_{i}}\big{)}\right)^{\omega\omega}\otimes\left(\bigwedge^{n}\big{(}\pi_{*}\mathcal{O}_{X|U_{i}}\big{)}\right)^{\omega\omega}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{=}$$\scriptstyle{\left(\bigwedge^{n}(\pi_{*}\lambda_{i})\right)^{\omega\omega}\otimes\operatorname{id}}$$\scriptstyle{\simeq}$${\simeq}$$\scriptstyle{\operatorname{id}\otimes\left(\bigwedge^{n}(\pi_{*}\lambda_{i}^{-1})\right)^{\omega\omega}}$

i.e.
$\alpha_{i}:=\left(\bigwedge^{n}\big{(}\pi_{*}\lambda_{i}\big{)}\right)^{\omega\omega}\otimes\left(\bigwedge^{n}\big{(}\pi_{*}\lambda_{i}^{-1}\big{)}\right)^{\omega\omega}$.
Since $\bigwedge^{n}\big{(}\pi_{*}\\_\big{)}$ and $(\\_)^{\omega\omega}$ are
functorial, the obstruction $\alpha_{i}\circ\alpha_{j}^{-1}$ is trivial on any
$V_{i}\cap V_{j}$, whence the $\alpha_{i}$’s glue together to a global
isomorphism $\alpha$. ∎

######  Corollary 2.3.11.

Let $\mathcal{L}$ be a line bundle on $X$. Then,

$\operatorname{Nm}_{\pi}(\mathcal{L})=[\pi_{*}](\mathcal{L}).$

###### Proof.

By Lemma 2.1.7,

$\operatorname{Nm}_{\pi}(\mathcal{L})\simeq\det(\pi_{*}\mathcal{L})\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

On the other side, by Proposition 2.3.10,

$[\pi_{*}](\mathcal{L})\simeq\left(\det(\pi_{*}\mathcal{L})\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

Since $\mathcal{L}$ is locally free, $\det(\pi_{*}\mathcal{L})$ is a line
bundle and in particular is a pure coherent sheaf. Then,
$\det(\pi_{*}\mathcal{L})\simeq\left(\det(\pi_{*}\mathcal{L})\right)^{\omega\omega}$,
proving the thesis. ∎

######  Corollary 2.3.12.

Let $\mathcal{L}$ be a generalized line bundle on $X$. Suppose that $Y$ is
smooth. Then:

$[\pi_{*}](\mathcal{L})\simeq\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

###### Proof.

First note that, by Proposition 2.3.10,

$[\pi_{*}](\mathcal{L})\simeq\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}\right)^{\omega\omega}\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

Second, observe that the pure sheaf $\pi_{*}\mathcal{L}$ is locally free of
rank $n$ since $Y$ is smooth (see [HL10, Example 1.1.16]), and
$\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}$ is a line bundle.
Then,

$\left(\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}\right)^{\omega\omega}\simeq\textstyle{\bigwedge^{n}}\big{(}\pi_{*}\mathcal{L}\big{)}.$

Combining these two facts, we have proved the thesis. ∎

####  2.3.3. The inverse image

In this subsection, we define the inverse image for generalized divisors and
generalized line bundles and we study the relation of the inverse image with
the direct image. We start from the case of effective divisors.

######  Definition/Lemma 2.3.13.

(Inverse image of an effective generalized divisor) Let
$D\in\operatorname{GDiv}^{+}(Y)$ be an effective generalized divisor with
ideal sheaf $\mathcal{I}\subseteq\mathcal{O}_{Y}$. The inverse image of $D$
relative to $\pi$, denoted $\pi^{*}(D)$, is the effective generalized divisor
with ideal sheaf $\pi^{-1}(\mathcal{I})\cdot\mathcal{O}_{X}$.

###### Proof.

The inverse image ideal $\pi^{-1}(\mathcal{I})$ is an ideal sheaf of
$\pi^{-1}(\mathcal{O}_{Y})$-modules and can be extended to
$\mathcal{O}_{X}$-modules via the canonical map of sheaves of rings
$\pi^{\sharp}:\pi^{-1}(\mathcal{O}_{Y})\rightarrow\mathcal{O}_{X}$. It is
coherent since $\mathcal{I}$ is coherent. If $\eta$ is a generic point of $X$,
then $\pi(\eta)$ is a generic point of $Y$, hence

$(\pi^{-1}(\mathcal{I})\cdot\mathcal{O}_{X})_{\eta}=\mathcal{I}_{\pi(\eta)}\cdot\mathcal{O}_{X,\eta}=\mathcal{O}_{X,\eta}$

since $\mathcal{I}_{\pi(\eta)}=\mathcal{O}_{Y,\pi(\eta)}$. ∎

######  Remark 2.3.14.

In the setting of Definition 2.3.13, consider the short exact sequence:

$0\rightarrow\mathcal{I}\rightarrow\mathcal{O}_{Y}\rightarrow\mathcal{O}_{D}\rightarrow
0.$

Since $\pi$ is flat and surjective, the pullback functor
$\\_\otimes_{\pi^{-1}\mathcal{O}_{Y}}\mathcal{O}_{X}$ is exact as well as the
inverse image functor $\pi^{-1}$. Then, the previous exact sequence induces
the following exact sequence:

$0\rightarrow\pi^{*}\mathcal{I}\rightarrow\pi^{*}\mathcal{O}_{Y}\rightarrow\pi^{*}\mathcal{O}_{D}\rightarrow
0.$

Since $\pi^{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$, the pullback sheaf
$\pi^{*}(\mathcal{I})$ has a canonical injection in $\mathcal{O}_{X}$, whose
image is exactly the inverse image ideal
$\pi^{-1}(\mathcal{I})\cdot\mathcal{O}_{X}$

######  Lemma 2.3.15.

Let $D\in\operatorname{GDiv}^{+}(Y)$ be a generalized effective divisor and
$E\in\operatorname{CDiv}^{+}(Y)$ be an effective Cartier divisor on $Y$. Then,
the inverse image divisor $\pi^{*}(E)$ is Cartier and
$\pi^{*}(D+E)=\pi^{*}(D)+\pi^{*}(E)$.

###### Proof.

Let $\mathcal{I}$ and $\mathcal{J}$ be the ideal sheaves of $D$ and $E$
respectively. The ideal sheaf $\mathcal{J}$ is locally principal, hence its
inverse image $\pi^{-1}(\mathcal{J})\cdot\mathcal{O}_{X}$ is again locally
principal; then, $\pi^{*}(E)$ is Cartier. The generalized divisor $D+E$ is
defined by the ideal sheaf $\mathcal{I}\cdot\mathcal{J}$, whose inverse image
is:

$\displaystyle\pi^{-1}(\mathcal{I}\cdot\mathcal{J})\cdot\mathcal{O}_{X}$
$\displaystyle=(\pi^{-1}(\mathcal{I})\cdot\pi^{-1}(\mathcal{J}))\cdot\mathcal{O}_{X}=$
$\displaystyle=(\pi^{-1}(\mathcal{I})\cdot\mathcal{O}_{X})\cdot(\pi^{-1}(\mathcal{J})\cdot\mathcal{O}_{X}),$

which is the defining ideal of $\pi^{*}(D)+\pi^{*}(E)$. ∎

Now, we can extend the definition of inverse image to any generalized divisor
and study its properties.

######  Definition/Lemma 2.3.16.

(Inverse image of a generalized divisor) Let $D\in\operatorname{GDiv}(Y)$ be
any generalized divisor and let $D=E-F$ with $E,F$ effective generalized
divisors and $F$ Cartier by Lemma 2.2.3. The inverse image of $D$ relative to
$\pi$, denoted $\pi^{*}(D)$, is the generalized divisor
$\pi^{*}(E)-\pi^{*}(F)$.

###### Proof.

To prove that it is well defined, let
$D=D^{\prime}-E=\widetilde{D}^{\prime}-\widetilde{E}$, with
$D^{\prime},\widetilde{D}^{\prime}$ effective and $E,\widetilde{E}$ effective
Cartier. Since $E,\widetilde{E}$ are Cartier,
$D^{\prime}+\widetilde{E}=\widetilde{D}^{\prime}+E$; then, by Lemma 2.3.15 we
have:

$\pi^{*}(D^{\prime})+\pi^{*}(\widetilde{E})=\pi^{*}(\widetilde{D}^{\prime})+\pi^{*}(E).$

Again by Lemma 2.3.15 $\pi^{*}(E)$ and $\pi^{*}(\widetilde{E})$ are Cartier,
so they can be subtracted from each side in order to obtain:

$\pi^{*}(D^{\prime})-\pi^{*}(E)=\pi^{*}(\widetilde{D}^{\prime})-\pi^{*}(\widetilde{E}),$

so $\pi^{*}(D)$ does not depend on the choice of $D^{\prime}$ and $E$. ∎

######  Proposition 2.3.17.

(Properties of inverse image)

  1. (1)

Let $D\in\operatorname{CDiv}(Y)$ be a Cartier divisor on $Y$. Then,
$\pi^{*}(D)$ is a Cartier divisor and $\pi^{*}(-D)=-\pi^{*}(D)$. Moreover,
$\pi^{*}(D)$ coincides with $\pi^{*}(D)$ of Definition 2.1.3.

  2. (2)

Let $D,E\in\operatorname{GDiv}(Y)$ be generalized divisors, such that $E$ is
Cartier. Then, $\pi^{*}(D+E)=\pi^{*}(D)+\pi^{*}(E)$.

  3. (3)

Let $V\subset Y$ be an open subset, and denote with $\pi_{U}$ the restriction
of $\pi$ to $U=\pi^{-1}(V)\subset X$. Let $D\in\operatorname{GDiv}(Y)$ be a
generalized divisor on $Y$. Then,

$(\pi_{U})^{*}(D_{|V})=\pi^{*}(D)_{|U}$

  4. (4)

Let $D,D^{\prime}\in\operatorname{GDiv}(Y)$ be generalized divisors such that
$D\sim D^{\prime}$. Then, $\pi^{*}(D)\sim\pi^{*}(D^{\prime})$.

###### Proof.

To prove (1), consider $D=E-F$ with $E,F$ effective and $F$ Cartier by Lemma
2.2.3 on $Y$. Since $D$ is Cartier and $E=D+F$, then also $E$ is Cartier. By
Definition 2.3.16,

$\pi^{*}(D)=\pi^{*}(E)-\pi^{*}(F).$

By Lemma 2.3.15, it is a difference of Cartier divisors and hence it is
Cartier. To compute $\pi(-D)$, note that since $F$ is Cartier then $-D=F-E$,
and it is a difference of effective Cartier divisors; then, apply Definition
2.3.16 to obtain

$\displaystyle\pi^{*}(-D)$ $\displaystyle=\pi^{*}(F)-\pi^{*}(E)=$
$\displaystyle=-\pi^{*}(D).$

To compare $\pi^{*}(D)$ with Definition 2.1.3, let $\mathcal{I}$ and
$\mathcal{J}$ be the ideal sheaves of $E$ and $F$ and let $\\{V_{i}\\}_{i\in
I}$ be an open cover of $Y$ such that $\mathcal{I}_{|V_{i}}and$
$\mathcal{J}_{|V_{i}}$ are principal ideals of $\mathcal{O}_{Y|V_{i}}$-modules
generated by regular sections $s_{i}$ and $t_{i}$ of
$\Gamma(V_{i},\mathcal{O}_{Y})$, respectively on each $i\in I$. The fractional
ideal of $D$ is generated on each $V_{i}$ by the meromorphic regular section
$u_{i}=s_{i}/t_{i}$ of $\Gamma(V_{i},\mathcal{K}_{Y})$. By Definition 2.3.13,
the ideal sheaves of $\pi^{*}(E)$ and $\pi^{*}(F)$ are generated on each
$U_{i}=\pi^{-1}(V_{i})$ by $\pi_{U_{i}}^{\sharp}(s_{i})$ and
$\pi_{U_{i}}^{\sharp}(t_{i})$ respectively. Then, by Definition 2.3.16, the
fractional ideal of $\pi^{*}(D)$ is generated on each $U_{i}$ by the
meromorphic regular section
$\pi_{U_{i}}^{\sharp}(s_{i})/\pi_{U_{i}}^{\sharp}(t_{i})$ of
$\Gamma(U_{i},\mathcal{K}_{X})$. These are exactly the local generators for
$\pi^{*}(D)$ as defined in Definition 2.1.3.

To prove (2), consider $D=D_{1}-D_{2}$ and $E=E_{1}-E_{2}$, with
$D_{1},D_{2},E_{1},E_{2}$ effective and $D_{2},E_{2}$ Cartier. Note that
$D+E=(D_{1}+E_{1})-(D_{2}+E_{2})$, and it is a difference of effective
divisors with $D_{2}+E_{2}$ Cartier. Then, applying Definition 2.3.16 and
Lemma 2.3.15, we obtain:

$\displaystyle\pi^{*}(D+E)$
$\displaystyle=\pi^{*}((D_{1}+E_{1})-(D_{2}+E_{2}))=$
$\displaystyle=\pi^{*}(D_{1}+E_{1})-\pi^{*}(D_{2}+E_{2})=$
$\displaystyle=\pi^{*}(D_{1})+\pi^{*}(E_{1})-\pi^{*}(D_{2})-\pi^{*}(E_{2})=$
$\displaystyle=(\pi^{*}(D_{1})-\pi^{*}(D_{2}))-(\pi^{*}(E_{1})-\pi^{*}(E_{2}))=$
$\displaystyle=\pi^{*}(D)+\pi^{*}(E).$

To prove (3), if $D$ is effective, the result follows the fact that the
inverse image functor $\pi^{-1}$ commutes with restrictions. Then, observe
that the operations of product and inverse of fractional ideals also commute
with restrictions.

To prove (4), let $f\in\Gamma(Y,\mathcal{K}_{Y})$ be a global section that
generates a principal divisor $E=(f)\in\operatorname{Prin}(Y)$ such that
$D=D^{\prime}+E$. The inverse image $\pi^{*}(E)$ is the principal divisor
$(\pi^{\sharp}(f))\in\operatorname{Prin}(X)$. Then, by Lemma 2.3.15,
$\pi^{*}(D)=\pi^{*}(D^{\prime})+(\pi^{\sharp}(f))$; so $\pi^{*}(D)$ and
$\pi^{*}(D^{\prime})$ are linearly equivalent. ∎

We are now ready to define the inverse image for generalized line bundles.

######  Definition/Lemma 2.3.18.

(Inverse image for generalized line bundles) The inverse image for generalized
divisors induce a inverse image map between the sets of generalized line
bundles, defined as:

$\displaystyle[\pi^{*}]:\hskip 20.00003pt\operatorname{GPic}(Y)$
$\displaystyle\longrightarrow\operatorname{GPic}(X)$ $\displaystyle[D]$
$\displaystyle\mapsto[\pi^{*}(D)].$

###### Proof.

Recall that for any curve $X$, the set $\operatorname{GPic}(X)$ can be seen
equivalently as the set of generalized line bundles or as the set of
generalized divisors modulo linear equivalence. Here, $[\pi^{*}]$ is defined
in terms of generalized divisors modulo linear equivalence. By Proposition
2.3.17, the direct images of linearly equivalent divisors are linearly
equivalent, hence $[\pi^{*}]$ is well defined. ∎

######  Remark 2.3.19.

If $\mathcal{L}$ is a generalized line bundle on $Y$ and $D$ is a generalized
divisor with fractional ideal $\mathcal{I}$ isomorphic to $\mathcal{L}$, then
the inverse image divisor $\pi^{*}(D)$ has fractional ideal isomorphic to the
pullback sheaf $\pi^{*}(\mathcal{I})$ by Remark 2.3.14. Since
$\pi^{*}(\mathcal{I})$ is isomorphic to $\pi^{*}(\mathcal{L})$ as abstract
$\mathcal{O}_{X}$-modules, we conclude that the inverse image
$[\pi^{*}](\mathcal{L})$ of the generalized line bundle $\mathcal{L}$ is equal
to the pullback sheaf $\pi^{*}(\mathcal{L})$.

We prove now a result on the composition of the direct image with the inverse
image of generalized divisors.

