Dottorato di Ricerca in Matematica

XXXII Ciclo
Dottorato in Matematica

The Norm map on the compactified Jacobian,
the Prym stack
and
Spectral data for G-Higgs pairs

RAFFAELE CARBONE Nome e Cognome del dottorando Firma FILIPPO VIVIANI Docente
Guida/Tutor: Prof. Firma ANGELO LOPEZ Coordinatore: Prof. Firma

###### Contents

  1. 0.1 Preliminaries
  2. 0.2 Direct image of generalized divisors and Norm map
  3. 0.3 Spectral data for $G$-Higgs pairs
    1. 0.3.1 $\boldsymbol{G=\mathrm{SL}(r,\mathbb{C})}$
    2. 0.3.2 $\boldsymbol{G=\mathrm{PGL}(r,\mathbb{C})}$
    3. 0.3.3 $\boldsymbol{G=\mathrm{Sp}(2r,\mathbb{C})}$
    4. 0.3.4 $\boldsymbol{G=\mathrm{GSp}(2r,\mathbb{C})}$
    5. 0.3.5 $\boldsymbol{G=\mathrm{PSp}(2r,\mathbb{C})}$
  4. 1 Preliminaries
    1. 1.1 Rank and degree of coherent sheaves
    2. 1.2 Moduli spaces of torsion-free sheaves “of rank 1”
    3. 1.3 Moduli spaces of Higgs pairs
      1. 1.3.1 Classical Higgs pairs
      2. 1.3.2 G-Higgs pairs
    4. 1.4 The spectral correspondence for Higgs pairs
  5. 2 The Norm map on the compactified Jacobian
    1. 2.1 Review of the Norm map
      1. 2.1.1 Direct and inverse image of Cartier divisors
      2. 2.1.2 Norm of line bundles
    2. 2.2 Review of generalized divisors
      1. 2.2.1 Generalized divisors
      2. 2.2.2 Degree of generalized divisors
      3. 2.2.3 Moduli of generalized divisors and the Abel Map
    3. 2.3 The direct and inverse image for generalized divisors and generalized line bundles
      1. 2.3.1 The Fitting ideal
      2. 2.3.2 The direct image
      3. 2.3.3 The inverse image
      4. 2.3.4 Relation with the degree
    4. 2.4 The direct and inverse image  for families of generalized divisors
    5. 2.5 The Norm and the inverse image  for families of torsion-free rank-1 sheaves
  6. 3 The fibers of the Norm map and the Prym stack
    1. 3.1 The fibers of the Hilbert-Chow morphism
    2. 3.2 The fibers of the direct image between Hilbert schemes
    3. 3.3 The fibers of the Norm map and $\overline{\Pr}(X,Y)$
  7. 4 Spectral data for $G$-Higgs pairs
    1. 4.1 $\mathrm{SL}(r,\mathbb{C})$-Higgs pairs
    2. 4.2 $\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs
    3. 4.3 $\mathrm{Sp}(2r,\mathbb{C})$-Higgs pairs
    4. 4.4 $\mathrm{GSp}(2r,\mathbb{C})$-Higgs pairs
    5. 4.5 $\mathrm{PSp}(2r,\mathbb{C})$-Higgs pairs

## Introduction and statement of results

Let $C$ be a fixed smooth and projective curve over $\mathbb{C}$. A
($L$-twisted) Higgs pair on $C$ is a pair $(E,\Phi)$ of a vector bundle $E$ on
$C$ and a endomorphism $\Phi:E\rightarrow E\otimes L$ with values in a line
bundle $L$ on $C$. Higgs pairs were introduced first in the case $L=K_{C}$
under the name of Higgs bundles by N. Hitchin, in his study about
dimensionally reduced self-duality equations of Yang-Mills gauge
theory([Hit87b], [Hit87a]), and then by C. Simpson, in his study of nonabelian
Hodge theory ([Sim92], [Sim94a], [Sim94b]). $L$-twisted Higgs pairs were
introduced by N. Nitsure [Nit92].

If $E$ has a $G$-structure, where $G$ is some complex reductive group, and
$\Phi$ satisfies some extra conditions depending on $G$, we speak of $G$-Higgs
pairs and $G$-Higgs bundles. Hitchin [Hit87a], followed by R. Donagi [Don93],
showed that the moduli space $M_{G}$ of semistable $G$-Higgs bundles is
endowed with a proper map $H_{G}$ to a vector space, whose generic fiber is a
complex Lagrangian torus and an abelian variety. Such map is called
$G$-Hitchin fibration and makes $M_{G}$ an algebraically completely integrable
system; moreover, the smooth locus of $M_{G}$ has the structure of a
hyperkähler manifold. More generally, the Hitchin fibration $H_{G}$ on the
moduli space $M_{G}^{L}$ of semistable $G$-Higgs pairs was introduced by N.
Nitsure (loc. cit.); when $G=\mathrm{GL}$ and $L\otimes K_{C}^{-1}$ is
effective, E. Markman [Mar94] and F. Bottacin [Bot95] proved that $M_{G}^{L}$
is endowed with a Poisson structure (depending upon the choice of a section of
$L\otimes K_{C}^{-1}$) with respect to which $H_{G}$ becomes an algebraically
completely integrable system.

T. Hausel and M. Thaddeus [HT03] related the $G$-Hitchin fibration to mirror
symmetry, while A. Kapustin and E. Witten [KW07] pointed out on physical
grounds that Hitchin’s system for a complex reductive Lie group $G$ is dual to
Hitchin’s system for the Langlands dual group ${}^{L}G$. This was proved in an
algebro-geometric setting by R. Donagi and T. Pantev [DP12]; they also
conjectured the classical limit of the geometric Langlands correspondence as a
canonical equivalence between the derived categories of coherent sheaves over
the moduli stacks of $G$-Higgs bundles and ${}^{L}G$-Higgs bundles, which
intertwines the action of the classical limit tensorization functors with the
action of the classical limit Hecke functors. Recently, M. Groechenig, D. Wyss
and P. Ziegler [GWZ17] proved, using an arithmetic perspective, a conjecture
by Hausel-Thaddues stating that the moduli spaces of $\mathrm{SL}$ and
$\mathrm{PGL}$-Higgs bundles are mirror partners and that appropriately
defined Hodge numbers of such spaces agree.

The connection between Hitchin system and geometric Langlands program led to
the study of Lagrangian submanifolds of the moduli space of Higgs bundles
supporting holomorphic sheaves (A-branes), and their dual objects (B-branes).
This study was introduced by A. Kapustin and E. Witten in loc. cit. (see also
E. Witten [Wit15]), followed by L. Schaposhnik and D. Baraglia ([BS14],
[BS16]) and L. Branco [Bra18].

More recently, M. de Cataldo, T. Hausel and L. Migliorini [dCHM12] followed by
J. Shen and Z. Zhang [SZ18] and M. de Cataldo, D. Maulik and J. Shen [dCMS19]
studied the “P=W” conjecture, stating that the canonical isomorphism

$H^{*}(\mathcal{M}_{B},\mathbb{Q})\simeq H^{*}(\mathcal{M}(r),\mathbb{Q})$

induced by canonical diffeomorphism between the moduli space $\mathcal{M}(r)$
of rank $r$ Higgs bundle and the corresponding character variety
$\mathcal{M}_{B}$ of rank $r$ stable local systems, identifies the weight
filtration and the perverse filtration associated with the Hitchin fibration.

The generic fiber of the $\mathrm{GL}$-Hitchin morphism is the Jacobian of a
curve associated to the fiber, called the spectral curve; when
$G=\mathrm{SL},\mathrm{Sp},\mathrm{SO}$, the generic fiber of the $G$-Hitchin
morphisms are Prym varieties associated to certain morphisms from the spectral
curve. This fact, also known as spectral correspondence or abelianization
process, has been pointed out first in [Hit87a], followed by [Hit07] and
[Sch13]. The duality of Hitchin systems manifests itself in the statement that
the dual of the abelian variety for the generic fiber of $\mathcal{H}_{G}$ is
the abelian variety for the generic fiber of $\mathcal{H}_{{}^{L}G}$. More
recently, N. Hitchin and L. Schaposnik introduced in [HS14] a non-
abelianization process in order to study Higgs bundles that correspond (by
solving the gauge-theoretic Higgs bundle equations) to flat connections on $C$
with holonomy in some real Lie groups.

The study of the Hitchin morphism restricted to the fibers whose associated
spectral curve is integral played a crucial role in B. C. Ngô’s proof of the
fundamental lemma ([Ngô06] and [Ngô10]); more generally, the study of the
Hitchin morphism restricted to the fibers whose associated spectral curve is
reduced was a key ingredient in Chaudard-Laumon’s proof of the weighted
fundamental lemma ([CL10] and [CL12]).

For $G=\mathrm{GL}$ the spectral correspondence has been generalized to non-
generic fibers with integral spectral curve by A. Beauville, M. Narasimhan and
S. Ramanan in [BNR89], and to any fiber by D. Schaub [Sch98], followed by
P.-H. Chaudouard and G. Laumon [CL16] and M. A. De Cataldo [dC17]. The
spectral correspondence for non-generic fibers involves a wider moduli space
than the Jacobian scheme, namely the compactified Jacobian parametrizing
torsion-free rank-1 sheaves.

For $G$ different from $\mathrm{GL}$, the spectral correspondence for non-
generic fibers has been studied by [HP12] for the case of $\mathrm{SL}$, and
by [Bra18] for the Hitchin map associated to some semisimple real Lie groups.

This thesis is divided in two parts, introduced by a preliminary Chapter about
torsion-free rank-1 sheaves and Higgs pairs (Chapter 1). In the first part
(Chapter 2 and 3), motivated by the spectral correspondence for
$G=\mathrm{SL}$, we study the Norm map $\operatorname{Nm}_{\pi}$ on the
compactified Jacobian associated to a finite, flat morphism
$X\xrightarrow{\pi}Y$ between projective curves; the Norm map happens to be
well-defined only if $Y$ is smooth. In such case, we define the Prym stack of
$X$ over $Y$ as the (stacky) fiber $\operatorname{Nm}^{-1}(\mathcal{O}_{Y})$.
In the case that $X$ is reduced with locally planar singularities, we show
that the usual Prym scheme is contained in the Prym stack as an open and dense
subset. In the second part (Chapter 4) we study the spectral correspondence
for $G$-Higgs pairs, in the case of $G=\mathrm{SL}(r,\mathbb{C})$,
$\mathrm{PGL}(r,\mathbb{C})$, $\mathrm{Sp}(2r,\mathbb{C})$,
$\mathrm{GSp}(2r,\mathbb{C})$, $\mathrm{PSp}(2r,\mathbb{C})$, over any fiber.

In the future work, we aim to study the spectral data also for other classical
groups such as $\mathrm{SO}(2r,\mathbb{C})$ and
$\mathrm{SO}(2r+1,\mathbb{C})$. Moreover, we aim to study the geometric
properties of the moduli loci arising in the description of the spectral data.
Finally, we are interested in characterizing, for various $G$, the spectral
data corresponding to semistable pairs, and to consider also such data up to
$S$-equivalence of the corresponding pairs; this would allow to describe
spectral data for the scheme-theoretic version of the Hitchin morphism.

Acknowledgements. First, I would like to thank my advisor, Filippo Viviani,
for suggesting the problem and for his continuous guidance and proofreading. I
would also like to thank Edoardo Sernesi for helpful comments and discussions,
together with Eduardo Esteves and Margarida Melo. Finally, I thank my
collegues Fabrizio Anella and Daniele Di Tullio for their help and patient
listening during these years.

###  0.1. Preliminaries

In Chapter 1 we first introduce generalized line bundles and torsion-free
rank-1 sheaves, giving the following general definitions.

######  Definition 0.1.1.

Let $X$ be a Noetherian scheme of pure dimension 1. A coherent sheaf
$\mathcal{F}$ on $X$ is said:

  1. (1)

torsion-free if the support of $\mathcal{F}$ has dimension 1 and the maximal
subsheaf $T(\mathcal{F})\subset\mathcal{F}$ of dimension 0 is equal to 0;

  2. (2)

rank-1 if for any generic point $\xi\in X$, the length of $\mathcal{F}_{\xi}$
as an $\mathcal{O}_{X,\xi}$-module is equal to the length of
$\mathcal{O}_{X,\xi}$ as a module over itself.