######  Proposition 2.3.20.

(Composition of the direct image with the inverse image) Let
$D\in\operatorname{GDiv}(Y)$ be a generalized divisor on $Y$. Then,

$\pi_{*}(\pi^{*}(D))=n\cdot D.$

###### Proof.

Since both of the terms are linear with respect to the sum of Cartier divisor,
we can suppose that $D$ is effective with ideal sheaf
$\mathcal{I}\subseteq\mathcal{O}_{Y}$. First note that, from the exact
sequence:

$0\rightarrow\pi^{*}\mathcal{I}\rightarrow\pi^{*}\mathcal{O}_{Y}\rightarrow\pi^{*}\mathcal{O}_{D}\rightarrow
0$

together with Remark 2.3.14, we obtain
$\mathcal{O}_{X}/(\pi^{-1}\mathcal{I}\cdot\mathcal{O}_{X})=\mathcal{O}_{X}/\pi^{*}\mathcal{I}=\pi^{*}(\mathcal{O}_{Y}/\mathcal{I})$.
To prove the thesis, we show that the equality

$\operatorname{Fitt}_{0}\big{(}\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})\big{)}=\mathcal{I}^{n}$

holds locally around any point $y\in Y$. Let $V\subseteq Y$ be an open
neighborhood of $y$ such that
$(\pi_{*}\mathcal{O}_{X})_{|V}\simeq(\mathcal{O}_{Y|V})^{\oplus n}$ and
$\mathcal{I}_{|V}$ is generated by sections $s_{1},\dots,s_{r}$ of
$\Gamma(V,\mathcal{I})$. Then, consider the following presentation:

$\mathcal{O}_{Y|V}^{\oplus r}\xrightarrow{(\cdot s_{1},\dots,\cdot
s_{r})}\mathcal{O}_{Y|V}\rightarrow(\mathcal{O}_{Y}/\mathcal{I})_{|V}\rightarrow
0.$

Pulling back with $\pi^{*}$, we obtain the following exact sequence on
$U=\pi^{-1}(V)$:

$\mathcal{O}_{X|U}^{\oplus r}\xrightarrow{(\cdot s_{1},\dots,\cdot
s_{r})}\mathcal{O}_{X|U}\rightarrow(\mathcal{O}_{X}/\pi^{*}\mathcal{I})_{|U}\rightarrow
0.$

In order to compute
$\operatorname{Fitt}_{0}\big{(}\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})\big{)}_{|V}$,
we consider then the pushforward sequence:

$\big{(}\pi_{*}\mathcal{O}_{X}^{\oplus r}\big{)}_{|V}\xrightarrow{(\cdot
s_{1},\dots,\cdot
s_{r})}\big{(}\pi_{*}\mathcal{O}_{X}\big{)}_{|V}\rightarrow\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})_{|V}\rightarrow
0.$

Since
$\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})_{|V}\simeq\big{(}\mathcal{O}_{Y}^{\oplus
n}\big{)}_{|V}$, the map on the left is represented by the following $n\times
nr$ matrix with entries in $\Gamma(V,\mathcal{O}_{Y})$:

$M=\left[\begin{array}[]{c|c|c}\begin{matrix}s_{1}&&\\\ &\ddots&\\\
&&s_{1}\end{matrix}&\dots&\begin{matrix}s_{n}&&\\\ &\ddots&\\\
&&s_{n}\end{matrix}\end{array}\right].$

Now,
$\operatorname{Fitt}_{0}\big{(}\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})\big{)}_{|V}$
is generated by the $n\times n$ minors of $M$, i.e. all the possible products
of $n$ generators of $\mathcal{I}$ on $V$, with ripetitions. This shows that
$\operatorname{Fitt}_{0}\big{(}\pi_{*}(\mathcal{O}_{X}/\pi^{*}\mathcal{I})\big{)}_{|V}=\mathcal{I}_{|V}^{n}$.
∎

####  2.3.4. Relation with the degree

In this subsection we show that $\pi_{*}$ preserves the degree of divisors
under the condition that $Y$ is smooth over $k$. In general, however, the
direct image of a generalized divisor $D$ may not have the same degree of $D$,
as the following example shows. Since we are dealing with degrees, we assume
that $X$ and $Y$ are projective curves over a base field $k$.

######  Example 2.3.21.

Fix $k$ an algebraically closed field. Let $A=k[x,y]/(y^{2}-x^{4})$ be the
affine coordinate ring of a curve $X=\operatorname{Spec}A$ with a tacnode at
the point $P$ corresponding to the maximal ideal $\mathfrak{p}=(x,y)$.

The involution $\sigma$ on $A$ defined by $x\mapsto-x,y\mapsto y$ induces a
involution $\sigma_{X}$ on the curve $X$. The geometric quotient
$Y=X/\sigma_{X}$ is an affine curve with coordinate ring equal to the ring of
invariants $A^{\sigma}=k[x^{2},y]/(y^{2}-x^{4})$, that is isomorphic to
$B=k[s,t]/(t^{2}-s^{2})$ putting $s\mapsto x^{2}$ and $t\mapsto y$. The
quotient curve $Y$ has a simple node at the point $Q$ corresponding to the
maximal ideal $\mathfrak{q}=(s,t)$.

The inclusion map $A^{\sigma}\subset A$ gives to $A$ the structure of free
$B$-module, with basis $\\{1,x\\}$; so, the corresponding morphism of curve
$\pi:X\rightarrow Y=X/\sigma_{X}$ is a finite, locally free map of degree $2$
sending $P$ to $Q$.

Let $D$ be the generalized divisor on $X$ defined by the ideal
$I=(x^{2},y)\subset A$; note that $D$ is supported only on the tacnode $P$.
Since we want to compare $\deg(D)=\deg_{P}(D)$ with
$\deg(\pi_{*}(D))=\deg_{Q}(\pi_{*}(D))$, we can restrict to work locally
around $P$ and $Q$. Let $B_{\mathfrak{q}}$ be the local ring of $Y$ at $Q$.
The induced map $B_{\mathfrak{q}}\rightarrow A_{\mathfrak{p}}$ makes
$A_{\mathfrak{p}}$ a free $B_{\mathfrak{q}}$-module of rank 2.

Let $E=A_{\mathfrak{p}}/I_{\mathfrak{p}}\simeq k[x]/(x^{2})$ be the local ring
of $D$ at the $P$. We have:

$\deg_{P}(D)=\ell(E)=2.$

Observe that $E$ has the following free presentation as
$A_{\mathfrak{p}}$-module:

$A_{\mathfrak{p}}^{\oplus 2}\xrightarrow{(\cdot x^{2},\cdot
y)}A_{\mathfrak{p}}\rightarrow E\rightarrow 0.$

Since $A_{\mathfrak{p}}$ is a free $B_{\mathfrak{q}}$-module of rank 2, $E$
has also a presentation as $B_{\mathfrak{q}}$-module:

$B_{\mathfrak{q}}^{\oplus 4}\xrightarrow{\varphi}B_{\mathfrak{q}}^{\oplus
2}\rightarrow E\rightarrow 0$

where

$\varphi=\left[\begin{array}[]{cc|cc}s&0&t&0\\\ 0&s&0&t\end{array}\right].$

Then, the 0-th Fitting ideal of $E$ as $B_{\mathfrak{q}}$-module is
$F_{0}(E)=(s^{2},t^{2},st)\subset B_{\mathfrak{q}}$. We have:

$\deg_{Q}(\pi_{*}(D))=\ell(B_{\mathfrak{q}}/F_{0}(E))=3.$

Note that there are divisors of degree 2 on $X$, supported at $P$, whose
direct image has degree $2$ on $Y$. For example, take $D^{\prime}=(x)$.

######  Remark 2.3.22.

The previous example shows, in particular, that Proposition 2.33 in [Vas04] is
false.

We now prove that the degree is preserved if the direct image is Cartier. We
show first a proposition that computes the degree at any point where the
direct image is locally principal.

######  Proposition 2.3.23.

(Degree of the direct image of a generalized divisor) Let
$D\in\operatorname{GDiv}(X)$ be a generalized divisor on $X$ and let $y\in Y$
be a point in codimension 1 of the support of $\pi_{*}(D)$. Suppose that
$\pi_{*}(D)$ is locally principal at $y$. Then,

$\deg_{y}(\pi_{*}(D))=\sum_{\pi(x)=y}\deg_{x}(D).$

###### Proof.

First, suppose that $D$ is effective. Let $V=\operatorname{Spec}(B)$ be an
affine open neighborhood of $y$ with affine pre-image
$U=\pi^{-1}(V)=\operatorname{Spec}(A)$, and let $I\subset A$ denote the ideal
of $D$ restricted to $U$. The coordinate ring of $D$ on $U$ is the Artin ring
$A/I$ whose spectrum is equal to
$\operatorname{Spec}(A)\cap\operatorname{Supp}(D)=\\{\mathfrak{p}_{1},\dots,\mathfrak{p}_{s}\\}$;
hence we have:

$A/I=\prod_{i=1}^{s}(A/I)_{\mathfrak{p}_{i}}.$

Let $\mathfrak{q}\subset B$ denote the maximal ideal corresponding to $y$ in
$\operatorname{Spec}(B)$ and let $B_{\mathfrak{q}}$ be the associated local
ring of dimension 1. Then, the localization of $A/I$ at $\mathfrak{q}$,
denoted $E$, is the coordinate ring of $D$ restricted to the fiber of $y$:

$E=(A/I)_{\mathfrak{q}}=\prod_{\pi^{-1}(\mathfrak{p}_{i})=\mathfrak{q}}(A/I)_{\mathfrak{p}_{i}}.$

Since $\pi_{*}(D)$ is effective, the degree of $\pi_{*}(D)$ at $y$ is:

$\displaystyle\deg_{y}(\pi_{*}(D))$
$\displaystyle=\ell\big{(}\mathcal{O}_{Y,y}/F_{0}(\pi_{*}\mathcal{O}_{D})_{y}\big{)}[\kappa(y):k]=$
$\displaystyle=\ell\big{(}B_{\mathfrak{q}}/F_{0}(E)\big{)}[\kappa(y):k].$

By hypothesis $F_{0}(E)$ is an invertible module, so by [Vas04, Proposition
2.32], we have $\ell\big{(}B_{\mathfrak{q}}/F_{0}(E)\big{)}=\ell(E)$. On the
other hand, thanks to [Sta19, https://stacks.math.columbia.edu/tag/02M0Tag
02M0], note that:

$\displaystyle\ell(E)$
$\displaystyle=\ell\left(\prod_{\pi^{-1}(\mathfrak{p}_{i})=\mathfrak{q}}(A/I)_{\mathfrak{p}_{i}}\right)=$
$\displaystyle=\sum_{\pi^{-1}(\mathfrak{p}_{i})=\mathfrak{q}}\ell(A_{\mathfrak{p}_{i}}/I_{\mathfrak{p}_{i}})][k(\mathfrak{p}_{i}):k(\mathfrak{q})]=$
$\displaystyle=\sum_{\pi(x)=y}\ell\big{(}\mathcal{O}_{X,x}/\mathcal{I}_{x}\big{)}[\kappa(x):\kappa(y)]=$
$\displaystyle=\sum_{\pi(x)=y}\deg_{x}(D)[\kappa(x):\kappa(y)].$

Putting everything together we have:

$\displaystyle\deg_{y}(\pi_{*}(D))$ $\displaystyle=\ell(E)[\kappa(y):k]=$
$\displaystyle=\left(\sum_{\pi(x)=y}\deg_{x}(D)[\kappa(x):\kappa(y)]\right)[\kappa(y):k]=$
$\displaystyle=\sum_{\pi(x)=y}\deg_{x}(D)[\kappa(x):k]=$
$\displaystyle=\sum_{\pi(x)=y}\deg_{x}(D)$

for $D$ effective generalized divisor on $X$.

Let now $D=E-F$ be a generalized divisor, written as a difference of effective
generalized divisors with $F$ Cartier by Lemma 2.2.3. Since $\pi_{*}(F)$ is
Cartier, using the result for effective divisors estabilished before, we have:

$\displaystyle\deg_{y}(\pi_{*}(D))$
$\displaystyle=\deg_{y}(\pi_{*}(E)-\pi_{*}(F))=$
$\displaystyle=\deg_{y}(\pi_{*}(E))-\deg_{y}(\pi_{*}(F))=$
$\displaystyle=\sum_{\pi(x)=y}\left[\deg_{x}(E)-\deg_{x}(F)\right]=$
$\displaystyle=\sum_{\pi(x)=y}\deg_{x}(D).$

∎

######  Corollary 2.3.24.

Let $D\in\operatorname{GDiv}(X)$ be a generalized divisor on $X$ such that
$\pi_{*}(D)$ is Cartier. Then,

$\deg_{Y}(\pi_{*}(D))=\deg_{X}(D).$

###### Proof.

Applying Definition 2.2.4 and Proposition 2.3.23, we get:

$\displaystyle\deg_{Y}(\pi_{*}(D))$ $\displaystyle=\sum_{y\textrm{ cod
1}}\deg_{y}(\pi_{*}(D))=$ $\displaystyle=\sum_{y\textrm{ cod
1}}\sum_{\pi(x)=y}\deg_{x}(D).$

By the properties of the Fitting image, $\pi_{*}(D)$ is supported on the set-
theoretic image of the support of $D$; hence, the $x$ appearing in the last
sum are all the $x$ for which $\deg_{x}(D)$ is not zero. Then, the previous
sum gives:

$\displaystyle\deg_{Y}(\pi_{*}(D))$ $\displaystyle=\sum_{x\textrm{ cod
1}}\deg_{x}(D)=\deg_{X}(D).$

∎

######  Corollary 2.3.25.

Suppose that $Y$ is smooth. Then, for any generalized divisor $D$ on $X$,

$\deg_{Y}(\pi_{*}(D))=\deg_{X}(D).$

###### Proof.

Let $D$ be a generalized divisor on $X$. The direct image $\pi_{*}(D)$ is a
generalized divisor on $Y$ smooth, hence Cartier. Then, apply Corollary
2.3.24. ∎

Proposition 2.3.23 yields another useful corollary about the surjectivity of
the direct image morphism.

######  Corollary 2.3.26.

(Surjectivity of the direct image for effective divisors) Suppose that $Y$ is
smooth and $k$ is algebraically closed. Then, the direct image for effective
divisors:

$\pi_{*}^{+}:\operatorname{GDiv}^{+}(X)\rightarrow\operatorname{CDiv}^{+}(Y)$

is surjective.

###### Proof.

Let $E\in\operatorname{CDiv}^{+}(Y)$ be an effective Cartier divisor on $Y$
and let $V=\operatorname{Spec}(R)\subseteq Y$ be an affine open subset of $Y$
such that $E$ is supported on $V$. Let
$\operatorname{Supp}(E)=\\{y_{1},\dots,y_{r}\\}$ be the support of $E$ and for
each $i=1,\dots,r$ let

$d_{i}=\deg_{y_{i}}(E)/[\kappa(y_{i}):k]=\ell_{\mathcal{O}_{Y,y_{i}}}(\mathcal{O}_{E,y_{i}}).$

Let $U=\operatorname{Spec}(S)=\pi^{-1}(V)$ be the preimage of $V$. For each
$i$, pick one element $x_{i}\in\pi^{-1}(y_{i})$ in the finite fiber of $y_{i}$
and let $\mathfrak{M}_{i}$ be the maximal ideal of $S$ corresponding to the
point $x_{i}$. Then, the ideal
$I=\mathfrak{M}_{1}^{d_{1}}\cdot\dots\cdot\mathfrak{M}_{r}^{d_{r}}$ in $S$
defines a divisor $D$ on $U$ (and hence of $X$) such that
$\deg_{y_{i}}(E)=\deg_{x_{i}}(D)$. Looking at its direct image $\pi_{*}(D)$,
it is an effective divisor on $Y$ with the same support of $E$ and the same
degree at any point by Propostion 2.3.23. Since $Y$ is smooth, we conclude
that $\pi_{*}(D)=E$. ∎

###  2.4. The direct and inverse image
for families of generalized divisors

Let $\pi:X\rightarrow Y$ be a finite, flat map of degree $n$ between
projective curves over a field $k$. In the present section, we discuss the
definition of direct and inverse image for families of effective generalized
divisors. Under suitable conditions, recalling Definition 2.2.7, we aim to
define a pair of geometric morphisms:

$\displaystyle\pi_{*}$
$\displaystyle:\operatorname{Hilb}_{X}\rightarrow\operatorname{Hilb}_{Y}$
$\displaystyle\pi^{*}$
$\displaystyle:\operatorname{Hilb}_{Y}\rightarrow\operatorname{Hilb}_{X}$

that, on $k$-valued points, coincide with the direct and inverse image between
$\operatorname{GDiv}^{+}(X)$ and $\operatorname{GDiv}^{+}(Y)$.

######  Remark 2.4.1.

Giving a definition of the direct image for families of effective generalized
divisors is not possible in general when the curve $Y$ is not smooth over $k$.
Consider, for example, the setting of Example 2.3.21; since $X$ and $Y$ are
reduced curves with planar singularities, their Hilbert schemes of generalized
divisors of given degree are connected (see [AIK77], [BGS81]). The effective
divisors $D_{1}$ and $D_{2}$ on $X$ defined by $(x^{2},y)$ and $(x)$ on $X$
have both degree $2$, but their direct images on the quotient node $Y$ have
degree respectively equal to $3$ and $2$. Then, their $k$-points $D_{1},D_{2}$
in the connected component $\operatorname{Hilb}_{X}^{2}$ of
$\operatorname{Hilb}_{X}$ are sent to different connected components of
$\operatorname{Hilb}_{Y}$. This shows that the direct image of divisors in
general is not defined as a geometric map.