######  Definition 0.1.2.

Let $X$ be a Noetherian scheme of pure dimension 1. A generalized line bundle
is a torsion-free sheaf $\mathcal{F}$ on $X$ such that for any generic point
$\xi\in X$,the stalk $\mathcal{F}_{\xi}$ is isomorphic to
$\mathcal{O}_{X,\xi}$ as an $\mathcal{O}_{X,\xi}$-module.

In the case of $X$ projective over a field, generalized line bundles and
torsion-free rank-1 sheaves are both particular cases of torsion-free sheaves
with polarized rank 1 with respect to any fixed polarization.

######  Definition 0.1.3.

Let $X$ be a projective scheme of pure dimension 1 over a base field $k$ and
let $H$ be a polarization of degree $\deg H=\delta$. Let $\mathcal{F}$ be a
torsion-free sheaf on $X$. The polarized rank of $\mathcal{F}$ is the rational
number $r_{H}(\mathcal{F})$ determined by the Hilbert polinomial of
$\mathcal{F}$ with respect to $H$:

$P(\mathcal{F},n,H):=\chi(\mathcal{F}\otimes\mathcal{O}_{X}(nH))=\delta
r_{H}(\mathcal{F})n+\chi(X,\mathcal{F}).$

The polarized rank of a torsion-free sheaf is related with its rank at the
generic points.

######  Theorem 0.1.4.

Let $X$ be a projective scheme of pure dimension 1 over a field, with
irreducible components $X_{1},\dots,X_{s}$ and let $\mathcal{F}$ be a torsion-
free sheaf on $X$. For each $i$, let $\xi_{i}$ be the generic point of $X_{i}$
and let
$\operatorname{rk}_{X_{i}}(\mathcal{F}):=\ell_{\mathcal{O}_{X,\xi_{i}}}(\mathcal{F}_{\xi_{i}})/\ell_{\mathcal{O}_{X,\xi_{i}}}(\mathcal{O}_{X,\xi_{i}})$
be the rank of $\mathcal{F}$ at $X_{i}$. Let $H$ be a polarization on $X$.
Then, the following formula for $r_{H}(\mathcal{F})$ holds:

$r_{H}(\mathcal{F})=\frac{\sum_{i=1}^{s}\operatorname{rk}_{X_{i}}(\mathcal{F})\deg
H_{|X_{i}}}{\sum_{i=1}^{s}\deg H_{|X_{i}}}.$

In particular, if $\mathcal{F}$ is a torsion-free rank-1 sheaf, then
$r_{H}(\mathcal{F})=1$ for any polarization $H$ on $X$.

We consider the following moduli spaces for torsion-free sheaves.

######  Definition 0.1.5.

Let $X$ be a projective scheme of pure dimension 1 over a field $k$ and let
$d\in\mathbb{Z}$ be an integer number.

  1. (1)

The Jacobian scheme of degree $d$ on $X$ is the algebraic scheme $J^{d}(X)$
representing the sheafification of the functor that associates to any
$k$-scheme $T$ the set of isomorphism classes of line bundles of degree $d$ on
$X\times_{k}T$. The union of the Jacobians of all degrees is denoted as
$J(X)$.

  2. (2)

The generalized Jacobian stack of degree $d$ on $X$ is the algebraic stack
$\operatorname{\textit{G}\mathbb{J}}(X,d)$ such that, for any $k$-scheme $T$,
$\operatorname{\textit{G}\mathbb{J}}(X,d)(T)$ is the groupoid of $T$-flat
coherent sheaves on $X\times_{k}T$ whose fibers over $T$ are generalized line
bundles of degree $d$ on $X\simeq X\times_{k}\\{t\\}$.

  3. (3)

The compactified Jacobian stack of degree $d$ on $X$ is the algebraic stack
$\overline{\mathbb{J}}(X,d)$ such that for any $k$-scheme $T$,
$\overline{\mathbb{J}}(X,d)(T)$ is the groupoid of $T$-flat coherent sheaves
on $X\times_{k}T$ whose fibers over $T$ are torsion-free sheaves of rank 1 and
degree $d$ on $X\simeq X\times_{k}\\{t\\}$.

  4. (4)

Let $H$ be a polarization on $X$ The Simpson Jacobian stack of degree $d$ on
$X$ is the algebraic stack $\overline{\mathbb{J}}_{tf}(X,H,d)$ such that for
any $k$-scheme $T$, $\overline{\mathbb{J}}_{tf}(X,H,d)(T)$ is the groupoid of
$T$-flat coherent sheaves on $X\times_{k}T$ whose fibers over $T$ are torsion-
free sheaves of polarized rank 1 and polarized degree $d$ on $X\simeq
X\times_{k}\\{t\\}$ with respect to $H$.

The locus of semi-stable object can be defined in any of such moduli spaces,
depending on the choiche of the polarization $H$. Then, we have the following
good moduli spaces.

  1. (5)

The generalized Jacobian scheme $\operatorname{\textit{GJ}}(X,H,d)$
representing S-equivalence classes of $H$-semistable generalized line bundles
of degree $d$ on $X$.

  2. (6)

The compactified Jacobian scheme $\overline{J}(X,H,d)$ representing
S-equivalence classes of $H$-semistable torsion-free sheaves of rank 1 and
degree $d$ on $X$.

  3. (7)

The Simpson Jacobian $\overline{J}_{tf}(X,H,d)$ representing S-equivalence
classes of $H$-semistable torsion-free sheaves of polarized rank 1 and
polarized degree $d$ on $X$.

These moduli spaces satisfy the following chains of inclusion:

$J^{d}(X)\stackrel{{\scriptstyle(1)}}{{\subseteq}}\operatorname{\textit{G}\mathbb{J}}(X,d)\stackrel{{\scriptstyle(2)}}{{\subseteq}}\overline{\mathbb{J}}(X,d)\stackrel{{\scriptstyle(3)}}{{\subseteq}}\overline{\mathbb{J}}_{tf}(X,H,d),$

and:

$J^{d}(X)\stackrel{{\scriptstyle(1^{\prime})}}{{\subseteq}}\operatorname{\textit{GJ}}(X,H,d)\stackrel{{\scriptstyle(2^{\prime})}}{{\subseteq}}\overline{J}(X,H,d)\stackrel{{\scriptstyle(3^{\prime})}}{{\subseteq}}\overline{J}_{tf}(X,H,d).$

All the inclusions above are open embeddings, non strict in general;
inclusions $(3)$ and $(3^{\prime})$ are also closed. When $X$ satisfies
additional conditions, some of them are actually equalities.

  * •

If $X$ is irreducible, inclusion $(3)$ (resp. $(3)^{\prime}$) is an equality.

  * •

If $X$ is reduced, inclusion $(2)$ (resp. $(2^{\prime})$) is an equality.

  * •

If $X$ is integral and smooth, inclusions $(1)$, $(2)$ and $(3)$ (resp.
$(1^{\prime})$, $(2^{\prime})$ and $(3^{\prime})$) are equalities.

The moduli space $\overline{J}_{tf}(X,H,d)$ of semistable torsion-free sheaves
of polarized rank $1$ and degree $d$ is projective by the work of C. Simpson
([Sim94a] and [Sim94b]); hence, the compactified Jacobian
$\overline{J}(X,H,d)$ is a projective subscheme and union of connected
components.

The compactified Jacobian plays an important role in the spectral
correspondence for Higgs pairs. Let $C$ be a fixed smooth curve over the field
of complex numbers and let $L$ be a fixed line bundle on $C$ with degree
$\ell=\deg L$. The algebraic stack $\mathcal{M}(r,d)$ of all $L$-twisted Higgs
pairs $(E,\Phi)$ on $C$ of rank $r$ and degree $d$ is endowed with a morphism,
called the Hitchin morphism, defined as:

$\displaystyle\mathcal{H}_{r,d}:\mathcal{M}(r,d)$
$\displaystyle\rightarrow\mathcal{A}(r)=\bigoplus_{i=1}^{r}H^{0}(C,L^{i})$
$\displaystyle(E,\Phi)$
$\displaystyle\mapsto(a_{1}(E,\Phi),\dots,a_{r}(E,\Phi))$

where $L^{i}=L^{\otimes i}$,
$a_{i}(E,\Phi):=(-1)^{i}\operatorname{tr}(\wedge^{i}\Phi)$ and
$\mathcal{A}(r)$ is called the Hitchin base. Similarly, the good moduli space
$M(r,d)$ parametrizing S-equivalence classes of semistable $L$-twisted Higgs
pairs of rank $r$ and degree $d$ is endowed with a flat projective morphism,
called the Hitchin fibration:

$\displaystyle H_{r,d}:M(r,d)$
$\displaystyle\rightarrow\mathcal{A}(r)=\bigoplus_{i=1}^{r}H^{0}(C,L^{i})$
$\displaystyle(E,\Phi)$
$\displaystyle\mapsto(a_{1}(E,\Phi),\dots,a_{r}(E,\Phi)).$

Let ${\underline{a}}\in\mathcal{A}(r)$ be any characteristic. The spectral
curve $X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$ is the projective
scheme defined in the total space of $L$, $P=\mathbb{P}(\mathcal{O}_{C}\oplus
L^{-1})\xrightarrow{p}C$, by the homogeneous equation

$x^{r}+p^{*}(a_{1})x^{r-1}y+\dots+p^{*}(a_{r})y^{r}=0$

where $x$ is the section of $\mathcal{O}_{P}(1)\otimes p^{*}(L)$ whose
pushforward via $p$ corresponds to the constant section $(1,0)$ of
$L\otimes\mathcal{O}_{C}$ and $y$ is the section of $\mathcal{O}_{P}(1)$ whose
pushforward via $p$ corresponds to the constant section $(1,0)$ of
$\mathcal{O}_{C}\otimes L^{-1}$. The spectral curve $X_{\underline{a}}$ has
pure dimension 1 and canonical sheaf
$\omega_{X_{\underline{a}}}=\pi_{\underline{a}}^{*}(\omega_{c}\otimes
L^{r-1})$. The following spectral correspondence is a classical result.

######  Proposition 0.1.6.

(Spectral correspondence) Let ${\underline{a}}\in\mathcal{A}(r)$ be any
characteristic and let $X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$
be the associated spectral curve. Let $\mathcal{O}_{C}(1)$ be an ample line
bundle on $C$; let
$\mathcal{O}_{X_{\underline{a}}}(1):=\pi_{\underline{a}}^{*}\big{(}\mathcal{O}_{C}(1)\big{)}$
be the ample line bundle on $X_{\underline{a}}$ obtained by pullback and
denote with $H$ the associated polarization on $X_{\underline{a}}$.

  1. (1)

For any integer $d$, there is an isomorphism of stacks:

$\ \mathcal{H}_{r,d}^{-1}({\underline{a}})\xrightarrow[\Pi]{\hskip
10.00002pt{}_{\sim}\hskip
10.00002pt}\overline{\mathbb{J}}(X_{\underline{a}},d^{\prime}),$

where
$d^{\prime}=d+r(1-g)-\chi(\mathcal{O}_{X_{\underline{a}}})=d+\frac{r(r-1)}{2}\ell$
and $g=g(C)$ is the genus of $C$. If $\mathcal{M}$ is a torsion-free sheaf of
rank 1 on $X_{\underline{a}}$, then $\Pi(\mathcal{M}):=(E,\Phi)$ where

$\displaystyle E$ $\displaystyle=\pi_{{\underline{a}},*}(\mathcal{M})$
$\displaystyle\Phi$ $\displaystyle=\pi_{{\underline{a}},*}(\cdot
x):\pi_{{\underline{a}},*}(\mathcal{M})\rightarrow
L\otimes\pi_{{\underline{a}},*}(\mathcal{M})\simeq\pi_{{\underline{a}},*}(\pi_{\underline{a}}^{*}L\otimes\mathcal{M}).$

  2. (2)

A torsion-free sheaf of rank 1 $\mathcal{M}$ on $X_{\underline{a}}$ is
$H$-semistable if and only if the associated Higgs pair $\Pi(\mathcal{M})$ is
semistable on $C$. Hence, the above correspondence yields an isomorphism of
schemes:

$H_{r,d}^{-1}({\underline{a}})\simeq\overline{J}(X_{\underline{a}},H,d^{\prime}).$

The inverse direction in the spectral correspondence is resumed by the
following proposition.