In the rest of the section, consider $\pi:X\rightarrow Y$ a finite, flat map
of degree $n$ between projective curves over $k$, and suppose that $Y$ is
smooth over $k$. Recall that, in such case,
$\operatorname{Hilb}_{Y}=\operatorname{{}^{\ell}Hilb}_{Y}$.

######  Definition/Lemma 2.4.2.

(Direct image for families of effective generalized divisors) Let $T$ be any
$k$-scheme. The direct image map for the Hilbert scheme of effective
generalized divisors is defined on the $T$-valued points as:

$\displaystyle\pi_{*}(T):\hskip 20.00003pt\operatorname{Hilb}_{X}(T)$
$\displaystyle\longrightarrow\operatorname{{}^{\ell}Hilb}_{Y}(T)$
$\displaystyle D\subseteq X\times_{k}T$
$\displaystyle\longmapsto\mathcal{Z}\left(\operatorname{Fitt}_{0}(\pi_{T,*}(\mathcal{O}_{D}))\right)$

where $\pi_{T}:X\times_{k}T\rightarrow Y\times_{k}T$ is the morphism induced
by base change of $\pi$.

###### Proof.

Let $D\subseteq X\times_{k}T$ be a $T$-flat family of effective divisors of
$X$, defined by an ideal sheaf
$\mathcal{I}\subseteq\mathcal{O}_{X\times_{k}T}$ such that
$\mathcal{O}_{D}=\mathcal{O}_{X\times_{k}T}/\mathcal{I}$ is flat over $S$.
From the exact sequence

$0\rightarrow\mathcal{I}\rightarrow\mathcal{O}_{X\times_{k}T}\rightarrow\mathcal{O}_{D}\rightarrow
0$

we deduce that also $\mathcal{I}$ is flat over $T$. Since $\pi$ is finite and
flat, $\pi_{T,*}(\mathcal{I})$ is also flat over $T$, fiberwise locally free
since $Y$ is smooth and hence locally free on $Y\times_{k}T$ by [HL10, Lemma
2.1.7]. Moreover, it fits the exact sequence

$0\rightarrow\pi_{T,*}(\mathcal{I})\xrightarrow{\varphi}\pi_{T,*}(\mathcal{O}_{X\times_{k}T})\rightarrow\pi_{T,*}(\mathcal{O}_{D})\rightarrow
0.$

Then, by Definition 2.3.1, the $0$-th Fitting ideal of
$\pi_{T,*}(\mathcal{O}_{D})$ is the image of the canonical injection

$\det\left(\pi_{T,*}(\mathcal{I})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}\xhookrightarrow{\det(\varphi)}\mathcal{O}_{Y\times_{k}T}$

and this is locally free over $Y\times_{k}T$, hence flat over $T$. Then, it
defines a $T$-flat family of effective divisors of $Y$. ∎

######  Remark 2.4.3.

For any $T$-family of effective generalized divisors $D\subseteq X\times_{k}T$
and for any point $t\in T$, the fiber $\pi_{*}(T)(D)_{t}$ is equal to the
direct image $\pi_{*}(D_{t})$ defined for the effective divisor $D_{t}$ on
$X$. Moreover, since $Y$ is smooth, by Corollary 2.3.25 we have:

$\deg_{Y}(\pi_{*}(D_{t}))=\deg_{X}(D_{t}).$

Then, for any $d\geq 0$, $\pi_{*}$ restricts to a map:

$\pi_{*}^{d}:\operatorname{Hilb}_{X}^{d}\longrightarrow\operatorname{{}^{\ell}Hilb}_{Y}^{d}.$

######  Definition/Lemma 2.4.4.

(Inverse image for families of effective generalized divisors) Let $T$ be any
$k$-scheme. The inverse image map for the Hilbert scheme of effective Cartier
divisors is defined on the $T$-valued points as:

$\displaystyle\pi^{*}(T):\hskip 20.00003pt\operatorname{{}^{\ell}Hilb}_{Y}(T)$
$\displaystyle\longrightarrow\operatorname{{}^{\ell}Hilb}_{X}(T)$
$\displaystyle D\subseteq Y\times_{k}T$
$\displaystyle\longmapsto\mathcal{Z}\left(\pi_{T}^{-1}(\mathcal{I}_{D})\cdot\mathcal{O}_{X\times_{k}T}\right)$

where $\pi_{T}:X\times_{k}T\rightarrow Y\times_{k}T$ is the morphism induced
by base change of $\pi$ and
$\mathcal{I}_{D}\subseteq\mathcal{O}_{Y\times_{k}T}$ is the ideal sheaf of
$D$.

###### Proof.

Since $\mathcal{I}$ is locally principal,
$\pi_{T}^{-1}(\mathcal{I})\cdot\mathcal{O}_{X\times_{k}T}\subseteq\mathcal{O}_{X\times_{k}T}$
is locally principal. The restriction of $\pi^{*}(T)(D)$ to the fiber over any
$t\in T$ has ideal sheaf
$\pi^{-1}(\mathcal{I}_{t})\cdot\mathcal{O}_{X\times_{k}t}$, which is equal to
the defining ideal of $\pi^{*}(D_{t})$ by Definition 2.3.16. Hence,
$\pi^{*}(T)(D)$ is a locally principal subscheme of $X\times_{k}T$, such that
all fibers over $T$ are effective Cartier divisors by Lemma 2.3.15. Then,
$\pi^{*}(T)(D)$ is $T$-flat by [Sta19,
https://stacks.math.columbia.edu/tag/062YTag 062Y], hence it is a $T$-family
of Cartier divisors. ∎

######  Remark 2.4.5.

For any integer $d\geq 0$, $\pi^{*}$ restricts to a map:

$\pi^{*}_{d}:\operatorname{{}^{\ell}Hilb}_{Y}^{d}\longrightarrow\operatorname{{}^{\ell}Hilb}_{X}^{nd}.$

We study now some properties of the direct and inverse image for families of
effective generalized divisors. With a slight abuse of notation, we will write
$\pi_{*}$ and $\pi^{*}$ instead of $\pi_{*}(T)$ and $\pi^{*}(T)$, when it is
clear that we are working on $T$-points.

######  Proposition 2.4.6.

(Properties of the direct and inverse image for families of effective
generalized divisors) Let $T$ be any $k$-scheme. Then, the following fact
holds.

  1. (1)

Let $D,E$ be $T$-families of effective divisors over $X$ such that $E$ is a
family of Cartier divisors. Then, $\pi_{*}(D+E)=\pi_{*}(D)+\pi_{*}(E)$.

  2. (2)

Let $F,G$ be $T$-families of effective divisors over $Y$. Then,
$\pi^{*}(F+G)=\pi^{*}(F)+\pi^{*}(G)$.

  3. (3)

If $F$ is a $T$-family of effective divisors over $Y$, then
$\pi_{*}(\pi^{*}(F))=nF$.

###### Proof.

The proof of parts (1) and (2) follows the proofs of the second part of Lemma
2.3.4 and 2.3.15 respectively. The proof of part (3) follows the proof of
Proposition 2.3.20. ∎

###  2.5. The Norm and the inverse image
for families of torsion-free rank-1 sheaves

In the present section, we provide the definition of the Norm map for families
torsion-free sheaves of rank 1 and the related inverse image map. Since we are
dealing with families of sheaves, we assume that $X$ and $Y$ are projective
curves over a field $k$.

In order to be compatible with the direct image for generalized line bundles,
our definition of the Norm map on $\overline{\mathbb{J}}(X)$ will be inspired
by the sheaf-theoretic formula of Proposition 2.1.7. By Proposition 2.3.10 and
its corollaries, the generalization of such formula to generalized divisors
and torsion-free sheaves involves taking the double $\omega$-dual of the
exterior power of the pushforward of generalized line bundles. In general,
this operation does not behave well in families if $Y$ is not smooth.

Then, in accordance with the previous section, we suppose that $Y$ is smooth
over $k$; in such case, the moduli space $\overline{\mathbb{J}}(Y)$ is
actually equal to the Jacobian $J(Y)$. Then, the aim of this chapter is to
provide a pair of geometric morphism:

$\displaystyle\operatorname{Nm}_{\pi}:\overline{\mathbb{J}}(X)$
$\displaystyle\rightarrow J(Y)$ $\displaystyle\pi^{-1}:J(Y)$
$\displaystyle\rightarrow J(X)\subseteq\overline{\mathbb{J}}(X)$

that, on $k$-valued points, coincide with the direct and inverse image maps
between $\operatorname{GPic}(X)$ and
$\operatorname{GPic}(Y)=\operatorname{Pic}(Y)$.

Recall also (Definition 2.2.8) that, for any line bundle $M$ on $X$, the
compactified Jacobian of $X$ is related to the Hilbert scheme of effective
generalized divisors via the twisted Abel map $\mathcal{A}_{M}$. We will show
that the direct image map and the Norm map are compatible as well as the
inverse image maps, meaning that for any $M\in J(X)$ and $N\in J(Y)$ there are
commutative diagrams of $k$-schemes:

${\operatorname{Hilb}_{X}}$${\operatorname{{}^{\ell}Hilb}_{Y}}$${\operatorname{\textit{G}\mathbb{J}}(X)}$${\subseteq}$${\overline{\mathbb{J}}(X)}$${J(Y)}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathcal{A}_{M}}$$\scriptstyle{\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}}$$\scriptstyle{\operatorname{Nm}_{\pi}}$
${\operatorname{{}^{\ell}Hilb}_{Y}}$${\operatorname{{}^{\ell}Hilb}_{X}}$${J(Y)}$${J(X).}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\mathcal{A}_{N}}$$\scriptstyle{\mathcal{A}_{\pi^{*}(N)}}$$\scriptstyle{\pi^{*}}$

We give first the definition for the Norm map on $\overline{\mathbb{J}}(X)$.

######  Definition/Lemma 2.5.1.

(Norm map for torsion-free rank-1 sheaves) Let $T$ be any $k$-scheme. The Norm
map between compactified Jacobians associated to $\pi$ is defined on the
$T$-valued points as:

$\displaystyle\operatorname{Nm}_{\pi}(T):\hskip
20.00003pt\overline{\mathbb{J}}(X)(T)$ $\displaystyle\longrightarrow J(Y)(T)$
$\displaystyle\mathcal{L}$
$\displaystyle\longmapsto\det\left(\pi_{T,*}(\mathcal{L})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}.$

###### Proof.

Let $\mathcal{L}$ be a $T$-family of torsion-free sheaves of rank 1 on $X$,
i.e. a $T$-flat coherent sheaf on $X\times_{k}T$, whose fibers over $T$ are
torsion-free sheaves of rank 1. The push-forward $\pi_{T,*}(\mathcal{L})$ is a
$T$-flat coherent sheaf on $Y\times_{k}T$ such that, for any $t\in T$, the
fiber $(\pi_{T,*}\mathcal{L})_{t}$ equals $\pi_{*}(\mathcal{L}_{t})$ on
$Y\simeq Y\times_{k}{t}$, . Since $Y$ is smooth, $\pi_{*}(\mathcal{L}_{t})$ is
a locally free sheaf of rank $n$ for any $t\in T$. Then, by [HL10, Lemma
2.1.7], $\pi_{T,*}(\mathcal{L})$ is a locally sheaf of rank $n$ on
$Y\times_{k}T$. Its determinant bundle is a line bundle on $Y\times_{k}T$,
hence flat over $T$. ∎

The definition of the inverse image is standard but we give it for the sake of
completeness.

######  Definition/Lemma 2.5.2.

(Inverse image map for line bundles) Let $T$ be any $k$-scheme. The inverse
image map between Jacobians associated to $\pi$ is defined on the $T$-valued
points as:

$\displaystyle\pi^{*}(T):\hskip 20.00003ptJ(Y)(T)$
$\displaystyle\longrightarrow J(X)\subseteq\overline{\mathbb{J}}(X)(T)$
$\displaystyle\mathcal{N}$ $\displaystyle\longmapsto\pi_{T}^{*}(\mathcal{N}).$

###### Proof.

By hypothesis, $\mathcal{N}$ is a $T$-flat coherent sheaf on $Y\times_{k}T$
that is a line bundle on any fiber over $T$; then, by [HL10, Lemma 2.1.7], it
is a line bundle on $Y\times_{k}T$. We conclude that $\pi^{*}\mathcal{N}$ is a
line bundle on $X\times_{k}T$ and hence a $T$-flat family of line bundles over
$T$. ∎

######  Remark 2.5.3.

When there is no ambiguity, we will write $\operatorname{Nm}_{\pi}$ and
$\pi^{*}$ instead of $\operatorname{Nm}_{\pi}(T)$ and $\pi^{*}(T)$. The Norm
and the inverse image for generalized line bundles define morphisms of
algebraic stacks

$\displaystyle\operatorname{Nm}_{\pi}:\hskip
20.00003pt\overline{\mathbb{J}}(X)$ $\displaystyle\longrightarrow J(Y)$
$\displaystyle\pi^{*}:\hskip 20.00003ptJ(Y)$ $\displaystyle\longrightarrow
J(X).$

For any integer $d$, they restricts to:

$\displaystyle\operatorname{Nm}_{\pi}^{d}:\hskip
20.00003pt\overline{\mathbb{J}}(X,d)$ $\displaystyle\longrightarrow J^{d}(Y)$
$\displaystyle\pi^{*}_{d}:\hskip 20.00003ptJ^{d}(Y)$
$\displaystyle\longrightarrow J^{nd}(X).$

Moreover, the Norm for torsion-free rank-1 sheaves, restricted to the locus of
line bundles $J(X)\subseteq\overline{\mathbb{J}}(X)$, coincides with the
classical Norm map from $J(X)$ to $J(Y)$ of Definition 2.1.9.

We study now some properties of the Norm map on $\overline{\mathbb{J}}(X)$ and
the inverse image map. First, we need a technical lemma.

######  Lemma 2.5.4.

Let $T$ be a fixed $k$-scheme. For any $T$-flat family $\mathcal{L}$ of
torsion-free rank-1 sheaves on $X\times_{k}T$ and any line bundle
$\mathcal{M}$ on $X\times_{k}T$, there is an isomorphism:

$\displaystyle\det\left(\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)\simeq$
$\displaystyle\simeq\det\left(\pi_{T,*}(\mathcal{L})\right)\otimes\det\left(\pi_{T,*}(\mathcal{M})\right).$

###### Proof.

The proof is similar to the second part of the proof of Proposition 2.3.10.

Since $\pi_{T}:X\times_{k}T\rightarrow Y\times_{k}T$ is finite and flat,
$\pi_{T,*}(\mathcal{L})$ is a $T$-flat coherent sheaf on $Y\times_{k}T$, that
is locally free on any fiber over $T$ since $Y$ is smooth; then,
$\pi_{T,*}(\mathcal{L})$ is locally free of rank $n$ by [HL10, Lemma 2.1.7].
The same holds for $\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})$. On the other
hand, by [Gro61, Proposition 6.1.12], $\pi_{T,*}\mathcal{M}$ is a locally free
$\pi_{T,*}\mathcal{O}_{X}$-module of rank $1$. In particular, there is open
cover $\\{V_{i}\\}_{I}$ of $Y$ such that $\mathcal{M}$ is a trivial
$\mathcal{O}_{X|U_{i}}$-module on each $U_{i}=\pi_{T}^{-1}(V_{i})$, i.e. there
are isomorphisms
$\lambda_{i}:\mathcal{M}_{|U_{i}}\xrightarrow{{}_{\sim}}\mathcal{O}_{X|U_{i}}$
for each $i\in I$. On the intersections $U_{i}\cap U_{j}$, the collection
$\\{\lambda_{i}\circ\lambda_{j}^{-1}\\}$ of automorphisms of
${\mathcal{O}_{X\times_{k}T}}_{|U_{i}\cap U_{j}}$ is a cochain that measures
the obstruction for the $\lambda_{i}$’s to glue to a global isomorphism. We
define now an isomorphism

$\alpha:\det\left(\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)\longrightarrow\det\left(\pi_{T,*}(\mathcal{L})\right)\otimes\det\left(\pi_{T,*}(\mathcal{M})\right)$

by glueing a collection of isomorphisms $\alpha_{i}$ defined on each $V_{i}$.
To do so, we define each $\alpha_{i}$ as the following composition of arrows:

${\left(\det\big{(}\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})\big{)}\otimes\det(\pi_{T,*}\mathcal{O}_{X})\right)_{|V_{i}}}$${\left(\det\big{(}\pi_{T,*}\mathcal{L}\big{)}\otimes\det\big{(}\pi_{T,*}\mathcal{M}\big{)}\right)_{|V_{i}}}$${\det\big{(}\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})_{|U_{i}}\big{)}\otimes\det(\pi_{T,*}\mathcal{O}_{X|U_{i}})}$${\det\big{(}\pi_{T,*}\mathcal{L}_{|U_{i}}\big{)}\otimes\det\big{(}\pi_{T,*}\mathcal{M}_{|U_{i}}\big{)}}$${\det\big{(}\pi_{T,*}\mathcal{L}_{|U_{i}}\big{)}\otimes\det(\pi_{T,*}\mathcal{O}_{X|U_{i}})}$${\det\big{(}\pi_{T,*}\mathcal{L}_{|U_{i}}\big{)}\otimes\det\big{(}\pi_{T,*}\mathcal{O}_{X|U_{i}}\big{)}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{=}$$\scriptstyle{\det\big{(}\pi_{T,*}\lambda_{i}\big{)}\otimes\operatorname{id}}$$\scriptstyle{=}$${=}$$\scriptstyle{\operatorname{id}\otimes\det\big{(}\pi_{T,*}\lambda_{i}^{-1}\big{)}}$

i.e.
$\alpha_{i}:=\det\big{(}\pi_{T,*}\lambda_{i}\big{)}\otimes\det\big{(}\pi_{T,*}\lambda_{i}^{-1}\big{)}$.
Since $\det\big{(}\pi_{T,*}\\_\big{)}$ is functorial, the obstruction
$\alpha_{i}\circ\alpha_{j}^{-1}$ is trivial on any $V_{i}\cap V_{j}$, whence
the $\alpha_{i}$’s glue together to a global isomorphism $\alpha$. ∎

######  Proposition 2.5.5.

(Properties of the Norm and the inverse image map) Let $T$ be any $k$-scheme.
Then, the following facts hold.