######  Proposition 0.1.7.

Let ${\underline{a}}\in\mathcal{A}(r)$ be any characteristic and let
$X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$ be the associated
spectral curve. Let
$\mathcal{M}\in\overline{\mathbb{J}}(X_{\underline{a}},d^{\prime})$ be a
torsion-free rank-1 sheaf on $X_{\underline{a}}$ corresponding to the Higgs
pair $(E,\Phi)\in\mathcal{H}_{r,d}^{-1}({\underline{a}})$ on $C$. Then, the
following exact sequence holds:

$0\rightarrow\mathcal{M}\otimes\pi_{\underline{a}}^{*}(L^{1-r})\rightarrow\pi_{\underline{a}}^{*}E\xrightarrow{\pi_{\underline{a}}^{*}(\Phi)-x}\pi_{\underline{a}}^{*}E\otimes\pi_{\underline{a}}^{*}L\xrightarrow{\operatorname{ev}}\mathcal{M}\otimes\pi_{\underline{a}}^{*}L\rightarrow
0$

where $\operatorname{ev}$ is induced by the evaluation map
$\pi_{\underline{a}}^{*}\pi_{{\underline{a}},*}(\mathcal{M})\rightarrow\mathcal{M}$.

###  0.2. Direct image of generalized divisors and Norm map

Let $X\xrightarrow{\pi}Y$ be a finite, flat morphism between projective
schemes of pure dimension 1 over a field, such that $Y$ is smooth. In Chapter
2 we introduce two pairs of important morphisms associated to $\pi$.

######  Definition 0.2.1.

  1. (1)

The direct and inverse image map between the Hilbert schemes
$\operatorname{Hilb}_{X}^{d}$ and $\operatorname{Hilb}_{Y}^{d}$ parametrizing
0-dimensional subschemes of $X$ (resp. of $Y$) with Hilbert polynomial equal
to $d\geq 0$ are defined on the $T$-valued points as:

$\displaystyle\pi_{*}(T):\hskip 20.00003pt\operatorname{Hilb}_{X}^{d}(T)$
$\displaystyle\longrightarrow\operatorname{Hilb}_{Y}^{d}(T)$ $\displaystyle
D\subseteq X\times_{k}T$
$\displaystyle\longmapsto\mathcal{Z}\left(\operatorname{Fitt}_{0}(\pi_{T,*}(\mathcal{O}_{D}))\right)$
$\displaystyle\pi^{*}(T):\hskip 20.00003pt\operatorname{Hilb}_{Y}^{d}(T)$
$\displaystyle\longrightarrow\operatorname{Hilb}_{X}^{d}(T)$ $\displaystyle
D\subseteq Y\times_{k}T$
$\displaystyle\longmapsto\mathcal{Z}\left(\pi_{T}^{-1}(\mathcal{I}_{D})\cdot\mathcal{O}_{X\times_{k}T}\right)$

where $\pi_{T}:X\times_{k}T\rightarrow Y\times_{k}T$ is the morphism induced
by base change of $\pi$, $\operatorname{Fitt}_{0}$ denotes the $0$-th Fitting
ideal of a sheaf of modules and $\mathcal{Z}$ denotes the closed subscheme
defined by a sheaf of ideals.

  2. (2)

The Norm and the inverse image map between the compactified Jacobians of any
degree $d$ on $X$ and $Y$ are defined on the $T$-valued points as:

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

Recall that, for any line bundle $M$ of degree $e$ on $X$, the $M$-twisted
Abel map relates $\operatorname{Hilb}_{X}^{d}$ with the generalized Jacobian
of degree $-d+e$:

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

######  Proposition 0.2.2.

Let $X\xrightarrow{\pi}Y$ be a finite, flat morphism between projective
schemes of pure dimension 1 over a field, such that $Y$ is smooth. The direct
image between Hilbert schemes and the Norm map between compactified Jacobians
are related for any $d\geq 0$ by the commutative diagram:

${\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_{*}}$$\scriptstyle{\mathcal{A}_{M}}$$\scriptstyle{\mathcal{A}_{\operatorname{Nm}_{\pi}(M)}}$$\scriptstyle{\operatorname{Nm}_{\pi}}$

Similarly, the inverse image between Hilbert schemes and the inverse image
between compactified Jacobians are related for any $d\geq 0$ by the
commutative diagram:

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

The Norm and the inverse image map between compactified Jacobians satisfy some
expected properties.

######  Proposition 0.2.3.

Let $X\xrightarrow{\pi}Y$ be a finite, flat morphism between projective
schemes of pure dimension 1 over a field, such that $Y$ is smooth. Let
$\operatorname{Nm}_{\pi}$ and $\pi^{*}$ be the Norm and inverse image map
between the compactified Jacobans of $X$ and $Y$ and let $T$ be any
$k$-scheme.

  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
$\operatorname{Nm}_{\pi}(\pi^{*}(\mathcal{N}))\simeq\mathcal{N}^{\otimes n}$.

The fibers of the direct image and of the Norm map are studied in Chapter 3,
in the case that $X$ is reduced with locally planar singularities.

######  Proposition 0.2.4.

Let $X\xrightarrow{\pi}Y$ be a finite, flat morphism between projective
schemes of pure dimension 1 over a field, such that $Y$ is smooth and $X$ is
reduced with locally planar singularities.

  1. (1)

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, the locus of Cartier divisors in
$\pi_{*}^{-1}(E)$ is an open and non-empty dense subset of $\pi_{*}^{-1}(E)$.

  2. (2)

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
subset.

In particular, the fiber of $\operatorname{Nm}_{\pi}$ over $\mathcal{O}_{Y}$
is called the Prym stack of $X$ associated to $\pi$. When $Y$ is smooth and
$X$ is reduced with locally planar singularities, the Prym variety contained
in the Prym stack of $X$ associated to $\pi$ as an open and dense subset.

###  0.3. Spectral data for $G$-Higgs pairs

In Chapter 4 we study the spectral correspondence 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 $\ell=\deg L$.

######  Definition 0.3.1.

Let $G$ be a complex reductive Lie group. A ($L$-twisted) $G$-Higgs pair is a
pair $(P,\phi)$ where $P$ is a principal $G$-bundle over $C$ and the $G$-Higgs
field $\phi$ is a section of $H^{0}(\operatorname{ad}(P)\otimes L)$.

Let $\delta\in\pi_{1}(G)$ and let $p_{1},\dots,p_{k}$ be a homogeneous basis
for the algebra of invariant polynomials of the Lie algebra $\mathfrak{g}$ of
$G$, such that $p_{i}$ has degree $d_{i}$. Then, the algebraic moduli stack
$\mathcal{M}_{G}(\delta)$ of all $L$-twisted $G$-Higgs pairs $(P,\phi)$ such
that $P$ has topological type $\delta$ is endowed with a morphism, called the
$G$-Hitchin morphism, defined as:

$\displaystyle\mathcal{H}_{G,\delta}^{p_{1},\dots,p_{k}}=\mathcal{H}_{G,\delta}:\mathcal{M}_{G}(\delta)$
$\displaystyle\rightarrow\mathcal{A}_{G}=\bigoplus_{i=1}^{k}H^{0}(C,L^{d_{i}})$
$\displaystyle(P,\phi)$ $\displaystyle\mapsto(p_{1}(\phi),\dots,p_{k}(\phi)).$

If $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$, any $G$-Higgs pair
$(P,\phi)$ gives rise via the associated bundle construction to a classical
Higgs pair $(E,\Phi)$, where $E=P\times_{\rho}\mathbb{C}^{r}$ and $\Phi$ is
the image $\phi$ with respect to the morphism
$\operatorname{ad}(P)\rightarrow\operatorname{End}(E)$ induced by the
embedding $\rho$; such $E$ has rank $r$ and degree $d$ corresponding to
$\delta$ under the map
$\pi_{1}(G)\xrightarrow{\pi_{1}(\rho)}\pi_{1}(\mathrm{GL}(r,\mathbb{C}))$
induced by $\rho$. In this case, we denote as

$\mathcal{M}_{G}(r,d):=\coprod_{\pi_{1}(\rho)(\delta)=d}\mathcal{M}_{G}(\delta)$

the moduli space of $G$-Higgs pairs whose associated vector bundle is of rank
$r$ and degree $d$; the $G$-Hitchin morphism on $\mathcal{M}_{G}(r,d)$ is
denoted as $\mathcal{H}_{G,r,d}$.

####  0.3.1. $\boldsymbol{G=\mathrm{SL}(r,\mathbb{C})}$

In this case, 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 $r$ on
$C$ with trace zero, and $\lambda$ is a trivialization of $\det E$. The
isomorphism $\det E\simeq\mathcal{O}_{C}$ implies in particular that $E$ has
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
$p_{i}(P,\phi):=(-)^{i}\operatorname{tr}(\wedge^{i}\phi)$. The
$\mathrm{SL}$-Hitchin morphism of rank $r$ can be defined as:

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

######  Proposition 0.3.2.

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 $J(Y,d^{\prime})$ induced by $\pi$
for $d^{\prime}=\frac{r(r-1)}{2}\ell$. If the spectral curve
$X_{\underline{a}}$ is reduced, then

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

####  0.3.2. $\boldsymbol{G=\mathrm{PGL}(r,\mathbb{C})}$

Any $\mathrm{PGL}(r,\mathbb{C})$-Higgs pair admits a lifting
$(\tilde{P},\tilde{\phi})$ to a $\mathrm{GL}(r,\mathbb{C})$-Higgs pair,
corresponding to a Higgs pair $(E,\Phi)$ 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:

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

Denote with $\mathcal{M}^{\operatorname{tr}=0}(r)$ the closed substack of
$\mathcal{M}(r)=\coprod_{d\in\mathbb{Z}}\mathcal{M}(r,d)$ 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:

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

######  Definition 0.3.3.

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
$\overline{d}\in\mathbb{Z}/r\mathbb{Z}$.

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 $P$.
Moreover, up to the action of multiples of $\mathcal{O}_{C}(1)$ on
$(E,\Phi)$,the $J(C)$-orbit $[(E,\Phi)]$ corresponding to a
$\mathrm{PGL}$-Higgs pair with degree $\overline{d}\in\mathbb{Z}/r\mathbb{Z}$
is the orbit of a Higgs pair of degree $d$ with $d\in\\{0,\dots,r-1\\}$. Then,
a $\mathrm{PGL}$-Higgs pair of degree $\overline{d}$ is identified uniquely
with the orbit in $\mathcal{M}^{\operatorname{tr}=0}(r,d)$ of a Higgs pair of
trace zero and degree $d$ with respect to the action of line bundles of degree
$0$ on $C$.

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
$p_{i}(P,\phi):=(-)^{i}\operatorname{tr}(\wedge^{i}\phi)$. Hence, we have the
following $\mathrm{PGL}$-Hitchin morphism of rank $r$ and degree
$\overline{d}$:

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

######  Proposition 0.3.4.

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

The case of $\mathrm{PGL}(r,\mathbb{C})$-Higgs pairs of degree $0$ deserves
special attention.

######  Proposition 0.3.5.

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:

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

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

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

####  0.3.3. $\boldsymbol{G=\mathrm{Sp}(2r,\mathbb{C})}$

In this case, 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
$p_{2i}(P,\phi):=\operatorname{tr}(\wedge^{2i}\phi)$. 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 $X_{\underline{a}}$ has an involution $\sigma$ defined by
$\sigma(x)=-x$.