  1. (1)

Let $\mathcal{L},\mathcal{M}\in\overline{\mathbb{J}}(X)(T)$ such that
$\mathcal{M}$ is a $T$-flat family of line bundles. Then,
$\operatorname{Nm}_{\pi}(\mathcal{L}\otimes\mathcal{M})\simeq\operatorname{Nm}_{\pi}(\mathcal{L})\otimes\operatorname{Nm}_{\pi}(\mathcal{M})$.

  2. (2)

Let $\mathcal{N},\mathcal{N}^{\prime}\in J(Y)(T)$. Then,
$\pi^{*}(\mathcal{N}\otimes\mathcal{N}^{\prime})\simeq\pi^{*}(\mathcal{N})\otimes\pi^{*}(\mathcal{N}^{\prime})$.

  3. (3)

Let $\mathcal{N}\in J(Y)(T)$. Then, $\pi^{*}(\mathcal{N})$ is a $T$-flat
family of line bundles over $X$ and
$\operatorname{Nm}_{\pi}(\pi^{*}(\mathcal{N}))\simeq\mathcal{N}^{\otimes n}$.

###### Proof.

To prove (1), note that $\mathcal{M}$ is a $T$-flat coherent sheaf on
$X\times_{k}T$ that is a line bundle on any fiber over $T$; hence it is a line
bundle on $X\times_{k}T$ by [HL10, Lemma 2.1.7]. Then, applying Definition
2.5.1 and Lemma 2.5.4, we have:

$\displaystyle\operatorname{Nm}_{\pi}(\mathcal{L}\otimes\mathcal{M})$
$\displaystyle=\det\left(\pi_{T,*}(\mathcal{L}\otimes\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}\simeq$
$\displaystyle\simeq\det\left(\pi_{T,*}(\mathcal{L})\right)\otimes\det\left(\pi_{T,*}(\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-2}\simeq$
$\displaystyle\simeq\operatorname{Nm}_{\pi}(\mathcal{L})\otimes\operatorname{Nm}_{\pi}(\mathcal{M}).$

Part (2) follows from the associative properties of the tensor product.

To prove (3), compute by Definitions 2.5.1 and 2.5.2:

$\displaystyle\operatorname{Nm}_{\pi}(\pi^{*}(\mathcal{N}))=\det\left(\pi_{T,*}(\pi_{T}^{*}(\mathcal{N}))\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}.$

By the projection formula [Sta19, https://stacks.math.columbia.edu/tag/01E8Tag
01E8] and the standard properties of determinants, we have:

$\displaystyle\operatorname{Nm}_{\pi}(\pi^{*}(\mathcal{N}))$
$\displaystyle=\det\left(\mathcal{N}\otimes\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}\simeq$
$\displaystyle\simeq\mathcal{N}^{n}\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}\simeq$
$\displaystyle\simeq\mathcal{N}^{n}.$

∎

We compare now the Norm and the inverse image maps between the compactified
Jacobians with the direct and inverse image maps between the Hilbert schemes
of divisors, respectively.

######  Proposition 2.5.6.

(Comparison of the direct image and the Norm map via the Abel map) For any
line bundle $M$ of degree $e$ on $X$ and for any $d\geq 0$, there is a
commutative diagram of algebraic stacks over $k$:

${\operatorname{Hilb}_{X}^{d}}$${\operatorname{{}^{\ell}Hilb}_{Y}^{d}}$${\operatorname{\textit{G}\mathbb{J}}(X,-d+e)}$${\subseteq}$${\overline{\mathbb{J}}(X,-d+e)}$${J^{-d+e}(Y)}$$\scriptstyle{\pi_{*}^{d}}$$\scriptstyle{\mathcal{A}_{M}^{d}}$$\scriptstyle{\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}^{d}}$$\scriptstyle{\operatorname{Nm}_{\pi}^{-d+e}}$

###### Proof.

Let $T$ be any $k$-scheme, let $D$ be a $T$-flat family of effective divisors
of degree $d$ on $X$ with ideal sheaf $\mathcal{I}$, and denote with
$\mathcal{M}$ the pullback of $M$ to $X\times_{k}T$. Following the bottom-left
side of the square, combining Definitions 2.2.8 and 2.5.1 we get:

$\operatorname{Nm}_{\pi}^{-d+e}(\mathcal{A}_{M}^{d}(D))=\det\left(\pi_{T,*}(\mathcal{I}\otimes\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}.$

Following the top-right side of the square, combining definitions 2.4.2, 2.5.1
and 2.2.8 we get:

$\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}^{d}(\pi_{*}^{d}(D))=\det\left(\pi_{T,*}(\mathcal{I})\right)\otimes\det\left(\pi_{T,*}(\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-2}.$

We are left to prove that:

$\displaystyle\det\left(\pi_{T,*}(\mathcal{I}\otimes\mathcal{M})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)\simeq$
$\displaystyle\simeq\det\left(\pi_{T,*}(\mathcal{I})\right)\otimes\det\left(\pi_{T,*}(\mathcal{M})\right).$

Now, $\mathcal{I}$ is a $T$-flat family of generalized line bundles by
hypothesis and $\mathcal{M}$ is a line bundle on $X\times_{k}T$ since it is
the pull-back of a line bundle on $X$. Then, the assertion is true by Lemma
2.5.4. ∎

######  Proposition 2.5.7.

(Comparison of the inverse image maps via the Abel map) For any line bundle
$N$ of degree $e$ on $Y$ and any $d\geq 0$, there is a commutative diagram of
algebraic stacks over $k$:

${\operatorname{{}^{\ell}Hilb}_{Y}^{d}}$${\operatorname{{}^{\ell}Hilb}_{X}^{d}}$${J(Y,-d+e)}$${J(X,-d+e)}$$\scriptstyle{\pi^{*}_{d}}$$\scriptstyle{\mathcal{A}_{N}^{d}}$$\scriptstyle{\mathcal{A}_{\pi^{*}(N)}^{d}}$$\scriptstyle{\pi^{*}_{-d+e}}$

###### Proof.

Let $T$ be any $k$-scheme, let $D$ be a $T$-flat family of divisors of degree
$d$ on $Y$ with ideal sheaf $\mathcal{I}\subseteq Y\times_{k}T$, and denote
with $\mathcal{N}$ the pullback of $N$ to $Y\times_{k}T$. By Definition 2.2.8,
Proposition 2.5.5(2) and Remark 2.3.19, we have:

$\displaystyle\pi^{*}_{d}(\mathcal{A}_{N}^{d}(D))$
$\displaystyle=\pi_{T}^{*}(\mathcal{I}\otimes\mathcal{N})\simeq\pi_{T}^{*}(\mathcal{I})\otimes\pi_{T}^{*}(\mathcal{N})\simeq$
$\displaystyle\simeq\mathcal{A}_{\pi^{*}(N)}^{d}(\pi^{*}_{-d+e}(\mathcal{N})).$

∎

##  Chapter 3 The fibers of the Norm map and the Prym stack

In the present chapter we study the fibers of the Norm map defined in Chapter
2. The purpose is to generalize the Prym scheme associated to a finite, flat
morphism between projective curves to the context of torsion-free sheaves of
rank 1. Fix an algebraically closed field $k$ of characteristic 0. Let $X,Y$
be projective curves over $k$, such that $Y$ is smooth, and let
$\pi:X\rightarrow Y$ be a finite, flat morphism of degree $n$. We start by
recalling the following definition.

######  Definition 3.0.1.

The Prym scheme of $X$ associated to $\pi$ is the locus $\Pr(X,Y)\subseteq
J(X)$ of line bundles whose Norm with respect to $\pi$ is trivial. In other
words:

$\Pr(X,Y)=\\{\mathcal{L}\in
J(X):\operatorname{Nm}_{\pi}(\mathcal{L})\simeq\mathcal{O}_{Y}\\}.$

We can now extend the definition of the Prym stack to the context of torsion-
free rank 1 sheaves using the Norm map defined in the previous chapter.

######  Definition 3.0.2.

The Prym stack of $X$ associated to $\pi$ is the locus
$\overline{\Pr}(X,Y)\subseteq\overline{\mathbb{J}}(X)$ of torsion-free rank-1
sheaves whose Norm with respect to $\pi$ is trivial. In other words:

$\overline{\Pr}(X,Y):=\\{(\mathcal{L},\lambda):\mathcal{L}\in\overline{\mathbb{J}}(X)\
\mathrm{and}\
\lambda:\operatorname{Nm}_{\pi}(\mathcal{L})\xrightarrow{{}_{\sim}}\mathcal{O}_{Y}\\}.$

######  Remark 3.0.3.

It follows from the definition that:

$\Pr(X,Y)=\overline{\Pr}(X,Y)\cap J(X).$

Since being locally-free is an open condition, $\Pr(X,Y)$ is open in
$\overline{\Pr}(X,Y)$.

The Prym stack is a fiber of the Norm map associated to $\pi$. In the present
chapter, we study the fibers of the Norm map assuming that the curve $X$ is
reduced with locally planar singularities. The study of such fibers involves
the study of the fibers of the direct image map for effective divisors defined
in Chapter 2, and the study of the fibers of the Hilbert-Chow morphism. We
start from the last.

###  3.1. The fibers of the Hilbert-Chow morphism

In this section, we refer to [Ber12] for notations and known results.

######  Definition 3.1.1.

Let $X$ be a projective curve over $k$ and let $d$ be a positive integer. The
$d$-symmetric power $X^{(d)}$ of $X$ is the quotient $X^{d}/\Sigma_{d}$ of the
Cartesian product $X^{n}$ by the action of the symmetric group $\Sigma_{d}$ in
$d$-letters permuting the factors. Note that, for $d=1$, $X^{(1)}=X$. For
$d=0$, we put $X^{(0)}=\\{0\\}$. The scheme of effective $0$-cycles (or of
effective Weil divisors) of $X$ is the algebraic scheme:

$\operatorname{WDiv}^{+}(X)=\bigsqcup_{d\geq 0}X^{(d)}=\bigsqcup_{d\geq
0}\operatorname{WDiv}^{d}(X).$

The Hilbert-Chow morphism of degree $d$ associated to $X$ is the map of
schemes

$\displaystyle\rho_{X}^{d}:\operatorname{Hilb}_{X}^{d}$
$\displaystyle\rightarrow X^{(d)}$ $\displaystyle D$
$\displaystyle\mapsto\sum_{x\textrm{ cod 1}}\deg_{x}(D)\cdot[x]$

Denote with $\operatorname{{}^{\ell}\rho}_{X}^{d}$ the restriction of
$\rho_{X}^{d}$ to the subscheme
$\operatorname{{}^{\ell}Hilb}_{X}^{d}\subseteq\operatorname{Hilb}_{X}^{d}$
parametrizing Cartier divisors. The collection of Hilbert-Chow morphisms in
non-negative degrees gives rise to the Hilbert-Chow morphism in any degree:

$\rho_{X}:\operatorname{Hilb}_{X}\rightarrow\operatorname{WDiv}^{+}(X).$

Finally, denote with $\operatorname{{}^{\ell}\rho}_{X}$ the restriction of
$\rho_{X}$ to
$\operatorname{{}^{\ell}Hilb}_{X}\subseteq\operatorname{Hilb}_{X}$.

######  Remark 3.1.2.

If $X$ is a smooth projective curve, then $\rho_{X}$ is an isomorphism and
coincides with $\operatorname{{}^{\ell}\rho}_{X}$.

We study now the fibers of the Hilbert-Chow morphism in the case when $X$ is
reduced with locally planar singularities.

######  Proposition 3.1.3.

Suppose that $X$ is reduced with locally planar singularities and that $Y$ is
smooth. Let $W\in\operatorname{WDiv}^{d}(X)$ be an effective Weil divisor of
degree $d$ on $X$ and let $\rho_{X}^{-1}(W)$ be the corresponding fiber in
$\operatorname{Hilb}_{X}$. The locus
$\operatorname{{}^{\ell}\rho}_{X}^{-1}(W)$ of Cartier divisors in
$\rho_{X}^{-1}(W)$ is an open and dense subset of $\rho_{X}^{-1}(W)$.

###### Proof.

First, note that
$\operatorname{{}^{\ell}\rho}_{X}^{-1}(W)=\rho_{X}^{-1}(W)\cap\operatorname{{}^{\ell}Hilb}_{X}$.
Since $\operatorname{{}^{\ell}Hilb}_{X}\subseteq\operatorname{Hilb}_{X}$ is
open (see for example [MRV17a, Fact 2.4]),
$\operatorname{{}^{\ell}\rho}_{X}^{-1}(W)$ is open in $\rho_{X}^{-1}(W)$.

To prove that it is dense, let $W=\sum_{i=1}^{s}n_{i}\cdot[x_{i}]$ with the
$x_{i}$’s are $s$ distinct points. For each $i$, the fiber of $\rho_{X}$ over
the cycle $n_{i}\cdot[x_{i}]$ is equal by definition to the punctual Hilbert
scheme $\operatorname{Hilb}_{X,x_{i}}^{n_{i}}$ parametrizing $0$-dimensional
subschemes of $X$ supported at $x_{i}$ having length $n_{i}$ over $k$. Since
the $x_{i}$ are distinct, we have:

$\displaystyle\rho_{X}^{-1}(W)$
$\displaystyle=\rho_{X}^{-1}\left(\sum_{i=1}^{s}n_{i}\cdot[x_{i}]\right)=$
$\displaystyle=\prod_{i=1}^{s}\rho_{X}^{-1}(n_{i}\cdot[x_{i}])=\prod_{i=1}^{s}\operatorname{Hilb}_{X,x_{i}}^{n_{i}}.$

For each $i$, let
$\operatorname{{}^{\ell}Hilb}_{X,x_{i}}^{n_{i}}=\operatorname{Hilb}_{X,x_{i}}^{n_{i}}\cap\operatorname{{}^{\ell}Hilb}_{X}$.
An effective divisor is Cartier if and only if it is Cartier at each point of
its support, hence:

$\operatorname{{}^{\ell}\rho}_{X}^{-1}(W)=\prod_{i=1}^{s}\left(\operatorname{Hilb}_{X,x_{i}}^{n_{i}}\cap\operatorname{{}^{\ell}Hilb}_{X}\right)=\prod_{i=1}^{s}\operatorname{{}^{\ell}Hilb}_{X,x_{i}}^{n_{i}}.$

For each $i$, $\operatorname{{}^{\ell}Hilb}_{X,x_{i}}^{n_{i}}$ is the smooth
locus of $\operatorname{Hilb}_{X,x_{i}}^{n_{i}}$ by [BGS81, Proposition 2.3],
so it is dense; hence $\operatorname{{}^{\ell}\rho}_{X}^{-1}(W)$ is dense in
$\rho_{X}^{-1}(W)$. ∎

###  3.2. The fibers of the direct image between Hilbert schemes

In the present section, we study the fibers of the direct image map $\pi_{*}$
defined between the Hilbert schemes of generalized divisors of $X$ and $Y$.

First, we introduce a similar notion for Weil divisors and we see how it
relates to the direct image for generalized divisors.

######  Definition 3.2.1.

The direct image for Weil divisors associated to $\pi$ is the morphism of
schemes:

$\pi_{*}:\operatorname{WDiv}^{+}(X)\rightarrow\operatorname{WDiv}^{+}(Y)$

given on the level of points by

$\sum_{i=1}^{d}n_{i}\cdot[x_{i}]\longmapsto\sum_{i=1}^{d}n_{i}\cdot[\pi(x_{i})].$

######  Proposition 3.2.2.

Assume that $Y$ is smooth. There is a commutative square of schemes over $k$:

${\operatorname{Hilb}_{X}}$${\operatorname{{}^{\ell}Hilb}_{Y}}$${\operatorname{WDiv}(X)}$${\operatorname{WDiv}(Y).}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\rho_{Y}}$$\scriptstyle{\pi_{*}}$

###### Proof.

Let $D$ be an effective generalized divisor on $X$. Following the bottom-left
side of the square, we obtain:

$\pi_{*}(\rho_{X}(D))=\sum_{x\textrm{ cod 1}}\deg_{x}(D)\cdot[\pi(x)],$

while on the top-right side of the square we have

$\rho_{Y}(\pi_{*}(D))=\sum_{y\textrm{ cod 1}}\deg_{y}(\pi_{*}(D))\cdot[y].$

Since $Y$ is smooth, $\pi_{*}(D)$ is Cartier locally at each point of $Y$. By
Proposition 2.3.23, for any point $y$ in codimension 1 of $Y$

$\deg_{y}(\pi_{*}(D))=\sum_{\pi(x)=y}\deg_{x}(D).$

Then we compute:

$\displaystyle\rho_{Y}(\pi_{*}(D))$ $\displaystyle=\sum_{y\textrm{ cod
1}}\deg_{y}(\pi_{*}(D))\cdot[y]=$ $\displaystyle=\sum_{y\textrm{ cod
1}}\left(\sum_{\pi(x)=y}\deg_{x}(D)\right)\cdot[y]=$
$\displaystyle=\sum_{y\textrm{ cod
1}}\sum_{\pi(x)=y}\deg_{x}(D)\cdot[\pi_{*}(x)]=$
$\displaystyle=\sum_{x\textrm{ cod
1}}\deg_{x}(D)\cdot[y]=\pi_{*}(\rho_{X}(D)).$

∎

We are now ready to study the fibers of the direct image for generalized
divisors.

######  Proposition 3.2.3.

Assume that $Y$ is smooth and $X$ is reduced with locally planar
singularities. Let $E\in\operatorname{Hilb}_{Y}$ be an effective divisor of
degree $d$ on $Y$ and let $\pi_{*}^{-1}(E)$ be the corresponding fiber in
$\operatorname{Hilb}_{X}$. Then,
$\pi_{*}^{-1}(E)\cap\operatorname{{}^{\ell}Hilb}_{X}$ is an open and non-empty
dense subset of $\pi_{*}^{-1}(E)$.

###### Proof.

By Proposition 3.2.2, there is a commutative diagram of schemes over $k$:

${\operatorname{Hilb}_{X}}$${\operatorname{{}^{\ell}Hilb}_{Y}}$${\operatorname{WDiv}(X)}$${\operatorname{WDiv}(Y).}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\rho_{Y}}$$\scriptstyle{\pi_{*}}$

First note that, since $Y$ is smooth, the map $\pi_{*}$ is surjective by
Corollary 2.3.26, hence $\pi_{*}^{-1}(E)$ is non-empty. Moreover, the Hilbert-
Chow morphism $\rho_{Y}$ is an isomorphism. Then,

$\pi_{*}^{-1}(E)=\rho_{X}^{-1}\pi_{*}^{-1}(\rho_{Y}(E)).$

Let $S=\\{y_{1},\dots,y_{r}\\}$ be the support of $E$ and let
$d_{i}=\deg_{y_{i}}E$ for each $i$; then

$\rho_{Y}(E)=\sum_{i=1}^{r}d_{i}\cdot[y_{i}].$

For each $i$, denote with $\\{x_{i}^{1},\dots,x_{i}^{n_{i}}\\}$ the discrete
fiber of $\pi$ over $y_{i}$. Since points can occur with multeplicity in the
geometric fibers, $n_{i}\leq n$ holds for each $i$. The fiber of
$\sum_{i=1}^{r}d_{i}\cdot[y_{i}]$ is then the discrete subset of
$\operatorname{WDiv}(X)$ given by:

$\pi_{*}^{-1}\left(\sum_{i=1}^{r}d_{i}\cdot[y_{i}]\right)=\left\\{\sum_{i=1}^{r}\sum_{j=1}^{n_{i}}c_{ij}\cdot[x_{i}^{j}]\
\middle|\ c_{ij}\in\mathbb{Z}_{\geq 0},\
\sum_{j=1}^{n_{i}}c_{ij}=d_{i}\textrm{ for any }i\right\\}.$

Denote with $Z$ the set of all tuples $(c_{ij})$ of positive integers with the
conditions that $\sum_{j=1}^{n_{i}}c_{ij}=d_{i}$ for any $i$. Then, we can
write the previous set as

$\pi_{*}^{-1}\left(\sum_{i=1}^{r}d_{i}\cdot[y_{i}]\right)=\bigsqcup_{(c_{ij})\in
Z}\left\\{\sum_{i=1}^{r}\sum_{j=1}^{n_{i}}c_{ij}\cdot[x_{i}^{j}]\right\\}.$

Then, applying $\rho_{*}^{-1}$, we otbain:

$\displaystyle\rho_{X}^{-1}\pi_{*}^{-1}(\rho_{Y}(E))$
$\displaystyle=\rho_{X}^{-1}\pi_{*}^{-1}\left(\sum_{i=1}^{r}d_{i}\cdot[y_{i}]\right)=$
$\displaystyle=\rho_{X}^{-1}\left(\bigsqcup_{(c_{ij})\in
Z}\left\\{\sum_{i=1}^{r}\sum_{j=1}^{n_{i}}c_{ij}\cdot[x_{i}^{j}]\right\\}\right)=$
$\displaystyle=\bigsqcup_{(c_{ij})\in
Z}\rho_{X}^{-1}\left(\sum_{i=1}^{r}\sum_{j=1}^{n_{i}}c_{ij}\cdot[x_{i}^{j}]\right),$

where the last disjoint union is a disjoint union of topological subspaces of
$\operatorname{Hilb}_{X}$ in the same connected component. Then,
$\pi_{*}^{-1}(E)$ is a non-emtpy disjoint uniont of a finite number of fibers
of the Hilbert-Chow morphism. By Proposition 3.1.3, the intersection of any
such fiber with the Cartier locus $\operatorname{{}^{\ell}Hilb}_{X}$ is open
and dense in the same fiber. Taking the disjoint union,
$\pi_{*}^{-1}(E)\cap\operatorname{{}^{\ell}Hilb}_{X}$ is a non-empty open and
dense subset of $\pi_{*}^{-1}(E)$. ∎

###  3.3. The fibers of the Norm map and $\overline{\Pr}(X,Y)$

In this section, assuming that $Y$ is smooth and $X$ is reduced with locally
planar singularities, we prove that the generalized Prym of $X$ with respect
to $Y$ is non-empty, open and dense in the Prym stack of $X$ with respect to
$Y$. The theorem is based upon the following two auxiliaries proposition.

######  Proposition 3.3.1.

Assume that $Y$ is smooth and let $\mathcal{N}\in J(Y)$ be any line bundle on
$Y$. Then, there is a line bundle $\mathcal{L}$ on $X$ such that
$\operatorname{Nm}_{\pi}(\mathcal{L})\simeq\mathcal{N}$.

###### Proof.

Let $D$ be a Cartier divisor on $Y$ such that $\mathcal{N}\in[D]$ and let
$D=E-F$ where $E,F$ are effective Cartier divisors by Lemma 2.2.3. By
Proposition 3.2.3 there are effective Cartier divisors $H,K$ on $X$ such that
$\pi_{*}(H)=E$ and $\pi_{*}(K)=F$. Then, by Lemma 2.3.4 $\pi_{*}(H-K)=D$. Set
$H-K=C$ and let $\mathcal{L}$ be the defining ideal of $C$, seen as a line
bundle. By Lemma 2.3.11 we conclude that
$\operatorname{Nm}_{\pi}(\mathcal{L})\simeq\mathcal{N}$. ∎

######  Proposition 3.3.2.

Assume that $Y$ is smooth and $X$ is reduced with locally planar singularities
and let $\mathcal{N}\in J(Y)$ be any line bundle on $Y$. Then, the fiber
$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})$ is non-empty and contains
$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})\cap J(Y)$ as an open and dense
substack.

###### Proof.

First, note that
$\overline{\mathbb{J}}(X)=\operatorname{\textit{G}\mathbb{J}}(X)$ since $X$ is
reduced. The fiber $\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})$ is non-empty by
Proposition 3.3.1 and the substack
$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})\cap J(Y)$ is open in
$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})$ since being locally free is an
open condition. To prove that it is dense, recall that for any fixed line
bundle $M\in J(X)$ there is a commutative diagram of stacks over $k$:

${\operatorname{Hilb}_{X}}$${\operatorname{{}^{\ell}Hilb}_{Y}}$${\overline{\mathbb{J}}(X)}$${J(Y)}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathcal{A}_{M}}$$\scriptstyle{\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}}$$\scriptstyle{\operatorname{Nm}_{\pi}}$

Moreover, by [MRV17a, Proposition 2.5] there is a cover of
$\overline{\mathbb{J}}(X)$ by $k$-finite type open subsets $\\{U_{\beta}\\}$
such that, for each $\beta$, there is a line bundle $M_{\beta}\in J(X)$ with
the property that
$\mathcal{A}_{M_{\beta}|V_{\beta}}:\mathcal{A}_{M_{\beta}}^{-1}(U_{\beta})=V_{\beta}\rightarrow
U_{\beta}$ is smooth and surjective.

Since density can be checked locally, fix $M=M_{\beta}$, $U=U_{\beta}$ and
$V=V_{\beta}$. Then, we have:

$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})\cap
U=\mathcal{A}_{M}\left(V\cap\pi_{*}^{-1}\left(\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}^{-1}(\mathcal{O}_{Y})\right)\right)$

and

$\operatorname{Nm}_{\pi}^{-1}(\mathcal{N})\cap J(X)\cap
U=\mathcal{A}_{M}\left(V\cap\pi_{*}^{-1}\left(\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}^{-1}(\mathcal{O}_{Y})\right)\cap\operatorname{{}^{\ell}Hilb}_{X}\right).$

Put
$\textbf{K}=\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}^{-1}(\mathcal{O}_{Y})$.
Then, the topological space underlying $\pi_{*}^{-1}(\textbf{K})$ contains the
topological union of fibers of the points in K, so that:

$\pi_{*}^{-1}(\textbf{K})\supseteq\bigcup_{E\in\textbf{K}}\pi_{*}^{-1}(E).$

By Proposition 3.2.3, $\pi_{*}^{-1}(E)\cap\operatorname{{}^{\ell}Hilb}_{X}$ is
a non-empty open and dense subscheme of $\pi_{*}^{-1}(E)$ for any closed point
$E\in\textbf{K}$. Hence,
$\pi_{*}^{-1}(\textbf{K})\cap\operatorname{{}^{\ell}Hilb}_{X}$ is non-empty,
open and dense in $\pi_{*}^{-1}(\textbf{K})$. Intersecting with $V$ and
composing with $\mathcal{A}_{M}$, we get the thesis. ∎

We finally come to the Prym stack of $X$ with respect to $Y$. Recall that by
definition:

$\displaystyle\Pr(X,Y)$
$\displaystyle=\operatorname{Nm}_{\pi}^{-1}(\mathcal{O}_{Y})\cap J(X),$
$\displaystyle\overline{\Pr}(X,Y)$
$\displaystyle=\operatorname{Nm}_{\pi}^{-1}(\mathcal{O}_{Y}).$

######  Corollary 3.3.3.

$\Pr(X,Y)$ is non-empty, open and dense in $\overline{\Pr}(X,Y)$.

###### Proof.

Set $\mathcal{N}=\mathcal{O}_{Y}$ and apply Proposition 3.3.2. ∎

##  Chapter 4 Spectral data for $G$-Higgs pairs

In this chapter we study how the spectral correspondence (see Chapter 1,
Section 1.4) specializes for $G$-Higgs pairs, where
$G=\mathrm{SL}(r,\mathbb{C})$, $\mathrm{PGL}(r,\mathbb{C})$,
$\mathrm{Sp}(2r,\mathbb{C})$, $\mathrm{GSp}(2r,\mathbb{C})$,
$\mathrm{PSp}(2r,\mathbb{C})$. Throughout this chapter, $C$ denotes a fixed
smooth curve over the field of complex numbers and $L$ a fixed line bundle on
$C$ with degree $\ell=\deg L$.

###  4.1. $\mathrm{SL}(r,\mathbb{C})$-Higgs pairs

The Special Linear Group $\mathrm{SL}(r,\mathbb{C})$ is defined as:

$\mathrm{SL}(r,\mathbb{C})=\\{M\in\mathrm{GL}(r,\mathbb{C}):\det M=1\\}.$

The Lie algebra associated to $\mathrm{SL}(r,\mathbb{C})$ is

$\mathfrak{sl}(r,\mathbb{C})=\left\\{X\in\mathfrak{gl}(r,\mathbb{C}):\operatorname{tr}X=0\right\\}.$

If $(P,\phi)$ is a $\mathrm{SL}(r,\mathbb{C})$-Higgs pair on $C$, the
associated vector bundle $E$ is endowed with a volume form $\lambda:\det
E\xrightarrow{{}_{\sim}}\mathcal{O}_{C}$ and the associated Higgs field $\Phi$
is a $L$-valued endomorphism with $\operatorname{tr}\Phi=0$; we say that
$(E,\Phi)$ is a Higgs pair with trace zero. Viceversa, let $(E,\Phi)$ be a
Higgs pair of rank $r$ such that $\operatorname{tr}\Phi=0$ and let
$\lambda:\det E\xrightarrow{{}_{\sim}}\mathcal{O}_{C}$ be a volume form on
$E$; then, the frame bundle $P=\operatorname{Fr}_{\mathrm{SL}}(E,\lambda)$ of
all ordered basis whose $\lambda$-volume equals 1 is a principal
$\mathrm{SL}(r,\mathbb{C})$-bundle, and the Higgs field $\Phi$ is the image of
a unique global section $\phi$ of $\operatorname{ad}P\otimes L$ with respect
to the morphism $\operatorname{ad}P\otimes
L\rightarrow\operatorname{End}E\otimes L$ induced by the canonical embedding
$\rho:\mathrm{SL}(r,\mathbb{C})\hookrightarrow\mathrm{GL}(r,\mathbb{C})$.

To sum up, the datum $(P,\phi)$ of a $\mathrm{SL}(r,\mathbb{C})$-Higgs pair on
$C$ corresponds univocally, via the associated bundle construction, to the
datum of $(E,\Phi,\lambda)$, where $(E,\Phi)$ is a Higgs pair of rank $r$ with
trace zero, and $\lambda$ is a trivialization of $\det E$. Note that the
isomorphism $\det E\simeq\mathcal{O}_{C}$ implies in particular that the Higgs
pairs associated to $\mathrm{SL}$-Higgs pairs have always degree equal to 0.

A basis for the invariant polynomials of $\mathfrak{sl}(r,\mathbb{C})$ is
given by $\\{p_{2},\dots,p_{r}\\}$ where
$p_{i}(P,\phi):=(-)^{i}\operatorname{tr}(\wedge^{i}\phi)$; in particular, if
$(E,\Phi)$ is the associated Higgs pair, then $p_{i}(P,\phi)=a_{i}(E,\phi)$.
Then, the $\mathrm{SL}$-Hitchin morphism of rank $r$ can be defined as:

$\displaystyle\mathcal{H}_{\mathrm{SL},r}:\mathcal{M}_{\mathrm{SL}}(r):=\mathcal{M}_{\mathrm{SL}}(r,0)$
$\displaystyle\longrightarrow\mathcal{A}_{\mathrm{SL}}(r)=\bigoplus_{i=2}^{r}H^{0}(C,L^{i})$
$\displaystyle(P,\phi)$
$\displaystyle\longmapsto(p_{2}(P,\phi),\dots,p_{r}(P,\phi))=$
$\displaystyle\hskip 20.00003pt=(a_{2}(E,\Phi),\dots,a_{r}(E,\Phi))$
$\displaystyle\hskip 25.00003pt\textrm{ for }(E,\Phi)\textrm{ associated to
}(P,\phi).$

######  Proposition 4.1.1.

(Spectral correspondence for $\mathrm{SL}(r,\mathbb{C})$) Let
${\underline{a}}\in\mathcal{A}_{\mathrm{SL}}(r)$ be any characteristic, let
$X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve, and
denote $B:=\det(\pi_{*}\mathcal{O}_{X})^{-1}$. The fiber
$\mathcal{H}_{\mathrm{SL},r}^{-1}({\underline{a}})$ of the
$\mathrm{SL}$-Hitchin morphism is isomorphic, via the spectral correspondence,
to the fiber $\operatorname{Nm}_{\pi}^{-1}(B)$ of the Norm map from
$\overline{\mathbb{J}}(X,d^{\prime})$ to $J(C,d^{\prime})$ induced by $\pi$
for $d^{\prime}=\frac{r(r-1)}{2}\ell$.