######  Proposition 0.3.6.

Let ${\underline{a}}\in\mathcal{A}_{\mathrm{Sp}}(r)$ 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

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

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

####  0.3.4. $\boldsymbol{G=\mathrm{GSp}(2r,\mathbb{C})}$

In this case, 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 $\omega:E\otimes
E\rightarrow M$ on $E$ 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

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

The affine space $\mathcal{A}_{\mathrm{Sp}}(2r)\oplus H^{0}(C,L)$ can be taken
as basis of a (translated) $\mathrm{GSp}$-Hitchin morphism
$\widetilde{\mathcal{H}}$, defined as:

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

where $(E,\Phi)$ is the Higgs pair associated to $(P,\phi)$ via the associated
bundle construction, $\mu$ is the global section
$\frac{\operatorname{tr}\Phi}{2r}\in H^{0}(C,L)$ and
$\Phi^{\prime}=\Phi-\mu\operatorname{id}_{E}$ is the translated Higgs field.

######  Proposition 0.3.7.

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 ${\underline{a}}^{\prime}$, with involution $\sigma:X\rightarrow
X$. Let $d^{\prime}=rd+r(2r-1)\ell$ and denote with
$\mathcal{P}(d^{\prime},n)=\overline{\mathbb{J}}(X,d^{\prime})\times J(C,d)$
the Cartesian product of the Simpson 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 $J(C,d)$
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
$\mathcal{E}_{{\underline{a}}^{\prime}}$.

####  0.3.5. $\boldsymbol{G=\mathrm{PSp}(2r,\mathbb{C})}$

In this case, any $\mathrm{PSp}(2r,\mathbb{C})$-Higgs pair $(P,\phi)$ has a
lifting $(\tilde{P},\tilde{\phi})$ to a $\mathrm{GSp}(r,\mathbb{C})$-Higgs
pair, corresponding via the associated bundle construction to the datum
$(E,\Phi,M,\omega)$ of a Higgs pair $(E,\Phi)$ 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 $d$ on $C$, and a Higgs field $\Phi$ with trace
zero satisfying the condition

$\omega(\Phi v,w)+\omega(v,\Phi w)=0.$

Then, the datum of $(P,\phi)$ corresponds uniquely to the datum of the
equivalence class $[(E,\Phi,M,\omega)]$ of Higgs pairs of rank $2r$ with a
non-degenerate symplectic form, under the equivalence relation $\sim_{J(C)}$
defined by:

$(E,\Phi,M,\omega)\sim_{J(C)}(E\otimes N,\Phi\otimes 1_{N},M\otimes
N^{2},\omega_{N})\hskip 20.00003pt\textrm{for any }N\in J(C)$

where

$\omega_{N}:(E\otimes N)\otimes(E\otimes N)\rightarrow M\otimes N^{2}$

is obtained by extension of scalars.

Let
$\mathcal{M}_{\mathrm{GSp}}(2r)=\coprod_{d\in\mathbb{Z}}\mathcal{M}_{\mathrm{GSp}}(2r,rd)$
be the moduli stack of $\mathrm{GSp}$-Higgs pairs of rank $2r$ in any degree
and denote with $\mathcal{M}_{\mathrm{GSp}}^{\operatorname{tr}=0}(2r)$ the
closed substack of $\mathcal{M}_{\mathrm{GSp}}(2r)$ given by
$\mathrm{GSp}$-Higgs pairs of rank $2r$ with trace zero. Then,
$[(E,\Phi,M,\omega)]$ is the orbit of $(E,\Phi,M,\omega)$ under the action of
$J(C)$ on $\mathcal{M}_{\mathrm{GSp}}^{\operatorname{tr}=0}(2r)$ defined by:

$\displaystyle\mathcal{M}_{\mathrm{GSp}}^{\operatorname{tr}=0}(2r)\times J(C)$
$\displaystyle\longrightarrow\mathcal{M}_{\mathrm{GSp}}^{\operatorname{tr}=0}(2r)$
$\displaystyle((E,\Phi,M,\omega),N)$ $\displaystyle\longmapsto(E\otimes
N,\Phi\otimes 1_{N},M\otimes N^{2},\omega_{N}).$

######  Definition 0.3.8.

Let $(P,\phi)$ be a $\mathrm{PSp}$-Higgs pair and let $(E,\Phi,M,\omega)$ be
the datum of a Higgs pair with trace zero and degree $rd$ endowed with a
$M$-valued symplectic form $\omega$, corresponding to a lifting
$(\tilde{P},\tilde{\phi})$ of $(P,\phi)$ to a $\mathrm{GSp}$-Higgs pair. The
degree of $(P,\phi)$ is the congruence class
$\overline{rd}\in\mathbb{Z}/2r\mathbb{Z}$.

Up to the action on $(E,\Phi,M,\omega)$ of a line bundle of degree $1$ on $C$,
it is straightforward to see that the $J(C)$-orbit $[(E,\Phi,M,\omega)]$
corresponding to a $\mathrm{PGL}$-Higgs pair with degree
$\overline{rd}\in\mathbb{Z}/2r\mathbb{Z}$ is always the orbit of a datum whose
Higgs pair has degree $0$ or $rd$. If we restrict the correspondence to Higgs
pairs with fixed degree $0$ or $rd$, we see that a $\mathrm{PGL}$-Higgs pair
of degree $\overline{rd}$ is identified uniquely by the orbit of a Higgs pair
with trace zero and degree $0$ or $rd$ with respect to the action of line
bundles of degree $0$ on $C$.

A basis for the invariant polynomials of
$\mathfrak{psp}(2r,\mathbb{C})=\mathfrak{sp}(2r,\mathbb{C})$ is given by
$\\{p_{2i}\\}_{i=1,\dots,r}$ where
$p_{2i}(P,\phi):=\operatorname{tr}(\wedge^{2i}\phi)$. The corresponding
$\mathrm{PSp}$-Hitchin morphism takes the form:

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

As in the case of $\mathrm{Sp}(2r,\mathbb{C})$-Higgs pairs, the curve
$X_{\underline{a}}$ has an involution defined by $\sigma(x)=-x$.

######  Proposition 0.3.9.

Let ${\underline{a}}\in\mathcal{A}_{\mathrm{PSp}}(2r)$ be any characteristic,
let $X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve
with involution $\sigma:X\rightarrow X$. Let $d\in\\{0,1\\}$,
$d^{\prime}=rd+r(2r-1)\ell$ 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

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

The group $J^{0}(C)$ of line bundles of degree $0$ on $C$ acts on
$\mathcal{E}_{\underline{a}}$ as follows:

$\displaystyle\mathcal{E}_{\underline{a}}\times J^{0}(C)$
$\displaystyle\longrightarrow\mathcal{E}_{\underline{a}}$
$\displaystyle((\mathcal{M},M,\lambda),N)$
$\displaystyle\longmapsto(\mathcal{M}\otimes\pi^{*}N,M\otimes
N^{2},\lambda_{N})$

where $\lambda_{N}$ is given by the composition of
$\lambda\otimes\operatorname{id}_{\pi^{*}N}$ with the canonical isomorphisms:

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

Then, the fiber
$\mathcal{H}_{\mathrm{PSp},2r,\overline{rd}}^{-1}({\underline{a}})$ of the
$\mathrm{PSp}$-Hitchin morphism is isomorphic, via the spectral
correspondence, to the quotient $\mathcal{E}_{\underline{a}}/J^{0}(C)$.

The case of $\mathrm{PSp}(2r,\mathbb{C})$-Higgs pairs of degree $0$ deserves
special attention.

######  Proposition 0.3.10.

Let ${\underline{a}}\in\mathcal{A}_{\mathrm{PSp}}(2r)$ be any characteristic,
let $X=X_{\underline{a}}\xrightarrow{\pi}C$ be the associated spectral curve
with involution $\sigma:X\rightarrow X$. Let $d^{\prime}=r(2r-1)\ell$ and let
$\mathcal{E}_{\underline{a}}$ be the equalizer stack of the two maps:

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

The group stack $J^{0}(C)[2]$ of 2-torsion line bundles on $C$ acts on
$\mathcal{E}_{\underline{a}}$ as follows:

$\displaystyle\mathcal{E}_{\underline{a}}\times\pi^{*}J^{0}(C)[2]$
$\displaystyle\longrightarrow\mathcal{E}_{\underline{a}}$
$\displaystyle((\mathcal{M},\lambda),(\pi^{*}N,\pi^{*}\epsilon))$
$\displaystyle\longmapsto(\mathcal{M}\otimes\pi^{*}N,\lambda\otimes\pi^{*}\epsilon).$

Then, the fiber
$\mathcal{H}_{\mathrm{PSp},2r,\overline{0}}^{-1}({\underline{a}})$ of the
$\mathrm{PSp}$-Hitchin morphism is isomorphic, via the spectral
correspondence, to the quotient
$\mathcal{E}_{\underline{a}}/\pi^{*}J^{0}(C)[2]$.

##  Chapter 1 Preliminaries

###  1.1. Rank and degree of coherent sheaves

Let $X$ be any Noetherian scheme of pure dimension $1$. In this section, we
study classes of coherent sheaves on $X$ for which the notions of rank on $X$
are well defined. Moreover, we discuss the notions of degree for sheaves with
well-defined rank.

The simplest class is clearly given by vector bundles or, equivalently,
locally free sheaves of constant rank. If $\mathcal{F}$ is a locally free
sheaf on $X$, we say that $\mathcal{F}$ has (free) rank $r$ if
$\mathcal{F}_{x}\simeq\mathcal{O}_{X,x}^{\oplus r}$ for any point $x$ of $X$.

For any coherent sheaf $\mathcal{F}$ on $X$, we give now the following
definitions.

######  Definition 1.1.1.

The support of $\mathcal{F}$ is the closed set

$\operatorname{Supp}(\mathcal{F})=\\{x\in X|\mathcal{F}_{x}\neq 0\\}\subseteq
X.$

The dimension of $\mathcal{F}$ is the dimension of its support and is denoted
$\dim(\mathcal{F})$.

######  Definition 1.1.2.

The torsion subsheaf of $\mathcal{F}$ is the maximal subsheaf
$T(\mathcal{F})\subseteq\mathcal{F}$ of dimension 0. If
$\mathcal{F}=T(\mathcal{F})$, we say that $\mathcal{F}$ is a torsion sheaf. If
$T(\mathcal{F})=0$ we say that $\mathcal{F}$ is torsion-free.

Equivalently, $\mathcal{F}$ is torsion-free if $\dim(\mathcal{E})=1$ for any
non-trivial coherent subsheaf $\mathcal{E}\subseteq\mathcal{F}$, i.e.
$\mathcal{F}$ is pure of dimension 1. For any sheaf $\mathcal{F}$, the
quotient sheaf $\mathcal{F}/T(\mathcal{F})$ is either zero or torsion-free.

The first class of torsion-free sheaves with well-defined rank is given by
generalized vector bundles.

######  Definition 1.1.3.

$\mathcal{F}$ is a generalized vector bundle if it is torsion-free and there
exists a positive integer $r$ such that
$\mathcal{F}_{\xi}\simeq\mathcal{O}_{\xi}^{\oplus r}$ for any generic point
$\xi$ of $X$, where $\mathcal{F}_{\xi}$ denotes the stalk of $\mathcal{F}$ at
$\xi$. The integer $r$ is called the rank of the generalized vector bundle
$\mathcal{F}$. A generalized vector bundle of rank $1$ is also called a
generalized line bundle.

Clearly, a vector bundle of rank $r$ is also a generalized vector bundle of
the same rank.

######  Remark 1.1.4.

Generalized line bundles were introduced in [BE00] in order to study limit
linear series on a ribbon. As pointed out in [EG95], generalized line bundles
are strictly related to generalized divisors introduced indipendently by
Hartshone in [Har86] (followed by [Har94] and [Har07]). See also Chapter 2,
2.2 for details. The generalization to any rank $r$ is due to [Sav19].

We introduce now the notion of rank for any coherent sheaf. Denote with
$X_{1},\dots,X_{s}$ the irreducible components of $X$ and with $\xi_{i}$ the
generic point of each irreducible component $X_{i}$. Denote with
$C_{i}=X_{i,red}$ the reduced subscheme underlying $X_{i}$ for each $i$.

######  Definition 1.1.5.

The multiplicity of $C_{i}$ in $X$ is defined as the positive integer:

$\operatorname{mult}_{X}(C_{i}):=\ell_{\mathcal{O}_{X,\xi_{i}}}(\mathcal{O}_{X,\xi_{i}}).$

where $\ell_{\mathcal{O}_{X,\xi_{i}}}$ denotes the length as
$\mathcal{O}_{X,\xi_{i}}$-module.