###### Proof.

The datum of a $\mathrm{SL}(r,\mathbb{C})$-Higgs pair $(P,\phi)$ with
characteristic ${\underline{a}}$ corresponds uniquely, via the associated
bundle construction, to the datum of $(E,\Phi,\lambda)$, where $(E,\Phi)$ is a
Higgs pair of rank $r$ and degree 0 with
$\mathcal{H}_{r,0}(E,\Phi)={\underline{a}}$ and $\lambda$ is a trivialization
of $\det E$.

The datum of the Higgs pair
$(E,\Phi)\in\mathcal{H}_{r,0}^{-1}({\underline{a}})$ corresponds uniquely, via
the spectral correspondence (Proposition 1.4.4), to the datum of a torsion-
free rank-1 sheaf $\mathcal{M}$ on $X$ of degree $d^{\prime}$. By Definition
2.5.1, $\operatorname{Nm}_{\pi}(\mathcal{M}):=\det(\pi_{*}\mathcal{M})\otimes
B$, so giving a isomorphism $\lambda$ between $\det\pi_{*}\mathcal{M}$ and
$\mathcal{O}_{C}$ is the same as giving a isomorphism $\epsilon$ between
$\operatorname{Nm}_{\pi}(\mathcal{M})$ and $B$.

To sum up, the datum of $(P,\phi)$ corresponds to the datum
$(\mathcal{M},\epsilon)$ of an element
$\mathcal{M}\in\overline{\mathbb{J}}(X,d^{\prime})$ and an isomorphism
$\epsilon:\operatorname{Nm}(\mathcal{M})\xrightarrow{{}_{\sim}}B$. This is an
element of the fiber product stack:

${\operatorname{Nm}^{-1}(B)}$${\overline{\mathbb{J}}(X,d^{\prime})}$${B}$${J(C,d^{\prime})}$${\lrcorner}$$\scriptstyle{\operatorname{Nm}_{\pi}}$

We conclude that
$\mathcal{H}_{\mathrm{SL},r}^{-1}({\underline{a}})\simeq\operatorname{Nm}^{-1}(B)$.
∎

######  Proposition 4.1.2.

Let ${\underline{a}}\in\mathcal{A}^{\textrm{reg}}_{\mathrm{SL}}(r)$ be a
characteristic such that the spectral curve $X$ is reduced. Then,

$\mathcal{H}_{\mathrm{SL},r}^{-1}(a)\simeq\overline{\Pr}(X,C).$

###### Proof.

Let $B=\det(\pi_{*}\mathcal{O}_{X})^{-1}\in J(C)$. Since $X$ is reduced with
locally planar singularities,
$\overline{\mathbb{J}}(X)=\operatorname{\textit{G}\mathbb{J}}(X)$ by Remark
1.2.9 and $\operatorname{Nm}_{\pi}^{-1}(B)$ contains at least a line bundle by
Proposition 3.3.2; denote it with $N$. Tensoring by $N$ in
$\overline{\mathbb{J}}(X)$ induces then by Proposition 2.5.5 a well-defined
isomorphism:

$\displaystyle\overline{\Pr}(X,C)\longrightarrow\operatorname{Nm}_{\pi}^{-1}(B)$
$\displaystyle\mathcal{L}\longmapsto\mathcal{L}\otimes N.$

∎

###  4.2. $\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs

We come now to the case of $\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs. Recall
first that the Projective Linear Group $\mathrm{PGL}(r,\mathbb{C})$ is defined
by the exact sequence:

(4.1) $\displaystyle
0\rightarrow\mathbb{C}^{*}\xrightarrow{\lambda\mapsto\lambda
I_{r}}\mathrm{GL}(r,\mathbb{C})\rightarrow\mathrm{PGL}(r,\mathbb{C})\rightarrow
0.$

From the exponential sequence
$0\rightarrow\mathbb{Z}\rightarrow\mathcal{O}_{C}\rightarrow\mathcal{O}_{C}^{*}\rightarrow
0$ we deduce that $H^{2}(C,\mathcal{O}_{C}^{*})=0$; hence, the sequence 4.1
applied to the structure sheaves:

$0\rightarrow\mathcal{O}_{C}^{*}\rightarrow\mathrm{GL}(r,\mathcal{O}_{C})\rightarrow\mathrm{PGL}(r,\mathcal{O}_{C})\rightarrow
0$

yields the cohomology exact sequence

(4.2) $\displaystyle H^{1}(C,\mathcal{O}_{C}^{*})\rightarrow
H^{1}(C,\mathrm{GL}(r,\mathcal{O}_{C}))\ext@arrow
0579\arrowfill@\relbar\relbar\twoheadrightarrow{}{q}H^{1}(C,\mathrm{PGL}(r,\mathcal{O}_{C}))\rightarrow
0.$

The sequence 4.2, read in terms of cocycles, means that
$\mathrm{PGL}(r,\mathbb{C})$-principal bundles are in one-to-one
correspondence with equivalence classes of
$\mathrm{GL}(r,\mathbb{C})$-principal bundles, with respect to the action on
associated bundles given by tensor product of line bundles on $C$. If $P$ is a
$\mathrm{PGL}(r,\mathbb{C})$-principal bundle and $\tilde{P}$ is a
$\mathrm{GL}(r,\mathbb{C})$-principal bundle such that $q(\tilde{P})=P$, we
say that $\tilde{P}$ is a lifting of $P$ to a
$\mathrm{GL}(r,\mathbb{C})$-principal bundle.

Moreover, the Lie algebra $\mathfrak{pgl}(r,\mathbb{C})$ associated to
$\mathrm{PGL}(r,\mathbb{C})$ is equal to the Lie algebra
$\mathfrak{sl}(r,\mathbb{C})$ of $\mathrm{SL}(r,\mathbb{C})$. If $P$ is any
$\mathrm{PGL}(r,\mathbb{C})$-principal bundle and $\tilde{P}$ is a lifting of
$P$ to a $\mathrm{GL}(r,\mathbb{C})$-principal bundle, then a section $\phi$
of $H^{0}(C,\operatorname{ad}P\otimes L)$ determines uniquely a section
$\tilde{\phi}$ of $H^{0}(C,\operatorname{ad}\tilde{P}\otimes L)$ with trace
equal to $0$, and viceversa. We say that $(\tilde{P},\tilde{\phi})$ is a
lifting of $(P,\phi)$ to a $\mathrm{GL}$-Higgs pair and with a slight abuse of
notation we write $q(\tilde{P},\tilde{\phi})=(P,\phi)$.

Tu sum up, any $\mathrm{PGL}(r,\mathbb{C})$-Higgs pair $(P,\phi)$ has a
lifting $(\tilde{P},\tilde{\phi})$ to a to a $\mathrm{GL}(r,\mathbb{C})$-Higgs
pair, corresponding to a Higgs pair $(E,\Phi)$ with trace zero via the
associated bundle construction. Then, the datum of $(P,\phi)$ corresponds
uniquely to the datum of the equivalence class $[(E,\Phi)]$ of Higgs pairs on
$C$ with trace zero, under the equivalence relation $\sim_{J(C)}$ defined by:

$(E,\Phi)\sim_{J(C)}(E\otimes N,\Phi\otimes 1_{N})\hskip 20.00003pt\textrm{for
any }N\in J(C).$

Let $\mathcal{M}(r)=\coprod_{d\in\mathbb{Z}}\mathcal{M}(r,d)$ be the moduli
stack of Higgs pairs of rank $r$ in any degree and denote with
$\mathcal{M}^{\operatorname{tr}=0}(r)$ the closed substack of $\mathcal{M}(r)$
given by Higgs pairs of rank $r$ with trace zero. Then, $[(E,\Phi)]$ is the
orbit of $(E,\Phi)$ under the action of $J(C)$ on
$\mathcal{M}^{\operatorname{tr}=0}(r)$ defined by:

$\displaystyle\mathcal{M}^{\operatorname{tr}=0}(r)\times J(C)$
$\displaystyle\longrightarrow\mathcal{M}^{\operatorname{tr}=0}(r)$
$\displaystyle((E,\Phi),N)$ $\displaystyle\longmapsto(E\otimes N,\Phi\otimes
1_{N}).$

The degree of Higgs pairs associated to a $\mathrm{PGL}$-Higgs pair is defined
only modulo $r$. Indeed, if $N$ is any line bundle on $C$, then

$\displaystyle\deg(E\otimes N)$ $\displaystyle=\deg(\det(E\otimes N))=$
$\displaystyle=\deg(\det(E)\otimes N^{r})=\deg(E)+r\deg(N)$

so the degree of $J(C)$-equivalent Higgs pairs may differ by multiples of $r$.
Hence, we can give the following definition.

######  Definition 4.2.1.

Let $(P,\phi)$ be a $\mathrm{PGL}$-Higgs pair and let $(E,\Phi)$ be a Higgs
pair with trace zero and degree $d\in\mathbb{Z}$ corresponding to a lifting
$(\tilde{P},\tilde{\phi})$ of $(P,\phi)$ to a $\mathrm{GL}$-Higgs pair. The
degree of $(P,\phi)$ is the congruence class
$\overline{d}\in\mathbb{Z}/r\mathbb{Z}$.

######  Remark 4.2.2.

Obviously, the same definition can be given just for $\mathrm{PGL}$-principal
bundles. Let $P$ be a $\mathrm{PGL}$-principal bundle ald let $E$ be the
vector bundle associated to any lifting of $P$ to a $\mathrm{GL}$-principal
bundle. The degree of a $\mathrm{PGL}$-principal bundles $P$ is the congruence
class $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$ of the degree $d=\deg E$.

As a matter of fact, the first homotopy group
$\pi_{1}(\mathrm{PGL}(r,\mathbb{C}))$ of $\mathrm{PGL}(r,\mathbb{C})$ is
isomorphic to $\mathbb{Z}/r\mathbb{Z}$ and the degree of $(P,\phi)$
characterizes uniquely the topological type of $P$.

Up to the action on $(E,\Phi)$ of a line bundle of degree $1$ on $C$, it is
straightforward to see that the $J(C)$-orbit $[(E,\Phi)]$ corresponding to a
$\mathrm{PGL}$-Higgs pair with degree $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$
is always the orbit of a Higgs pair of degree $d$ with
$d\in\\{0,\dots,r-1\\}$. If we restrict the correspondence to Higgs pairs with
fixed degree $d\in\\{0,\dots,r-1\\}$, then a $\mathrm{PGL}$-Higgs pair of
degree $\overline{d}$ is identified uniquely with the orbit of a Higgs pair of
trace zero and degree $d$ with respect to the action of line bundles of degree
$0$ on $C$. We have then the following proposition.

######  Proposition 4.2.3.

Let $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$ with $d\in\\{0,\dots,r-1\\}$.
Denote with $\mathcal{M}^{\operatorname{tr}=0}(r,d)$ the closed substack of
$\mathcal{M}(r,d)$ given by Higgs pairs with trace zero. The moduli stack
$\mathcal{M}_{\mathrm{PGL}}(r,\overline{d})$ of $\mathrm{PGL}$-Higgs pairs of
rank $r$ and degree $\overline{d}$ is isomorphic to the quotient stack

$\displaystyle q:\mathcal{M}^{\operatorname{tr}=0}(r,d)$
$\displaystyle\longrightarrow\mathcal{M}^{\operatorname{tr}=0}(r,d)/J^{0}(C)\xrightarrow{{}_{\sim}}\mathcal{M}_{\mathrm{PGL}}(r,\overline{d})$
$\displaystyle(E,\Phi)$
$\displaystyle\longmapsto[(E,\Phi)]\longmapsto(P,\phi):=q(\tilde{P},\tilde{\phi})$

where $(E,\Phi)$ is associated to the $\mathrm{GL}$-principal bundle
$(\tilde{P},\tilde{\phi})$.

A basis for the invariant polynomials of
$\mathfrak{pgl}(r,\mathbb{C})=\mathfrak{sl}(r,\mathbb{C})$ is given by
$p_{2},\dots,p_{r}$ where
$p_{i}(P,\phi):=(-)^{i}\operatorname{tr}(\wedge^{i}\phi)$; moreover, if
$[(E,\Phi)]$ is the associated $J^{0}(C)$-equivalence class of Higgs pairs
with zero trace, then $p_{i}(P,\phi)=a_{i}(E,\Phi)$ for any $(E,\Phi)$
associated to a lifting of $(P,\phi)$. Hence, we have the following
$\mathrm{PGL}$-Hitchin morphism of rank $r$ and degree $\overline{d}$:

$\displaystyle\mathcal{H}_{\mathrm{PGL},r,\overline{d}}:\mathcal{M}_{\mathrm{PGL}}(r,\overline{d})$
$\displaystyle\longrightarrow\mathcal{A}_{\mathrm{PGL}}(r)=\bigoplus_{i=2}^{r}H^{0}(C,L^{i})$
$\displaystyle(P,\phi)$
$\displaystyle\longmapsto(p_{2}(P,\phi),\dots,p_{r}(P,\phi)).$

The $\mathrm{PGL}$-Hitchin morphism fits the following commutative diagram:

${\mathcal{M}(r,d)}$${\mathcal{M}^{\operatorname{tr}=0}(r,d)}$${\mathcal{M}_{\mathrm{PGL}}(r,\overline{d})}$${\mathcal{A}(r)}$${\bigoplus_{i=2}^{r}H^{0}(C,L^{i})}$${=}$${\mathcal{A}_{\mathrm{PGL}}(r)}$$\scriptstyle{\mathcal{H}_{r,d}}$$\scriptstyle{\mathcal{H}_{r,d}^{\operatorname{tr}=0}}$$\scriptstyle{\mathcal{H}_{\mathrm{PGL},r,\overline{d}}}$

with $d\in\\{0,\dots,r-1\\}$ and $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$.

We are now ready to state the spectral correspondence for
$\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs.

######  Proposition 4.2.4.

(Spectral correspondence for $\mathrm{PGL}(r,\mathbb{C})$) Let
${\underline{a}}\in\mathcal{A}_{\mathrm{PGL}}(r)$ be any characteristic and
let $X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve.
Let $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$ with $d\in\\{0,\dots,r-1\\}$ be
any degree. The fiber
$\mathcal{H}_{\mathrm{PGL},r,\overline{d}}^{-1}({\underline{a}})$ of the
$\mathrm{PGL}$-Hitchin morphism is isomorphic, via the spectral
correspondence, to the quotient moduli space

$\overline{\mathbb{J}}(X,d^{\prime})/\pi^{*}J^{0}(C)$

of torsion-free sheaves of rank 1 and degree $d^{\prime}$ up to the action of
line bundles of degree $0$ on $C$ by tensor product, with
$d^{\prime}=d+\frac{r(r-1)}{2}\ell$.

###### Proof.

The datum of a $\mathrm{PGL}(r,\mathbb{C})$-Higgs pair $(P,\phi)$ with degree
$\overline{d}$ and characteristic ${\underline{a}}$ corresponds uniquely to
the datum of a $J^{0}(C)$-equivalence class $[(E,\Phi)]$ of Higgs pairs of
rank $r$ and degree $d$ with characteristic ${\underline{a}}$, where
$(E,\Phi)$ is the Higgs pair associated, via the vector bundle construction,
to a lifting of $(P,\phi)$ to a $\mathrm{GL}$-Higgs pair. By the spectral
correspondence, the datum of $(E,\Phi)$ corresponds uniquely to the datum of a
pure sheaf $\mathcal{M}\in\overline{\mathbb{J}}(X,d^{\prime})$ such that
$\pi_{*}\mathcal{M}=E$ and $d^{\prime}=d+\frac{r(r-1)}{2}\ell$. Let $N\in
J^{0}(C)$ be any line bundle of degree 0 on $C$. By the projection formula,

$\pi_{*}(\mathcal{M}\otimes\pi^{*}N)\simeq\pi_{*}\mathcal{M}\otimes N,$

and the following square commutes:

${\pi_{*}(\mathcal{M}\otimes\pi^{*}N)}$${\pi_{*}\mathcal{M}\otimes
N}$${\pi_{*}(\mathcal{M}\otimes\pi^{*}N)\otimes
L}$${(\pi^{*}\mathcal{M}\otimes N)\otimes L.}$∼$\scriptstyle{\pi_{*}(\cdot
x)}$$\scriptstyle{\Phi_{N}}$∼

Hence, the spectral correspondence is equivariant with respect to the action
of $\pi^{*}J^{0}(C)$ by tensor product on
$\overline{\mathbb{J}}(X,d^{\prime})$ and the action of $J^{0}(C)$ on
$\mathcal{M}^{\operatorname{tr}=0}(r,d)$. Then, passing to the quotient on
both sides, the datum of the $J^{0}(C)$-orbit $[(E,\Phi)]$ corresponds
uniquely to the datum of a $\pi^{*}J^{0}(C)$-orbit
$[\mathcal{M}]\in\overline{\mathbb{J}}(X,d^{\prime})/\pi^{*}J^{0}(C)$. ∎

We study now the special case of $\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs of
degree $0$. Since $\mathbb{C}$ is algebraically closed, the Projective Linear
group is equal to the Projective Special Linear group
$\mathrm{PSL}(r,\mathbb{C})$ defined by the exact sequence:

(4.3) $\displaystyle 0\rightarrow\mu_{r}\xrightarrow{\lambda\mapsto\lambda
I_{r}}\mathrm{SL}(r,\mathbb{C})\rightarrow\mathrm{PSL}(r,\mathbb{C})\rightarrow
0;$

here we denote with $\mu_{r}$ the group of $r$-th roots of unity, as defined
by the exact sequence:

$0\rightarrow\mu_{r}\rightarrow\mathbb{C}^{*}\xrightarrow{\lambda\mapsto\lambda^{r}}\mathbb{C}^{*}\rightarrow
0.$

In particular, $H^{2}(C,\mu_{r})=\mathbb{Z}/r\mathbb{Z}$, hence the exact
sequence 4.3, applied to structure sheaves, induces the cohomology exact
sequence:

(4.4) $\displaystyle H^{1}(C,\mu_{r})\rightarrow
H^{1}(C,\mathrm{SL}(r,\mathcal{O}_{C}))\ext@arrow
0579\arrowfill@\relbar\relbar\twoheadrightarrow{}{q}H^{1}(C,\mathrm{PSL}(r,\mathcal{O}_{C}))\xrightarrow{\deg}\mathbb{Z}/r\mathbb{Z}\rightarrow
0.$

The sequence 4.4, read in terms of cocycles, means that a
$\mathrm{PGL}(r,\mathbb{C})$-principal bundle $P$ can be lifted to a principal
$\mathrm{SL}(r,\mathbb{C})$-principal bundle $P_{0}$ if and only if $P$ has
degree $0\in\mathbb{Z}/r\mathbb{Z}$; any other lifting $P_{0}^{\prime}$ of $P$
differs from $P_{0}$ by the action on associated bundles of a $r$-th torsion
line bundle by tensor product. Moreover, a Higgs field $\phi$ on $P$
determines uniquely a Higgs field $\phi_{0}$ on $P_{0}$, and viceversa.

To sum up, any $\mathrm{PGL}(r,\mathbb{C})$-principal bundle $(P,\phi)$ of
degree zero has a lifting $(P_{0},\phi_{0})$ to a
$\mathrm{SL}(r,\mathbb{C})$-principal bundle, corresponding to the datum
$(E,\Phi,\lambda)$ of a Higgs pair $(E,\Phi)$ with trace zero and a volume
form $\lambda:\det E\xrightarrow{{}_{\sim}}\mathcal{O}_{C}$ via the associated
bundle construction. Then, the datum of $(P,\phi)$ corresponds uniquely to the
datum of the equivalence class $[(E,\phi,\lambda)]$ of Higgs pairs with trace
zero, under the equivalence relation $\sim_{J^{0}(C)[r]}$ defined by:

$(E,\Phi,\lambda)\sim_{J^{0}(C)[r]}(E\otimes N,\Phi\otimes
1_{N},\lambda_{N,\epsilon})\hskip 20.00003pt\textrm{for any }(N,\epsilon)\in
J^{0}(C)[r].$

Here, $J^{0}(C)[r]$ is the group stack of $r$-torsion line bundles $(N,\eta)$
on $C$, where $\eta$ is the isomorphism
$N^{r}\xrightarrow{{}_{\sim}}\mathcal{O}_{C}$, and $\lambda_{N,\eta}$ denote
the isomorphism:

$\lambda_{N,\eta}:\det(E\otimes N)\xrightarrow{{}_{\sim}}\det(E)\otimes
N^{r}\xrightarrow[\lambda\otimes\eta]{\sim}\mathcal{O}_{C}.$

We can prove now the following proposition.

######  Proposition 4.2.5.

(Spectral correspondence for $\mathrm{PGL}(r,\mathbb{C})$ of degree 0) Let
${\underline{a}}\in\mathcal{A}_{\mathrm{PGL}}(r)$ be any characteristic, let
$X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve and
denote $B:=\det(\pi_{*}\mathcal{O}_{X})^{-1}$. Let
$d^{\prime}=\frac{r(r-1)}{2}\ell$ and let $\operatorname{Nm}_{\pi}$ be the
Norm map induced by $\pi$ on $\overline{\mathbb{J}}(X,d^{\prime})$. Let
$J^{0}(C)[r]$ be the group stack of line bundles with $r$-th torsion on $C$,
acting on $\operatorname{Nm}_{\pi}^{-1}(B)$ as follows:

$\displaystyle\operatorname{Nm}_{\pi}^{-1}(B)\times\pi^{*}J^{0}(C)[r]$
$\displaystyle\longrightarrow\operatorname{Nm}_{\pi}^{-1}(B)$
$\displaystyle((\mathcal{M},\epsilon),(\pi^{*}N,\pi^{*}\eta))$
$\displaystyle\longmapsto(\mathcal{M}\otimes\pi^{*}N,\epsilon_{N,\eta})$

where $\epsilon_{N,\eta}$ is equal to the following composition:

$\displaystyle\epsilon_{N,\eta}:$
$\displaystyle\operatorname{Nm}(\mathcal{M}\otimes\pi^{*}N)\xrightarrow{{}_{\sim}}\operatorname{Nm}(\mathcal{M})\otimes\operatorname{Nm}(\pi^{*}N)\xrightarrow{{}_{\sim}}\operatorname{Nm}(\mathcal{M})\otimes
N^{r}\xrightarrow{\epsilon\otimes\eta}B.$

Then, the fiber
$\mathcal{H}_{\mathrm{PGL},r,\overline{0}}^{-1}({\underline{a}})$ of the
$\mathrm{PGL}$-Hitchin morphism is isomorphic, via the spectral
correspondence, to the quotient moduli space

$\operatorname{Nm}_{\pi}^{-1}(B)/\pi^{*}(J^{0}C)[r].$

###### Proof.

The datum of a $\mathrm{PGL}(r,\mathbb{C})$-Higgs pair $(P,\phi)$ with degree
$0$ and characteristic ${\underline{a}}$ corresponds uniquely to the datum of
a $J^{0}(C)[r]$-equivalence class $[(E,\Phi,\lambda)]$ of Higgs pairs of rank
$r$ and degree $0$ with characteristic ${\underline{a}}$, where
$(E,\Phi,\lambda)$ is the datum of a Higgs pair with a volume form $\lambda$
associated, via the vector bundle construction, to a lifting of $(P,\phi)$ to
a $\mathrm{SL}$-Higgs pair.

By Proposition 4.1.1, the datum of $(E,\Phi,\lambda)$ corresponds to the datum
$(M,\epsilon)$ of a torsion-free rank-1 sheaf $\mathcal{M}$ of degree
$d^{\prime}$ on $X$ and an isomorphism
$\epsilon:\operatorname{Nm}(\mathcal{M})\xrightarrow{{}_{\sim}}B$, i.e. an
element of $\operatorname{Nm}^{-1}(B)$. If $(N,\eta)$ is a line bundle on $C$
with $r$-th torsion, the datum of $(E\otimes N,\Phi_{N},\lambda_{N,\eta})$
corresponds by projection formula to the datum of
$(M\otimes\pi^{*}N,\epsilon_{N,\eta)}$. Then, the datum of the
$J^{0}(C)[r]$-equivalence class $[(E,\Phi,\lambda)]$ corresponds uniquely to
the datum of the $\pi^{*}J^{0}(C)[r]$-equivalence class
$[(\mathcal{M},\epsilon)]$.

∎

###  4.3. $\mathrm{Sp}(2r,\mathbb{C})$-Higgs pairs

The symplectic group $\mathrm{Sp}(2r,\mathbb{C})$ is defined as:

$\mathrm{Sp}(2r,\mathbb{C})=\\{M\in\mathrm{GL}(2r,\mathbb{C}):M\Omega
M^{t}=\Omega\\}$

where $\Omega=\left(\begin{matrix}0&I_{r}\\\ -I_{r}&0\end{matrix}\right)$.

The Lie algebra associated to $\mathrm{Sp}(2r,\mathbb{C})$ is

$\displaystyle\mathfrak{sp}(2r,\mathbb{C})$
$\displaystyle=\left\\{X\in\mathfrak{gl}(2r,\mathbb{C}):X\Omega+\Omega
X^{t}=0\right\\}.$

If $(P,\phi)$ is a $\mathrm{Sp}(2r,\mathbb{C})$-Higgs pair on $C$, the
associated vector bundle $E$ is endowed with a non-degenerate symplectic form
$\omega:E\otimes E\rightarrow\mathcal{O}_{C}$ and the associated Higgs field
$\Phi$ is satisfies the condition $\omega(\Phi v,w)=-\omega(v,\Phi w)$ for all
sections $v,w$ of $E$. The existence of a non-degenerate symplectic form
implies that $E$ has trivial determinant and, in particular, that $E$ has
degree $0$. Viceversa, let $(E,\Phi)$ be a Higgs pair of rank $2r$ and let
$\omega$ be a non-degenerate symplectic form on $E$ satisfying the condition
$\omega(\Phi v,w)=-\omega(v,\Phi w)$; then, the frame bundle
$P=\operatorname{Fr}_{\mathrm{Sp}}(E,\omega)$ of all ordered symplectic basis
of $(E,\omega)$ is a principal $\mathrm{Sp}(2r,\mathbb{C})$-bundle, and the
Higgs field $\Phi$ is the image of a unique global section $\phi$ of
$\operatorname{ad}P\otimes L$ with respect to the morphism
$\operatorname{ad}P\otimes L\rightarrow\operatorname{End}E\otimes L$ induced
by the canonical embedding
$\rho:\mathrm{Sp}(2r,\mathbb{C})\hookrightarrow\mathrm{GL}(2r,\mathbb{C})$.

To sum up, the datum of a $\mathrm{Sp}(2r,\mathbb{C})$-Higgs pair $(P,\phi)$
on $C$ corresponds univocally to the datum of $(E,\Phi,\omega)$ where
$(E,\Phi)$ is a Higgs pair of rank $2r$ and degree $0$ and $\omega:E\otimes
E\rightarrow\mathcal{O}_{C}$ is a non-degenerate symplectic form on $E$
satisfying the condition:

$\omega(\Phi v,w)=-\omega(v,\Phi w).$

A basis for the invariant polynomials of $\mathfrak{sp}(2r,\mathbb{C})$ is
given by $\\{p_{2i}\\}_{i=1,\dots,r}$ where
$p_{2i}(P,\phi):=\operatorname{tr}(\wedge^{2i}\phi)$; if $(E,\Phi)$ is the
associated Higgs pair, then $p_{2i}(P,\phi)=a_{2i}(E,\phi)$. The corresponding
$\mathrm{Sp}$-Hitchin morphism takes the form:

$\displaystyle\mathcal{H}_{\mathrm{Sp},2r}:\mathcal{M}_{\mathrm{Sp}}(2r):=\mathcal{M}_{\mathrm{Sp}}(2r,0)$
$\displaystyle\longrightarrow\mathcal{A}_{\mathrm{Sp}}(2r)=\bigoplus_{l=1}^{r}H^{0}(C,L^{2l})$
$\displaystyle(P,\phi)$
$\displaystyle\longmapsto(p_{2}(P,\phi),p_{4}(P,\phi),\dots,p_{2r}(P,\phi)).$

For any characteristic ${\underline{a}}\in\mathcal{A}_{\mathrm{Sp}}(2r)$, the
spectral curve $\pi:X_{\underline{a}}\rightarrow C$ is defined in the total
space $p:\mathbb{P}(\mathcal{O}_{C}\oplus L^{-1})\rightarrow C$ by the
equation

$x^{2r}y+a_{2}x^{2r-2}y^{2}+\dots+a_{2r-2}x^{2}y^{2r-2}+a_{2r}y^{2r}=0.$

Hence, the curve $X=X_{\underline{a}}$ has an involution $\sigma$ defined by
$\sigma(x)=-x$ and a quotient curve:

${X}$${X/\sigma=Y}$${C}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\scriptstyle{\overline{\pi}}$

The involution $\sigma$ induces by pullback an involution on the compactified
Jacobian of torsion-free rank-1 sheaves with any degree $d^{\prime}$:

$\displaystyle\sigma^{*}:\overline{\mathbb{J}}(X,d^{\prime})$
$\displaystyle\longrightarrow\overline{\mathbb{J}}(X,d^{\prime})$
$\displaystyle\mathcal{L}$ $\displaystyle\longmapsto\sigma^{*}\mathcal{L}.$

######  Proposition 4.3.1.

(Spectral correspondence for $\mathrm{Sp}(2r,\mathbb{C})$) Let
${\underline{a}}\in\mathcal{A}_{\mathrm{Sp}}(2r)$ be any characteristic and
let $X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve
with involution $\sigma:X\rightarrow X$. The fiber
$\mathcal{H}_{\mathrm{Sp},2r}^{-1}({\underline{a}})$ of the
$\mathrm{Sp}$-Hitchin morphism is isomorphic, via the spectral correspondence,
to the equalizer $\mathcal{E}_{\underline{a}}$ of the two maps

$\displaystyle\texttt{\char
95\relax}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}:=\mathcal{H}om_{\mathcal{O}_{X}}(\texttt{\char
95\relax},\mathcal{O}_{X}):\overline{\mathbb{J}}(X,d^{\prime})$
$\displaystyle\rightarrow\overline{\mathbb{J}}(X,-d^{\prime})$
$\displaystyle\sigma^{*}\texttt{\char
95\relax}\otimes\pi^{*}L^{1-2r}:\overline{\mathbb{J}}(X,d^{\prime})$
$\displaystyle\rightarrow\overline{\mathbb{J}}(X,-d^{\prime}),$

where $d^{\prime}=r(2r-1)\ell$.

###### Proof.

The dual of a torsion-free sheaf on a Gorenstein curve is torsion-free sheaf
with the same rank, hence the map
$\mathcal{H}om_{\mathcal{O}_{X}}(\texttt{\char 95\relax},\mathcal{O}_{X})$ is
well-defined. Let $(P,\phi)$ be any $\mathrm{Sp}$-Higgs pair with
characteristic ${\underline{a}}$ and let $(E,\Phi,\omega)$ be the associated
datum of a Higgs pair of rank $2r$ and degree 0 with characteristic
${\underline{a}}$ and symplectic form $\omega$. The torsion-free sheaf
$\mathcal{M}$ on $X$ associated to $(E,\Phi)$ by the spectral correspondence
fits into the exact sequence:

$0\rightarrow\mathcal{M}\otimes\pi^{*}L^{1-2r}\rightarrow\pi^{*}E\xrightarrow{\pi^{*}\Phi-x}\pi^{*}(E\otimes
L)\rightarrow\mathcal{M}\otimes\pi^{*}L\rightarrow 0.$

Taking the dualized sequence and tensoring by $\pi^{*}L$ gives a left exact
sequence:

$0\rightarrow\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\rightarrow\pi^{*}E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\xrightarrow{\pi^{*}\Phi^{t}-x}\pi^{*}E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes\pi^{*}L.$

On the other hand, applying $\sigma^{*}$ to the first exact sequence gives:

$0\rightarrow\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}\rightarrow\pi^{*}E\xrightarrow{\pi^{*}\Phi+x}\pi^{*}(E\otimes
L)\rightarrow\sigma^{*}\mathcal{M}\otimes\pi^{*}L\rightarrow 0.$

Now, the symplectic form $\omega$ induce an isomorphism $\omega_{E}:E\simeq
E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ and
hence a commutative diagram with vertical isomorphisms:

${0}$${\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}}$${\pi^{*}E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}}$${\pi^{*}E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes\pi^{*}L}$${0}$${\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}}$${\pi^{*}E}$${\pi^{*}(E\otimes
L)}$$\scriptstyle{\pi^{*}\Phi^{t}-x}$$\scriptstyle{\omega_{E}}$$\scriptstyle{\sim}$$\scriptstyle{-\pi^{*}\Phi-x}$$\scriptstyle{\omega_{E}}$$\scriptstyle{\sim}$

Hence, we conclude that the sheaf $\mathcal{M}$ comes with a sheaf isomorphism

$\lambda:\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\xrightarrow{{}_{\sim}}\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}.$

Viceversa, suppose that $\mathcal{M}$ is a torsion-free sheaf of rank 1 and
degree $d^{\prime}$ on $X$ with an isomorphism
$\lambda:\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\xrightarrow{{}_{\sim}}\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}$.
Let $(E,\Phi)$ be the $\mathrm{GL}$-Higgs pair with characteristic
${\underline{a}}$ corresponding to $\mathcal{M}$. By [HS93], the Relative
duality formula holds for $X/C$ with dualizing sheaf equal to the sheaf of
relative differentials $\omega_{X/C}$. By Lemma 1.4.1 and Riemann-Hurwitz we
obtain that $\omega_{X/C}\simeq\pi^{*}L^{2r-1}$. Applying Relative duality in
dimension 0 and the projection formula, we obtain an isomorphism:

$\displaystyle
E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}$
$\displaystyle=\pi_{*}(\mathcal{M})^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\simeq\pi_{*}(\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes\pi^{*}L^{2r-1})\simeq$
$\displaystyle\simeq\pi_{*}(\sigma^{*}(\mathcal{M}))=\pi_{*}(\sigma_{*}\sigma^{*}\mathcal{M})\simeq$
$\displaystyle\simeq\pi_{*}(\sigma_{*}\mathcal{O}_{X}\otimes\mathcal{M})=\pi_{*}(\mathcal{O}_{X}\otimes\mathcal{M})\simeq$
$\displaystyle\simeq\pi_{*}(\mathcal{M})=E;$

that induces the symplectic structure $\omega:E\otimes
E\rightarrow\mathcal{O}_{C}$.