######  Definition 1.1.6.

(Rank and multirank of a coherent sheaf) The rank of $\mathcal{F}$ on $X_{i}$
is defined as the positive rational number

$\operatorname{rk}_{X_{i}}(\mathcal{F})=\frac{\ell_{\mathcal{O}_{X,\xi_{i}}}(\mathcal{F}_{\xi_{i}})}{\operatorname{mult}_{X}(C_{i})}.$

The multirank of $\mathcal{F}$ on $X$ is the $n$-uple
${\underline{r}}=(r_{1},\dots,r_{n})$ where $r_{i}$ is the rank of
$\mathcal{F}$ at $X_{i}$. The sheaf $\mathcal{F}$ has rank $r$ on $X$ if it
has rank $r$ at each $X_{i}$.

Note that, since length of modules is additive in short exact sequences, the
rank and multirank of coherent sheaves are additive in short exact sequences
too.

######  Proposition 1.1.7.

A generalized vector bundle of rank $r$ on $X$ has rank $r$ as a coherent
sheaf. If $X$ is reduced, any torsion-free sheaf of rank $r$ is a generalized
vector bundle of the same rank.

###### Proof.

The first statement follows from 1.1.6 and the fact that isomorphic modules
have the same length. For the second assertion, note that on an integral curve
$X$ there exists for any torsion-free sheaf $\mathcal{F}$ an open dense subset
$U\subset X$ such that $\mathcal{F}_{|U}$ is locally free, hence the stalk of
$\mathcal{F}$ at the generic point of $X$ is a free
$\mathcal{O}_{X,\xi}$-module of rank $\operatorname{rk}_{X}(\mathcal{F})$. If
$X$ is reducible, the assertion follows by considering any irreducible
component. ∎

######  Remark 1.1.8.

There are in literature other notions for the rank of a coherent sheaf on a
irreducible component. A classical notion is the reduced rank of $\mathcal{F}$
on $X_{i}$:

$\operatorname{\textbf{rank}}_{X_{i}}(\mathcal{F})=\dim_{\kappa(\xi_{i})}(\mathcal{F}_{\xi_{i}}\otimes_{\mathcal{O}_{X,\xi}}\kappa(\xi_{i})).$

This definition computes the rank of the sheaf restricted to its reduced
support; it agrees with the other definitions for generalized vector bundles.
However, there are cases when the reduced rank of a torsion-free sheaf differs
from its rank, as it happens for quasi-locally free sheaves on a ribbon (see
[Sav19, §1.4]). They also provide examples of torsion-free sheaves which are
not generalized vector bundles, even if their rank is well defined.

The definition of $\operatorname{rk}_{H}(\mathcal{F})$ comes at least from
[Sch98, Définition 1.2], where it is given in the context of projective
$k$-schemes without embedded points. The notion of generalized rank introduced
in [Dre04] and [Sav19] is essentially equivalent. Finally, a common definition
in projective algebraic geometry is the polarized rank, that we will discuss
later on in the present section.

In order to introduce the degree of coherent sheaves, from now on we  assume
that $X$ is projective curve over a base field $k$. Recall that the Euler
characteristic of a coherent sheaf $\mathcal{F}$ is

$\chi(\mathcal{F}):=\sum(-)^{i}\dim_{k}H^{i}(X,\mathcal{F}).$

######  Definition 1.1.9.

(Degree of a coherent sheaf) Suppose that $\mathcal{F}$ has rank $r$ on $X$.
Then, the degree of $\mathcal{F}$ on $X$ is defined as the fractional number:

$\deg\mathcal{F}=\chi(X,\mathcal{F})-r\chi(X,\mathcal{O}_{X}).$

######  Remark 1.1.10.

If the rank of $\mathcal{F}$ is integer, the degree is integer as well. If $X$
is integral, the degree of a sheaf is an integer and coincides with the
classical notion of degree for sheaves on integral curves.

The following technical lemma is very useful.

######  Lemma 1.1.11.

Assume that $X$ is irreducible. Let $\mathcal{F}$ be a coherent sheaf of rank
$r$ on $X$ and let $\mathcal{E}$ be a locally free sheaf of rank $n$ on $X$.
Then

$\chi(X,\mathcal{F}\otimes\mathcal{E})=r\deg(\mathcal{E})+n\chi(X,\mathcal{F}).$

###### Proof.

The proof is inspired by [Sta19, https://stacks.math.columbia.edu/tag/0AYVTag
0AYV] and uses devissage for coherent sheaves as stated in [Sta19,
https://stacks.math.columbia.edu/tag/01YITag 01YI]. Let $\mathcal{P}$ be the
property of coherent sheaves $\mathcal{F}$ on $X$ expressing that the formula
of the Lemma holds. By additivity of rank and Euler characteristic in short
exact sequences, $\mathcal{P}$ satisfy the two-out-of-three property. The
integral subschemes $Z$ of $X$ are the reduced subscheme $C$ with support
equal to $X$, and the closed points of $X$. For $Z=C$, the formula of the
Lemma for $\mathcal{O}_{C}$ is:

$\chi(X,\mathcal{E}\otimes\mathcal{O}_{C})=\frac{\deg(\mathcal{E})}{\operatorname{mult}_{X}C}+n\chi(X,\mathcal{O}_{C}).$

This is true by definition of degree of
$\mathcal{E}\otimes\mathcal{O}_{C}=\mathcal{E}_{|C}$ on $C$ and the fact that
$\deg(\mathcal{E})=\operatorname{mult}_{X}(C)\deg(\mathcal{E}_{|C})$. Then
$\mathcal{P}(\mathcal{O}_{C})$ holds. If $i:Z\hookrightarrow X$ is a closed
point, the formula of the Lemma is true for $i_{*}\mathcal{O}_{Z}$ since it is
a torsion sheaf of rank 0 and $\chi(X,\mathcal{E}\otimes
i_{*}\mathcal{O}_{Z})=n\chi(X,i_{*}\mathcal{O}_{Z})$. Then
$\mathcal{P}(i_{*}\mathcal{O}_{Z})$ holds. ∎

Let $H$ be any polarization of $X$; then, the notion of polarized rank and
degree of sheaves can be defined. Denote with $H_{|C_{i}}$ the restriction of
$H$ to the closed subscheme $C_{i}\subseteq X$. Since $H$ is locally free of
rank $1$, the degree of $H$ on $X$ is related to the degrees of each
$H_{|C_{i}}$ on $C_{i}$ by the formula:

$\deg H=\sum_{i=1}^{s}\operatorname{mult}_{X}(C_{i})\deg H_{|C_{i}}.$

######  Definition 1.1.12.

(Polarized rank and degree of a torsion-free sheaf) Let $H$ be any
polarization of $X$ with degree $\deg H=\delta$ and let $\mathcal{F}$ be a
torsion-free sheaf on $X$. The polarized rank and degree of $\mathcal{F}$ are
the rational numbers $r_{H}(\mathcal{F})$ and $d_{H}(\mathcal{F})$ determined
by the Hilbert polinomial of $\mathcal{F}$ with respect to $H$:

$P(\mathcal{F},n,H):=\chi(\mathcal{F}\otimes\mathcal{O}_{X}(nH))=\delta
r_{H}(\mathcal{F})n+d_{H}(\mathcal{F})+r_{H}(\mathcal{F})\chi(\mathcal{O}_{X}).$

The polarized rank and degree of a sheaf depend strictly on the degrees of the
restrictions $H_{|C_{i}}$, as the following theorem shows.

######  Theorem 1.1.13.

Let $H$ be a polarization of $X$ and let $\mathcal{F}$ be a torsion-free sheaf
on $X$. The polarized rank of $\mathcal{F}$ is related to the multirank of
$\mathcal{F}$ by the formula

$r_{H}(\mathcal{F})=\frac{\sum_{i=1}^{s}\operatorname{rk}_{X_{i}}(\mathcal{F})\operatorname{mult}_{X}(C_{i})\deg
H_{|C_{i}}}{\sum_{i=1}^{s}\operatorname{mult}_{X}(C_{i})\deg H_{|C_{i}}}.$

###### Proof.

Since the Hilbert polynomial of $\mathcal{F}$ with respect to $H$ has degree
$1$, its leading term can be computed in terms of Euler characteristic as

$\chi(X,\mathcal{F}\otimes\mathcal{O}_{X}(nH))-\chi(X,\mathcal{F}).$

Since $\delta=\sum_{i=1}^{n}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H$ is
the degree of $H$ on $X$, by Definition 1.1.12 we have to prove:

$\chi(X,\mathcal{F}\otimes\mathcal{O}_{X}(nH))-\chi(X,\mathcal{F})=n\sum_{i=1}^{s}\operatorname{rk}_{X_{i}}(\mathcal{F})\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H.$

First, we reduce to the case of $X$ irreducible. Consider the exact sequence:

$0\rightarrow\mathcal{F}\rightarrow\bigoplus_{i}\mathcal{F}_{|X_{i}}\rightarrow
T\rightarrow 0$

where $T$ is a torsion sheaf supported only at the intersections of the
irreducible components. Tensoring by $nH$, we obtain another exact sequence:

$0\rightarrow\mathcal{F}\otimes\mathcal{O}_{X}(nH)\rightarrow\bigoplus_{i}(\mathcal{F}\otimes\mathcal{O}_{X}(nH))_{|X_{i}}\rightarrow
T\otimes\mathcal{O}_{X}(nH)\rightarrow 0.$

By additivity of the Euler characteristic with respect to exact sequences we
have:

$\displaystyle\sum_{i}$
$\displaystyle\left(\chi(X_{i},(\mathcal{F}\otimes\mathcal{O}_{X}(nH))_{|X_{i}})-\chi(X_{i},\mathcal{F}_{|X_{i}})\right)=$
$\displaystyle=\chi(X,\mathcal{F}\otimes\mathcal{O}_{X}(nH))-\chi(X,\mathcal{F})+\chi(X,T\otimes\mathcal{O}_{X}(nH))-\chi(X,T).$

Since $T$ has dimension 0, the characteristics
$\chi(X,T\otimes\mathcal{O}_{X}(nH))$ and $\chi(X,T)$ are the same, hence:

$\chi(X,\mathcal{F}\otimes\mathcal{O}_{X}(nH))-\chi(X,\mathcal{F})=\sum_{i}\left(\chi(X_{i},(\mathcal{F}\otimes\mathcal{O}_{X}(nH))_{|X_{i}})-\chi(X_{i},\mathcal{F}_{|X_{i}})\right).$

We are then reduced to prove the statement on each irreducible component
$X_{i}$. In other words we have to prove that, if $X$ is a irreducible
(possibly non-reduced) curve with reduced structure $C$, then:

$\chi(X,\mathcal{F}\otimes\mathcal{O}_{X}(nH))-\chi(X,\mathcal{F})=n\deg_{X}(H)\operatorname{rk}_{X}(\mathcal{F}).$

This is exactly the content of Lemma 1.1.11, with
$\mathcal{E}=\mathcal{O}_{X}(nH)$. ∎

######  Corollary 1.1.14.

Let $\mathcal{F}$ be any torsion-free sheaf with well-defined rank. Then, its
polarized rank $r_{H}(\mathcal{F})$ and degree $d_{H}(\mathcal{F})$ are equal
respectively to $\operatorname{rk}_{X}(\mathcal{F})$ and
$\deg_{X}(\mathcal{F})$ for any polarization $H$ on $X$. f $X$ is irreducible,
the rank and degree of a sheaf coincide with the polarized counterparts, for
any polarization.