To sum up, the datum of $(P,\phi)$ in the fiber
$\mathcal{H}_{\mathrm{Sp},2r}^{-1}({\underline{a}})$ corresponds uniquely to
the datum $(\mathcal{M},\lambda)$ of a torsion-free rank-1 sheaf
$\mathcal{M}\in\overline{\mathbb{J}}(X,d^{\prime})$ and an isomorphism
$\lambda:\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\xrightarrow{{}_{\sim}}\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}$,
i.e. an element of the equalizer stack $\mathcal{E}_{\underline{a}}$. ∎

###  4.4. $\mathrm{GSp}(2r,\mathbb{C})$-Higgs pairs

The General Symplectic group $\mathrm{GSp}(2r,\mathbb{C})$ is defined as:

$\mathrm{GSp}(2r,\mathbb{C})=\\{M\in\mathrm{GL}(2r,\mathbb{C}):M\Omega
M^{t}=\lambda\Omega\textrm{ for some }\lambda\in\mathbb{C}^{*}\\}$

where $\Omega=\left(\begin{matrix}0&I_{r}\\\ -I_{r}&0\end{matrix}\right)$.

The Lie algebra associated to $\mathrm{GSp}(2r,\mathbb{C})$ is

$\displaystyle\mathfrak{gsp}(2r,\mathbb{C})$
$\displaystyle=\left\\{A\in\mathfrak{gl}(2r,\mathbb{C}):A\Omega+\Omega
A^{t}=\frac{\operatorname{tr}A}{r}\Omega\right\\}\simeq$
$\displaystyle\simeq\mathfrak{sp}(2r,\mathbb{C})\oplus\mathbb{C}$

where the isomorphism is given by the decomposition
$A=X+\frac{\operatorname{tr}A}{2r}I_{2r}$, with
$X\in\mathfrak{sp}(2r,\mathbb{C})$ and $\operatorname{tr}(A)\in\mathbb{C}$.

If $(P,\phi)$ is a $\mathrm{GSp}(2r,\mathbb{C})$-Higgs pair on $C$, the
associated vector bundle $E$ is endowed with a non-degenerate symplectic form
$\omega:E\otimes E\rightarrow M$ with values in a line bundle $M$ on $C$ and
the associated Higgs field $\Phi$ satisfies the condition

$\omega(\Phi v,w)+\omega(v,\Phi w)=\frac{\operatorname{tr}\Phi}{r}\omega(v,w)$

for all sections $v,w$ of $E$. The existence of a non-degenerate symplectic
$M$-valued form implies that $E$ has determinant isomorphic to $M^{r}$ and, in
particular, that $E$ has degree equal to $r\deg M$. Viceversa, let $(E,\Phi)$
be a Higgs pair of rank $2r$ and let $\omega$ be a non-degenerate symplectic
$M$-valued form on $E$ satisfying the condition

$\omega(\Phi v,w)+\omega(v,\Phi
w)=\frac{\operatorname{tr}\Phi}{r}\omega(v,w);$

then, the frame bundle $P=\operatorname{Fr}_{\mathrm{GSp}}(E,\omega)$ of all
ordered symplectic basis of $(E,\omega)$ is a principal
$\mathrm{GSp}(2r,\mathbb{C})$-bundle, and the Higgs field $\Phi$ is the image
of a unique global section $\phi$ of $\operatorname{ad}P\otimes L$ with
respect to the morphism $\operatorname{ad}P\otimes
L\rightarrow\operatorname{End}E\otimes L$ induced by the canonical embedding
$\rho:\mathrm{Sp}(2r,\mathbb{C})\hookrightarrow\mathrm{GL}(2r,\mathbb{C})$.

Tu sum up, the datum of a $\mathrm{GSp}(2r,\mathbb{C})$-Higgs pair $(P,\phi)$
corresponds uniquely, via the associated bundle construction, to the datum
$(E,\Phi,M,\omega)$ of a Higgs pair $(E,\Phi)$ of rank $2r$ and degree $rd$
and a non-degenerate symplectic form $\omega:E\otimes E\rightarrow M$ with
values in a line bundle $M$ of degree $d$, and $\Phi\in
H^{0}(C,\operatorname{End}(E)\otimes L)$ satisfies

(4.5) $\displaystyle\omega(\Phi v,w)+\omega(v,\Phi
w)=\frac{\operatorname{tr}\Phi}{r}\omega(v,w).$

Since $\mathfrak{gsp}(2r,\mathbb{C})\subset\mathfrak{gl}(2r,\mathbb{C})$, the
$\mathrm{GSp}$-Hitchin morphism can be defined with the help of the
$\mathrm{GL}$-Hitchin morphism, as follows:

$\displaystyle\mathcal{H}_{\mathrm{GSp},2r,rd}:\mathcal{M}_{\mathrm{GSp}}(2r,rd)$
$\displaystyle\longrightarrow\mathcal{A}_{\mathrm{GSp}}(2r)\subseteq\bigoplus_{i=1}^{2r}H^{0}(C,L^{i})$
$\displaystyle(P,\phi)$
$\displaystyle\longmapsto(p_{1}(P,\phi),\dots,p_{2r}(P,\phi)).$

where $p_{i}(P,\phi)=(-)^{i}\operatorname{tr}(\wedge^{i}\phi)$ and
$\mathcal{A}_{\mathrm{GSp}}(2r)$ is defined as the locus of characteristics in
$\mathcal{A}_{\mathrm{GL}}(2r)$ resulting as characteristics of
$\mathrm{GSp}(2r,\mathbb{C})$-Higgs pairs.

In order to study the space of characteristics
$\mathcal{A}_{\mathrm{GSp}}(2r)$, let $(E,\Phi)$ be the Higgs pair associated
to any $\mathrm{GSp}$-Higgs pair $(P,\phi)$ and consider then the Higgs field

$\Phi^{\prime}=\Phi-\frac{\operatorname{tr}\Phi}{2r}\operatorname{id}_{E}\in
H^{0}(C,\operatorname{End}(E)\otimes L).$

Reformulating Equation 4.5, the following condition on $\Phi^{\prime}$ holds:

(4.6) $\displaystyle\omega(\Phi^{\prime}v,w)+\omega(v,\Phi^{\prime}w)=0.$

As in the case of $\mathrm{Sp}$-Higgs pairs, this condition implies that
$a_{i}(E,\Phi^{\prime})=0$ when $i=2l+1$. In particular, the vector
${\underline{a}}^{\prime}=(a_{2}(E,\Phi^{\prime}),a_{4}(E,\Phi^{\prime}),\dots,a_{2r}(E,\Phi^{\prime}))$
may assume any value in the affine space

$\bigoplus_{l=1}^{r}H^{0}(C,L^{2l})=\mathcal{A}_{\mathrm{Sp}}(2r)\subset\mathcal{A}_{\mathrm{GL}}(2r)=\mathcal{A}(2r).$

Let
${\underline{a}}=(a_{1}(E,\Phi),a_{2}(E,\Phi),\dots,a_{2r}(E,\Phi))\in\mathcal{A}(2r)$
be the characteristic of $(E,\Phi)$. Since $a_{i}(E,\Phi)=p_{i}(P,\phi)$ for
any $i=0,\dots,2r$, we have that ${\underline{a}}$ is equal to the
characteristic of $(P,\phi)$ in $\mathcal{A}_{\mathrm{GSp}}(2r)$. What is the
relation between ${\underline{a}}$ and ${\underline{a}}^{\prime}$? Denote with
$\chi_{\Phi}$ and $\chi_{\Phi^{\prime}}$ respectively the characteristic
polynomials of $\Phi$ and $\Phi^{\prime}$, and set
$\frac{\operatorname{tr}\Phi}{2r}=\mu\in H^{0}(C,L)$. By definition of
$\Phi^{\prime}$, the characteristic polynomials $\chi_{\Phi}$ and
$\chi_{\Phi^{\prime}}$ are related by the following equality:

(4.7)
$\displaystyle\chi_{\Phi}(t+\mu)=\det(\Phi-(t+\mu)\operatorname{id}_{E})=\det(\Phi^{\prime}-t\operatorname{id}_{E})=\chi_{\Phi^{\prime}}(t).$

Comparing the coefficients in Equation 4.7 and recalling that
$\operatorname{tr}\Phi=a_{1}(E,\Phi)$, it follows that the vector
$({\underline{a}}^{\prime},\mu)$ can be determined by a polynomial combination
${\underline{P}}({\underline{a}})$ of the entries of ${\underline{a}}$.
Viceversa, by the equality $\chi_{\Phi}(t)=\chi_{\Phi^{\prime}}(t-\mu)$, it
follows that the vector ${\underline{a}}$ can be determined back by a
polynomial combination of the entries of the vector ${\underline{a}}^{\prime}$
and the scalar $\mu$, denoted by
${\underline{Q}}({\underline{a}}^{\prime},\mu)$. In other words,
${\underline{P}}$ defines an isomorphism between
$\mathcal{A}_{\mathrm{GSp}}(2r)$ and the affine space
$\mathcal{A}_{\mathrm{Sp}}(2r)\oplus H^{0}(C,L)$, with inverse
${\underline{Q}}$.

By means of the previous discussion, the datum of a
$\mathrm{GSp}(2r,\mathbb{C})$-Higgs pair $(P,\phi)$ corresponds uniquely, via
the associated bundle construction and the translation of the Higgs field, to
the datum $(E,\Phi^{\prime},\omega,M,\mu)$ of a Higgs pair $(E,\Phi^{\prime})$
of rank $2r$ and degree $rd$, a non-degenerate symplectic form
$\omega:E\otimes E\rightarrow M$ with values in a line bundle $M$ of degree
$d$, and a global global section $\mu\in H^{0}(C,L)$, such that
$\Phi^{\prime}\in H^{0}(C,\operatorname{End}(E)\otimes L)$ satisfies

(4.8) $\displaystyle\omega(\Phi^{\prime}v,w)+\omega(v,\Phi^{\prime}w)=0.$

Moreover, the affine space $\mathcal{A}_{\mathrm{Sp}}(2r)\oplus H^{0}(C,L)$
can be taken as basis of another $\mathrm{GSp}$-Hitchin morphism
$\widetilde{\mathcal{H}}={\underline{P}}\circ\mathcal{H}$:

$\displaystyle\widetilde{\mathcal{H}}_{\mathrm{GSp},2r,rd}:\mathcal{M}_{\mathrm{GSp}}(2r,rd)$
$\displaystyle\longrightarrow\mathcal{A}_{\mathrm{Sp}}(2r)\oplus H^{0}(C,L)$
$\displaystyle(P,\phi)$
$\displaystyle\longmapsto({\underline{a}}^{\prime},\mu)={\underline{P}}(p_{1}(P,\phi),\dots,p_{2r}(P,\phi))=$
$\displaystyle\hskip
20.00003pt=(a_{2}(E,\Phi^{\prime}),a_{4}(E,\Phi^{\prime}),\dots,a_{2r}(E,\Phi^{\prime}),\frac{\operatorname{tr}\Phi}{2r}).$

######  Proposition 4.4.1.

(Spectral correspondence for $\mathrm{GSp}(2r,\mathbb{C})$) Let
${\underline{a}}^{\prime}\in\mathcal{A}_{\mathrm{Sp}}(2r)$ be any
characteristic and let $\mu\in H^{0}(C,L)$ be any section. Let
$X=X_{{\underline{a}}^{\prime}}\xrightarrow{\pi}C$ be the spectral curve
associated to ${\underline{a}}^{\prime}$ and let $\sigma$ be the involution
defined on $X$ as in Section 4.3. Let $d^{\prime}=rd+r(2r-1)\ell$ and denote
with $\mathcal{P}(d^{\prime},n)=\overline{\mathbb{J}}(X,d^{\prime})\times
J^{d}(C)$ the Cartesian product of the compactified Jacobian of degree
$d^{\prime}$ on $X$ and the Jacobian of degree $d$ on $C$, endowed with the
projection maps $p_{X}$ and $p_{C}$ on $\overline{\mathbb{J}}(X,d^{\prime})$
and $J^{d}(C)$ respectively. Let $\mathcal{E}_{{\underline{a}}^{\prime}}$ be
the equalizer of the two maps

$\displaystyle(\mathcal{H}om_{\mathcal{O}_{X}}(\texttt{\char
95\relax},\mathcal{O}_{X})\circ p_{X})\otimes(\pi^{*}\circ
p_{C}):\mathcal{P}(d^{\prime},d)$
$\displaystyle\rightarrow\overline{\mathbb{J}}(X,rd-r(2r-1)\ell)$
$\displaystyle(\sigma^{*}\circ
p_{X})\otimes\pi^{*}L^{1-2r}:\mathcal{P}(d^{\prime},d)$
$\displaystyle\rightarrow\overline{\mathbb{J}}(X,rd-r(2r-1)\ell).$

Then, the fiber
$\widetilde{\mathcal{H}}_{\mathrm{GSp},2r,rd}^{-1}({\underline{a}}^{\prime},\mu)$
is isomorphic, via the spectral correspondence, to
$\mathcal{E}_{{\underline{a}}^{\prime}}$.

###### Proof.

The datum of a $\mathrm{GSp}$-Higgs pair $(P,\phi)$ in the fiber
$\widetilde{\mathcal{H}}_{\mathrm{GSp},2r,rd}^{-1}({\underline{a}}^{\prime},\mu)$
corresponds uniquely to the datum $(E,\Phi^{\prime},\omega,M,\mu)$ of a Higgs
pair $(E,\Phi^{\prime})$ with rank $2r$ and characteristic
${\underline{a}}^{\prime}$, a non-degenerate symplectic form $\omega$ on $E$
with values in a line bundle $M$ of degree $d$ on $C$, and $\mu\in H^{0}(C,L)$
fixed. The torsion-free sheaf $\mathcal{M}$ on $X$ corresponding to
$(E,\Phi^{\prime})$ by the spectral correspondence fits into the exact
sequence:

$0\rightarrow\mathcal{M}\otimes\pi^{*}L^{1-2r}\rightarrow\pi^{*}E\xrightarrow{\pi^{*}\Phi-x}\pi^{*}(E\otimes
L)\rightarrow\mathcal{M}\otimes\pi^{*}L\rightarrow 0.$

Taking the dualized sequence and tensoring by $\pi^{*}(L\otimes M)$ gives a
left exact sequence:

$0\rightarrow\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M\rightarrow\pi^{*}(E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M)\xrightarrow{\pi^{*}\Phi^{t}-x}\pi^{*}(E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M)\otimes\pi^{*}L.$

On the other hand, applying $\sigma^{*}$ to the first exact sequence gives:

$0\rightarrow\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}\rightarrow\pi^{*}E\xrightarrow{\pi^{*}\Phi+x}\pi^{*}(E\otimes
L)\rightarrow\sigma^{*}\mathcal{M}\otimes\pi^{*}L\rightarrow 0.$

Now, the non-degenerate symplectic form $\omega$ induces an isomorphism
$\omega_{E}:E\rightarrow
E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M$ and hence a commutative diagram with vertical isomorphisms:

${0}$${\mathcal{M}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M}$${\pi^{*}(E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M)}$${\pi^{*}(E^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}\otimes
M)\otimes\pi^{*}L}$${0}$${\sigma^{*}\mathcal{M}\otimes\pi^{*}L^{1-2r}}$${\pi^{*}E}$${\pi^{*}(E\otimes
L)}$$\scriptstyle{\pi^{*}\Phi^{t}-x}$$\scriptstyle{\omega_{E}}$$\scriptstyle{\sim}$$\scriptstyle{-\pi^{*}\Phi-x}$$\scriptstyle{\omega_{E}}$$\scriptstyle{\sim}$

Hence, we conclude that the sheaf $\mathcal{M}$ comes with sheaf isomorphism