###### Proof.

The statement about the rank follows from Theorem 1.1.13, where $r_{i}=r$ for
each irreducible component $X_{i}$. To prove the statement about the degree,
note that for any polarization $H$:

$d_{H}(\mathcal{F})=\chi(X,\mathcal{F})-r_{H}(\mathcal{F})\chi(X,\mathcal{O}_{X}).$

If $r_{H}(\mathcal{F})=\operatorname{rk}(\mathcal{F})$, then
$d_{H}(\mathcal{F})=\deg(\mathcal{F})$ by Definition 1.1.9. The last statement
follows from the fact that the $\operatorname{rk}_{X}(\mathcal{F})$ is well-
defined for any torsion-free sheaf when $X$ has only one irreducible
component. ∎

######  Remark 1.1.15.

For reducible curves, the definitions of polarized rank and degree are the
most general. As pointed out in [Lop04, §2], being of polarized rank $1$ for a
torsion-free sheaf does not ensure that the sheaf is supported on the whole
curve. Moreover, there are cases of torsion-free sheaves of polarized rank $1$
whose polarized rank on the restrictions to the irreducible components is
different from $1$ and not an integer, even for reduced curves.

###  1.2. Moduli spaces of torsion-free sheaves “of rank 1”

In this section, $X$ is a (possibly reducible, non-reduced) projective curve
over a base field $k$, with irreducible components $X_{1},\dots,X_{s}$. We
introduce the definition of several moduli spaces parametrizing classes of
torsion-free sheaves “of rank 1” (with different meanings) that will play an
important role in the present work. We start by recalling the definition of
the moduli space parametrizing line bundles on $X$.

######  Definition 1.2.1.

Let $d\in\mathbb{Z}$ be an integer number. The Jacobian scheme of degree $d$
on $X$ is the algebraic scheme $J^{d}(X)$ representing the sheafification of
the functor that associates to any $k$-scheme $T$ the set of isomorphism
classes of line bundles of degree $d$ on $X\times_{k}T$. The union of the
Jacobians of all degrees is denoted as $J(X)$.

######  Remark 1.2.2.

The existence of the representing scheme $J^{d}(X)$ is a classical result in
algebraic geometry, see for example [Kle05, Corollary 4.18.3]. Following a
number of authors (see for example [MRV17a] or [Kas15]) we prefer to use the
term “Jacobian” instead of “Picard” even if we consider line bundles in any
degree; this is motivated only by the simpler notation.

Since we are dealing with moduli spaces, we recall also the definition of
slope and stability for torsion-free sheaves.

######  Definition 1.2.3.

Let $H$ be a polarization on $\mathcal{F}$ and $\mathcal{F}$ be a torsion-free
sheaf on $X$ with polarized rank $r_{H}(\mathcal{F})$ and polarized degree
$d_{H}(\mathcal{F})$.

The slope of $\mathcal{F}$ with respect to $H$ is defined as the rational
number $\mu_{H}(\mathcal{F})=d_{H}(\mathcal{F})/r_{H}(\mathcal{F})$.

A coherent sheaf $\mathcal{F}$ on $X$ is $H$-stable (respectively $H$-semi-
stable) if it is torsion-free and for any proper subsheaf
$\mathcal{E}\subset\mathcal{F}$ the equality
$\mu_{H}(\mathcal{E})<\mu_{H}(\mathcal{F})$ holds (respectively $\leq$).

We start from the more general moduli space, parametrizing for torsion-free
sheaves of polarized rank 1.

######  Definition/Lemma 1.2.4.

Let $H$ be a polarization on $X$ and let $d\in\mathbb{Z}$ be an integer
number. The Simpson Jacobian stack of degree $d$ on $X$ is the algebraic stack
$\overline{\mathbb{J}}_{tf}(X,H,d)$ such that for any $k$-scheme $T$,
$\overline{\mathbb{J}}_{tf}(X,d)(T)$ is the groupoid of $T$-flat coherent
sheaves on $X\times_{k}T$ whose fibers over $T$ are torsion-free sheaves of
polarized rank 1 and polarized degree $d$ on $X\simeq X\times_{k}\\{t\\}$ with
respect to $H$.

The Simpson Jacobian of degree $d$ is the projective scheme
$\overline{J}_{tf}(X,H,d)$ which is the good moduli space representing
S-equivalence classes of $H$-semistable torsion-free sheaves of polarized rank
1 and polarized degree $d$ on $X$.

The union of the Simpson Jacobians of all degrees is denoted
$\overline{\mathbb{J}}_{tf}(X,H)$ (resp. $\overline{J}_{tf}(X,H)$).

###### Proof.

The algebraicity of $\overline{\mathbb{J}}_{tf}(X,H,d)$ follows from the
algebraicity of the stack of coherent sheaves on $X$ (see [Lie06, 2.1] and
[CW17, Theorem 7.20]). The moduli space of semistable sheaves with fixed
polarized rank and degree was constructed by Simpson in [Sim94a]. ∎

In the following proposition, we show that the multirank of sheaves splits
$\overline{\mathbb{J}}_{tf}(X,H,d)$ in unions of connected components.

######  Lemma 1.2.5.

Let $H$ be a fixed polarization on $X$. For any
${\underline{r}}=(r_{1},\dots,r_{s})\in\mathbb{Q}_{\geq 0}^{s}$, let
$W_{\underline{r}}\subseteq\overline{\mathbb{J}}_{tf}(X,H,d)$ be the substack
parametrizing torsion-free sheaves with multirank equal to ${\underline{r}}$.
Then
$\\{W_{\underline{r}}|W_{\underline{r}}\neq\emptyset\\}_{{\underline{r}}\in\mathbb{Q}_{\geq
0}^{s}}$ is a finite collection of pairwise disjoint substacks of
$\overline{\mathbb{J}}_{tf}(X,H,d)$ that covers the whole space. Moreover,
each $W_{\underline{r}}$ is open and closed, hence union of connected
components of $\overline{\mathbb{J}}_{tf}(X,H,d)$.

###### Proof.

The multi-rank of a sheaf is well-defined, hence the substacks
$W_{\underline{r}}$ are pairwise disjoint.

Let $\mathcal{F}$ be a sheaf with polarized rank $r_{H}$ equal to $1$. By
Theorem 1.1.13 its multirank ${\underline{r}}$ must satisfy:

$1=r_{H}(\mathcal{F})=\frac{\sum_{i=1}^{s}r_{i}(t)\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}{\sum_{i=1}^{n}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}.$

Since each $r_{i}$ is a non-negative fraction with integer numerator and
denominator equal to $\operatorname{mult}_{X}(C_{i})$, the possible values for
${\underline{r}}$ form a finite subset of $\mathbb{Q}_{\geq 0}^{s}$, once that
$H$ and $r_{H}$ are fixed. Then, only a finite number of $W_{\underline{r}}$
is non-empty, hence $\\{W_{\underline{r}}|W_{\underline{r}}\neq\emptyset\\}$
is a finite collection.

We claim that each $W_{\underline{r}}$ is closed; since the collection is
finite and the $W_{\underline{r}}$’s are pairwise disjoint, this implies that
each $W_{\underline{r}}$ is open and closed, and hence union of connected
components.

To prove that $W_{\underline{r}}$ is closed, let $T$ be any $k$-scheme and let
$\mathcal{F}$ be any family of torsion-free sheaves of polarized rank $1$ on
$X\times_{k}T\xrightarrow{\pi}T$. Suppose that $\mathcal{F}$ has generically
multirank ${\underline{r}}$ on $X$, meaning that
$\operatorname{rk}_{X_{i}}(\mathcal{F}_{|\pi^{-1}(\eta)})=r_{i}$ for each
irreducible component $X_{i}$ of $X$ and any generic point $\eta$ of $T$. Let
$t$ be any point in the closure of $\\{\eta\\}$. Since the length is upper-
semicontinuous, the rank of $\mathcal{F}_{|\pi^{-1}(t)}$ at each $X_{i}$ is a
rational number $r_{i}(t)$ greater or equal than $r_{i}$. On the other hand,
$\mathcal{F}_{|\pi^{-1}(t)}$ have polarized rank $1$; hence, thanks to Theorem
1.1.13, we can write:

$\displaystyle 1=r_{H}(\mathcal{F})$
$\displaystyle=\frac{\sum_{i=1}^{s}r_{i}(t)\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}{\sum_{i=1}^{n}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}\geq$
$\displaystyle\geq\frac{\sum_{i=1}^{s}r_{i}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}{\sum_{i=1}^{n}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}=1.$

By difference we obtain:

$\frac{\sum_{i=1}^{s}(r_{i}(t)-r_{i})\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}{\sum_{i=1}^{n}\operatorname{mult}_{X}(C_{i})\deg_{C_{i}}H}=0.$

Since all the terms of the expression are non-negative, the only possibility
is that $r_{i}(t)=r_{i}$ for each $i$. We conclude that the whole family
$\mathcal{F}$ belongs to $W_{\underline{r}}$. ∎

We come now to torsion-free sheaves with well-defined rank 1 on $X$.

######  Definition/Lemma 1.2.6.

Let $d\in\mathbb{Z}$ be an integer number. The compactified Jacobian stack of
degree $d$ on $X$ is the algebraic stack $\overline{\mathbb{J}}(X,d)$ such
that for any $k$-scheme $T$, $\overline{\mathbb{J}}(X,d)(T)$ is the groupoid
of $T$-flat coherent sheaves on $X\times_{k}T$ whose fibers over $T$ are
torsion-free sheaves of rank 1 and degree $d$ on $X\simeq X\times_{k}\\{t\\}$.

Let $H$ be a polarization on $X$. The compactified Jacobian scheme of degree
$d$ is the good moduli space $\overline{J}(X,H,d)$ representing S-equivalence
classes of $H$-semistable torsion-free sheaves of rank 1 and degree $d$ on
$X$.

The union of the compactified Jacobians of all degrees is denoted
$\overline{\mathbb{J}}(X)$ (resp. $\overline{J}(X,H)$).

###### Proof.

The property of having rank $1$ on $X$ implies that the polarized rank is
equal to $1$ for any polarization; moreover, the degree is well defined and
equal to the polarized degree for any polarization. Fix any polarization $H$
on $X$. Then, $\overline{\mathbb{J}}(X,d)$ can be seen as the open and closed
substack $W_{(1,\dots,1)}$ of $\overline{\mathbb{J}}_{tf}(X,H,d)$ (Lemma
1.2.5), and hence is algebraic. The same argument holds for
$\overline{J}(X,H,d)$, with the difference that the notion of semi-stability
strictly depends on the choice of $H$. ∎

The following proposition shows that $\overline{J}(X,H,d)$ is actually a
compactification of $J^{d}(X)$.

######  Proposition 1.2.7.

For any polarization $H$ on $X$, $\overline{J}(X,H,d)$ is a projective scheme,
union of connected components of $\overline{J}_{tf}(X,H,d)$, and contains
$J^{d}(X)$ as an open subscheme.

###### Proof.

Since $\overline{\mathbb{J}}(X,d)$ is an open and closed substack of
$\overline{\mathbb{J}}_{tf}(X,H,d)$, then $\overline{J}(X,H,d)$ is an open and
closed subscheme of $\overline{J}_{tf}(X,H,d)$ and union of connected
components. It is non-empty, as it contains the structure sheaf
$\mathcal{O}_{X}$. Moreover $\overline{J}_{tf}(X,H,d)$ is projective, hence
$\overline{J}(X,H,d)$ is projective too. Finally, the condition of being
locally free is open and implies both stability and rank 1; hence, $J^{d}(X)$
is an open subscheme of $\overline{J}(X,H,d)$. ∎

We come now to the last moduli space, which parametrizes generalized line
bundles.

######  Definition/Lemma 1.2.8.

Let $d\in\mathbb{Z}$ be an integer number. The generalized Jacobian stack of
degree $d$ on $X$ is the algebraic stack
$\operatorname{\textit{G}\mathbb{J}}(X,d)$ such that, for any $k$-scheme $T$,
$\operatorname{\textit{G}\mathbb{J}}(X,d)(T)$ is the groupoid of $T$-flat
coherent sheaves on $X\times_{k}T$ whose fibers over $T$ are generalized line
bundles of degree $d$ on $X\simeq X\times_{k}\\{t\\}$.

Let $H$ be a polarization on $X$. The generalized Jacobian scheme of degree
$d$ is the good moduli space $\operatorname{\textit{GJ}}(X,H,d)$ representing
S-equivalence classes of $H$-semistable generalized line bundles of degree $d$
on $X$.

The union of the generalized Jacobians of all degrees is denoted
$\operatorname{\textit{G}\mathbb{J}}(X)$ (resp.
$\operatorname{\textit{GJ}}(X,H)$).

###### Proof.

Generalized line bundles are torsion-free sheaves of rank 1; moreover, the
condition of being generically locally free is open, hence
$\operatorname{\textit{G}\mathbb{J}}(X,d)$ can be seen as an open substack of
$\overline{\mathbb{J}}(X,d)$. In particular,
$\operatorname{\textit{G}\mathbb{J}}(X,d)$ is an algebraic stack. Similarly,
$\operatorname{\textit{GJ}}(X,H,d)$ is an open subscheme of
$\overline{J}(X,H,d)$. ∎

To sum up, we defined a number of moduli spaces satisfying the following chain
of inclusions:

$J^{d}(X)\stackrel{{\scriptstyle(1)}}{{\subseteq}}\operatorname{\textit{G}\mathbb{J}}(X,d)\stackrel{{\scriptstyle(2)}}{{\subseteq}}\overline{\mathbb{J}}(X,d)\stackrel{{\scriptstyle(3)}}{{\subseteq}}\overline{\mathbb{J}}_{tf}(X,H,d),$

and:

$J^{d}(X)\stackrel{{\scriptstyle(1^{\prime})}}{{\subseteq}}\operatorname{\textit{GJ}}(X,H,d)\stackrel{{\scriptstyle(2^{\prime})}}{{\subseteq}}\overline{J}(X,H,d)\stackrel{{\scriptstyle(3^{\prime})}}{{\subseteq}}\overline{J}_{tf}(X,H,d).$

######  Remark 1.2.9.

Inclusions $(1)$ and $(2)$ (resp. $(1^{\prime})$ and $(2^{\prime})$) above are
open embeddings; inclusion $(3)$ (resp. $(3^{\prime})$) is an open and closed
embedding. In general they are strict and not dense, in the sense that for any
of them there exists a (possibly non-reduced or reducible) curve $X$ such that
the closure of the smaller space in the bigger one is contained strictly.
Anyway, when $X$ satisfies additional conditions, some of the inclusions above
are actually equalities.

  * •

If $X$ is irreducible, inclusion $(3)$ (resp. $(3)^{\prime}$) is an equality.

  * •

If $X$ is reduced, inclusion $(2)$ (resp. $(2^{\prime})$) is an equality.

  * •

If $X$ is integral and smooth, inclusions $(1)$, $(2)$ and $(3)$ (resp.
$(1^{\prime})$, $(2^{\prime})$ and $(3^{\prime})$) are equalities.

###  1.3. Moduli spaces of Higgs pairs

In this section, following [Sch14, Chapter 2] and [MRV17b, §10], we review the
definitions of Higgs pairs and $G$-Higgs pairs and of the fibrations involving
their moduli spaces. Throughout this section, $C$ denotes a fixed smooth curve
over the field of complex numbers and $L$ a fixed line bundle on $C$ with
degree $\ell=\deg L$.

####  1.3.1. Classical Higgs pairs

We begin with the case of classical Higgs pairs.

######  Definition 1.3.1.

A ($L$-twisted) Higgs pair (or simply a Higgs pair) is a pair $(E,\Phi)$
consisting of a vector bundle $E$ on $C$ and a section $\Phi\in
H^{0}(\Sigma,\operatorname{End}(E)\otimes L)$. The map $\Phi$ is called the
Higgs field. The degree and the rank of the Higgs pair are the degree and the
rank of the underlying bundle $E$.

######  Definition 1.3.2.

The algebraic stack $\mathcal{M}(r,d)$ of all $L$-twisted Higgs pairs
$(E,\Phi)$ on $C$ of rank $r$ and degree $d$ is endowed with a morphism,
called the Hitchin morphism, defined as:

$\displaystyle\mathcal{H}_{r,d}=\mathcal{H}:\mathcal{M}(r,d)$
$\displaystyle\rightarrow\mathcal{A}(r)=\bigoplus_{i=1}^{r}H^{0}(C,L^{i})$
$\displaystyle(E,\Phi)$
$\displaystyle\mapsto(a_{1}(E,\Phi),\dots,a_{r}(E,\Phi))$

where $L^{i}=L^{\otimes i}$,
$a_{i}(E,\Phi):=(-1)^{i}\operatorname{tr}(\wedge^{i}\Phi)$ and
$\mathcal{A}=\mathcal{A}(r)$ is called the Hitchin base.

######  Definition 1.3.3.

A vector subbundle $F\subseteq E$ such that $\Phi(F)\subseteq F\otimes L$ is
said to be a $\Phi$-invariant subbundle of $E$.

A Higgs pair $(E,\Phi)$ is said stable (respectively semi-stable) if for each
proper $\Phi$-invariant subbundle $F\subset E$ one has $\mu(F)<\mu(E)$
(respectively $\mu(F)\leq\mu(E)$).

######  Definition 1.3.4.

The good moduli space $M(r,d)$ of S-equivalence classes of semistable
$L$-twisted Higgs pairs of rank $r$ and degree $d$ is endowed with a flat
projective morphism, called the Hitchin fibration:

$\displaystyle H_{r,d}=H:M(r,d)=M$
$\displaystyle\rightarrow\mathcal{A}=\mathcal{A}(r)=\bigoplus_{i=1}^{r}H^{0}(C,L^{i})$
$\displaystyle(E,\Phi)$
$\displaystyle\mapsto(a_{1}(E,\Phi),\dots,a_{r}(E,\Phi)).$

####  1.3.2. G-Higgs pairs

Let $G$ be a complex reductive Lie group.

######  Definition 1.3.5.

A ($L$-twisted) $G$-Higgs pair is a pair $(P,\phi)$ where $P$ is a principal
$G$-bundle over $C$ and the $G$-Higgs field $\phi$ is a section of
$H^{0}(\operatorname{ad}(P)\otimes L)$.

######  Recall 1.3.6.

When $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$, to any $G$-principal
bundle $P$ is associated a vector bundle of rank $r$, defined as:

$E:=P\times_{\rho}\mathbb{C}^{r}=(P\times\mathbb{C}^{r})/\sim$

where $(s\cdot g,v)\sim(s,\rho(g)\cdot v)$ for any section $s$ of $P$ and any
vector $v\in\mathbb{C}^{r}$. In terms of cocycles, if $\\{U_{\alpha}\\}_{A}$
is a trivializing cover for $P$ on $C$ and $\\{g_{\alpha\beta}\\}$ is a
collection of transition elements, then $E$ is the vector bundle defined by
the $\rho(g_{\alpha\beta})$ as cocycles.

When, $G=\mathrm{GL}(r,\mathbb{C})$ the construction can be reversed; indeed,
given any vector bundle $E$ of rank $r$ on $C$, the frame bundle of all
ordered basis $P=\operatorname{Fr}(E)$ is a
$\mathrm{GL}(r,\mathbb{C})$-principal bundle such that $E\simeq
P\times_{\rho}\mathbb{C}^{r}$.

######  Recall 1.3.7.

Let $P$ be any $G$-principal bundle and $E=P\times_{\rho}\mathbb{C}^{r}$ as
above. The adjoint bundle $\operatorname{ad}(P)$ is defined as
$P\times_{\operatorname{ad}_{G}}\mathfrak{g}$ where
$\mathfrak{g}=\operatorname{Lie}(G)$ is the Lie algebra associated to the Lie
group $G$ and
$\operatorname{ad}_{G}:G\rightarrow\operatorname{Aut}(\mathfrak{g})$ is the
adjoint representation. If $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$,
then the embedding $\rho$ induces at the level of Lie algebras a
$G$-equivariant map
$\psi:=\operatorname{ad}\rho:\mathfrak{g}\hookrightarrow\mathfrak{gl}(r,\mathbb{C})=\operatorname{End}(\mathbb{C}^{r})$;
hence, we obtain a morphism

$\operatorname{ad}P=P\times_{\operatorname{ad}_{G}}\mathfrak{g}\xrightarrow{1\times\psi}P\times_{\psi(\operatorname{ad}_{G})}\operatorname{End}(\mathbb{C}^{r})=\operatorname{End}(E).$

When $G=\mathrm{GL}(r,\mathbb{C})$, the adjoint bundle $\operatorname{ad}P$ is
canonically isomorphic to $\operatorname{End}(E)$.

The two remarks above allows us to formulate the following:

######  Proposition 1.3.8.

Let $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$ be a complex reductive
Lie group emebedding in $\mathrm{GL}(r,\mathbb{C})$. Then, any $G$-Higgs pair
$(P,\phi)$ gives rise via the associated bundle construction to a classical
Higgs pair $(E,\Phi)$ of rank $r$, where $E=P\times_{\rho}\mathbb{C}^{r}$ and
$\Phi$ is the image of $\phi$ with respect to the morphism
$\operatorname{ad}(P)\rightarrow\operatorname{End}(E)$ induced by $\rho$.

When $G=\mathrm{GL}(r,\mathbb{C})$, $G$-Higgs pairs are in one-to-one
correspondence, via the associated bundle construction, with classical Higgs
pairs of rank $r$.

Similarly to Higgs pairs, the moduli space of $G$-Higgs pairs admits a
morphism to an affine space, which plays an important role in its study.

######  Definition 1.3.9.

Let $G$ be a complex reductive Lie group, let $\delta\in\pi_{1}(G)$ and let
$p_{1},\dots,p_{k}$ be a homogeneous basis for the algebra of invariant
polynomials of the Lie algebra $\mathfrak{g}$ of $G$, such that $p_{i}$ has
degree $d_{i}$. Then, the algebraic moduli stack $\mathcal{M}_{G}(\delta)$ of
all $L$-twisted $G$-Higgs pairs $(P,\phi)$ such that $P$ has topological type
$\delta$ is endowed with a morphism, called the $G$-Hitchin morphism, defined
as:

$\displaystyle\mathcal{H}_{G,\delta}^{p_{1},\dots,p_{k}}=\mathcal{H}_{G,\delta}:\mathcal{M}_{G}(\delta)$
$\displaystyle\rightarrow\mathcal{A}_{G}=\bigoplus_{i=1}^{k}H^{0}(C,L^{d_{i}})$
$\displaystyle(P,\phi)$ $\displaystyle\mapsto(p_{1}(\phi),\dots,p_{k}(\phi)).$

When $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$, the associated bundle
construction induces a morphism of algebraic stack:

$\mathcal{M}_{G}(\delta)\xrightarrow{\alpha}\mathcal{M}(r,d).$

where $d$ corresponds to $\delta$ by the canonical map
$\pi_{1}(G)\xrightarrow{\pi_{1}(\rho)}\pi_{1}(\mathrm{GL}(r,\mathbb{C}))$.

######  Definition 1.3.10.

Let $G\xhookrightarrow{\rho}\mathrm{GL}(r,\mathbb{C})$. We denote as

$\mathcal{M}_{G}(r,d):=\coprod_{\pi_{1}(\rho)(\delta)=d}\mathcal{M}_{G}(\delta)$

the moduli space of $G$-Higgs pairs whose associated vector bundle is of rank
$r$ and degree $d$. The $G$-Hitchin morphism on $\mathcal{M}_{G}(r,d)$ is
denoted as $\mathcal{H}_{G,r,d}$. When $G=\mathrm{GL}(r,\mathbb{C})$, the
moduli space of $\mathrm{GL}$-Higgs pairs $\mathcal{M}_{\mathrm{GL}}(r,d)$ is
isomorphic via the associated bundle construction to the moduli space
$\mathcal{M}(r,d)$ of classical Higgs pairs of rank $r$ and degree $d$.

######  Remark 1.3.11.

For classical groups like $\mathrm{SL},\mathrm{Sp},\mathrm{SO}$, the basis
$p_{1},\dots,p_{k}$ can be chosen such that there is a commutative diagram:

${\mathcal{M}_{G}(r,d)}$${\mathcal{M}(r,d)}$${\mathcal{A}_{G}(r)}$${\subseteq}$${\mathcal{A}(r).}$$\scriptstyle{\alpha}$$\scriptstyle{\mathcal{H}_{G,r,d}}$$\scriptstyle{\mathcal{H}_{r,d}}$

Finally, the notion of stability can be defined for $G$-Higgs pairs, such that
it is compatible with the notion of stability for the associated Higgs pairs
when $G\subseteq\mathrm{GL}(r,\mathbb{C})$ (see [Sch13, Definition 2.4] for
details). Then, we have the following:

######  Definition 1.3.12.

Let $G$ be a complex reductive Lie group, let $\delta\in\pi_{1}(G)$ and denote
with $p_{i}$ for $i=1,\dots,k$ a homogeneous basis for the algebra of
invariant polynomials of the Lie algebra $\mathfrak{g}$ of $G$, with degrees
$d_{i}$. The good moduli space $M_{G}(\delta)$ of S-equivalence classes of
semistable $G$-Higgs pairs $(P,\phi)$ such that $P$ has topological type
$\delta$ is endowed with a flat projective morphism, called the $G$-Hitchin
fibration:

$\displaystyle H_{G,\delta}^{p_{1},\dots,p_{k}}=H_{G,\delta}:M_{G}(\delta)$
$\displaystyle\rightarrow\mathcal{A}_{G}=\bigoplus_{i=1}^{k}H^{0}(C,L^{d_{i}})$
$\displaystyle(P,\phi)$ $\displaystyle\mapsto(p_{1}(\phi),\dots,p_{k}(\phi)).$

###  1.4. The spectral correspondence for Higgs pairs

In this section, following [MRV17b], [CL16] and [dC17], we resume known facts
about the spectral correspondence, which describes any fiber of the Hitchin
morphism $\mathcal{H}$ in terms of torsion-free rank-1 sheaves on an
associated spectral curve.

Let ${\underline{a}}\in\mathcal{A}(r)$ be any characteristic. The spectral
curve $X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$ is the projective
scheme defined in the total space of $L$, $P=\mathbb{P}(\mathcal{O}_{C}\oplus
L^{-1})\xrightarrow{p}C$, by the homogeneous equation

$x^{r}+p^{*}(a_{1})x^{r-1}y+\dots+p^{*}(a_{r})y^{r}=0$

where $x$ is the section of $\mathcal{O}_{P}(1)\otimes p^{*}(L)$ whose
pushforward via $p$ corresponds to the constant section $(0,1)$ of
$L\oplus\mathcal{O}_{C}$ and $y$ is the section of $\mathcal{O}_{P}(1)$ whose
pushforward via $p$ corresponds to the constant section $(1,0)$ of
$\mathcal{O}_{C}\oplus L^{-1}$. Note that the restriction of $y$ to
$X_{\underline{a}}$ is everywhere non-zero and hence the restriction of
$\mathcal{O}_{P}(1)$ to $X_{\underline{a}}$ is trivial. Moreover, the
restriction of $x$ to $X_{\underline{a}}$ can be considered as a section of
$\pi_{\underline{a}}^{*}(L)$.

We can now compute the canonical sheaf of $X_{\underline{a}}$.

######  Lemma 1.4.1.

(Canonical sheaf of the spectral curve) The canonical sheaf of
$X_{\underline{a}}$ is equal to

$\omega_{X_{\underline{a}}}=\pi_{\underline{a}}^{*}(\omega_{c}\otimes
L^{r-1}).$

###### Proof.

By [Har77, Chap. V, Lemma 2.10] the canonical sheaf of $P$ is equal to

$\omega_{P}=\mathcal{O}_{P}(-2)\otimes p^{*}(\omega_{C}\otimes L^{-1})$

The canonical sheaf of $X_{\underline{a}}$ can be computed with the adjunction
formula

$\omega_{X_{\underline{a}}}=\left(\omega_{P}(X_{\underline{a}})\right)_{|X_{\underline{a}}}=\left(\mathcal{O}_{P}(r-2)+p^{*}(\omega_{c}\otimes
L^{r-1})\right)_{|X_{\underline{a}}}.$

Since $\mathcal{O}_{P}(1)_{|X_{\underline{a}}}$ is trivial, we conclude that

$\omega_{X_{\underline{a}}}=\pi_{\underline{a}}^{*}(\omega_{c}\otimes
L^{r-1}).$

∎

######  Corollary 1.4.2.

Denote with $g$ the genus of $C$ and with $\ell$ the degree of $L$ on $C$.
Then

$\chi(X_{\underline{a}})=r(1-g)-\frac{r(r-1)}{2}\ell.$

###### Proof.

The pullback of a line bundle by a finite map has degree equal to the degree
of the bundle times the degree of the morphism. Then

$\deg(\omega_{X_{\underline{a}}})=r\deg(\omega_{C})+r(r-1)\deg(L)=2r(g-1)+r(r-1)\ell.$

The statement follows then by the fact that
$\chi(X_{\underline{a}})=-\deg(\omega_{X_{\underline{a}}})/2$. ∎

######  Remark 1.4.3.

The spectral curve $X_{\underline{a}}$ has pure dimension 1 and is embedded in
the smooth surface $P$, hence it has only locally planar singularities. The
Hitchin base $\mathcal{A}$ admits three notable open subsets
$\mathcal{A}^{\textrm{sm}}\subseteq\mathcal{A}^{\textrm{ell}}\subseteq\mathcal{A}^{\textrm{reg}}\subseteq\mathcal{A}$,
called respectively the smooth locus, the elliptic locus and the regular
locus, defined as:

$\displaystyle\mathcal{A}^{\textrm{sm}}$
$\displaystyle:=\\{{\underline{a}}\in\mathcal{A}:X_{\underline{a}}\textrm{ is
smooth and connected}\\},$ $\displaystyle\mathcal{A}^{\textrm{ell}}$
$\displaystyle:=\\{{\underline{a}}\in\mathcal{A}:X_{\underline{a}}\textrm{ is
integral}\\},$ $\displaystyle\mathcal{A}^{\textrm{reg}}$
$\displaystyle:=\\{{\underline{a}}\in\mathcal{A}:X_{\underline{a}}\textrm{ is
reduced and connected}\\}.$

We come now to the spectral correspondence for Higgs pairs. The following
spectral correspondence is due to [Sch98], [CL16] and [dC17], as a
generalization of the spectral correspondence stated in [BNR89] for smooth
characteristics.

######  Proposition 1.4.4.

(Spectral correspondence) Let ${\underline{a}}\in\mathcal{A}(r)$ be any
characteristic and let $X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$
be the associated spectral curve. Let $\mathcal{O}_{C}(1)$ be an ample line
bundle on $C$; let
$\mathcal{O}_{X_{\underline{a}}}(1):=\pi_{\underline{a}}^{*}\big{(}\mathcal{O}_{C}(1)\big{)}$
be the ample line bundle on $X_{\underline{a}}$ obtained by pullback and
denote with $H$ the associated polarization on $X_{\underline{a}}$.

  1. (1)

For any integer $d$, there is an isomorphism of stacks:

$\ \mathcal{H}_{r,d}^{-1}({\underline{a}})\xrightarrow[\Pi]{\hskip
10.00002pt{}_{\sim}\hskip
10.00002pt}\overline{\mathbb{J}}(X_{\underline{a}},d^{\prime}),$

where
$d^{\prime}=d+r(1-g)-\chi(\mathcal{O}_{X_{\underline{a}}})=d+\frac{r(r-1)}{2}\ell$
and $g=g(C)$ is the genus of $C$. If $\mathcal{M}$ is a torsion-free sheaf of
rank 1 on $X_{\underline{a}}$, then $\Pi(\mathcal{M}):=(E,\Phi)$ where

$\displaystyle E$ $\displaystyle=\pi_{{\underline{a}},*}(\mathcal{M})$
$\displaystyle\Phi$ $\displaystyle=\pi_{{\underline{a}},*}(\cdot
x):\pi_{{\underline{a}},*}(\mathcal{M})\rightarrow
L\otimes\pi_{{\underline{a}},*}(\mathcal{M})\simeq\pi_{{\underline{a}},*}(\pi_{\underline{a}}^{*}L\otimes\mathcal{M}).$

  2. (2)

A torsion-free sheaf of rank 1 $\mathcal{M}$ on $X_{\underline{a}}$ is
$H$-semistable if and only if the associated Higgs pair $\Pi(\mathcal{M})$ is
semistable on $C$. Hence, the above correspondence yields an isomorphism of
schemes:

$H_{r,d}^{-1}({\underline{a}})\simeq\overline{J}(X_{\underline{a}},H,d^{\prime}).$

Finally, in the following proposition, inspired by [BNR89, Remark 3.7], we
show that the torsion-free sheaf $\mathcal{M}$ corresponding to an Higgs pair
$(E,\Phi)$ can be always resumed, up to twisting, as the eigenspace of
$\pi^{*}\Phi$ with eigenvalue $x$.

######  Proposition 1.4.5.

Let ${\underline{a}}\in\mathcal{A}(r)$ be any characteristic and let
$X_{\underline{a}}\xrightarrow{\pi_{\underline{a}}}C$ be the associated
spectral curve. Let
$\mathcal{M}\in\overline{\mathbb{J}}(X_{\underline{a}},d^{\prime})$ be a
torsion-free rank-1 sheaf on $X_{\underline{a}}$ corresponding to the Higgs
pair $(E,\Phi)\in\mathcal{H}_{r,d}^{-1}({\underline{a}})$ on $C$. Then, the
following exact sequence holds:

$0\rightarrow\mathcal{M}\otimes\pi_{\underline{a}}^{*}(L^{1-r})\rightarrow\pi_{\underline{a}}^{*}E\xrightarrow{\pi_{\underline{a}}^{*}(\Phi)-x}\pi_{\underline{a}}^{*}E\otimes\pi_{\underline{a}}^{*}L\xrightarrow{\operatorname{ev}}\mathcal{M}\otimes\pi_{\underline{a}}^{*}L\rightarrow
0$

where $\operatorname{ev}$ is induced by the evaluation map
$\pi_{\underline{a}}^{*}\pi_{{\underline{a}},*}(\mathcal{M})\rightarrow\mathcal{M}$.

###### Proof.

We first prove the exact sequence locally. Let
$U=\operatorname{Spec}R\subseteq C$ be any open affine subset where $L$ is
trivial. The preimage $V=\pi^{-1}(U)$ is then equal to
$\operatorname{Spec}(R[x]/P(x))$ where
$P(x)=x^{r}+a_{1}x^{r-1}+\dots+a_{r-1}x+a_{r}$ and the $a_{i}$’s denote (with
a slight abuse of notation) the elements in $R$ corresponding to the entries
of ${\underline{a}}$ after choosing the trivialization of $L$. Denote
$R[x]/P(x)=S$ and note that the inclusion $R\subset S$ makes $S$ into an
$R$-module with basis $1,x,\dots,x^{r-1}$.

Recall that $E=\pi_{{\underline{a}},*}(\mathcal{M})$ and
$\Phi=\pi_{{\underline{a}},*}(\cdot x)$. Denote with $M$ the $S$-module
corresponding to $\mathcal{M}$ on $V=\operatorname{Spec}S$. The pushforward
$\pi_{{\underline{a}},*}(\mathcal{M})$ corresponds on $V$ to the restriction
of scalars of $M$ from $S$ to $R$. Then, $\pi_{\underline{a}}^{*}E$ correspond
on $V$ to the $S$-module $S\otimes_{R}M$, with the structure of $S$-module
given by multiplication on the left-hand side of the tensor product. On the
other hand, any element $p(x)\in S$ acts on $S\otimes_{R}M$ also on the right-
hand side. We denote such action as

$\displaystyle\overline{p(x)}:S\otimes_{R}M$ $\displaystyle\longrightarrow
S\otimes_{R}M$ $\displaystyle q(x)\otimes m$ $\displaystyle\longmapsto
q(x)\otimes p(x)\cdot m.$

Regarding the evaluation map, it can be defined at the level of $S$-modules as

$\displaystyle\operatorname{ev}:S\otimes_{R}M$ $\displaystyle\longrightarrow
M$ $\displaystyle p(x)\otimes m$ $\displaystyle\longmapsto p(x)\cdot m.$

With this notation, we are left to prove that the following exact sequence of
$S$-modules holds:

$0\rightarrow M\rightarrow
S\otimes_{R}M\xrightarrow{\Psi=\overline{x}-x}S\otimes_{R}M\xrightarrow{\operatorname{ev}}M\rightarrow
0.$

The proof of the exactness on the right follows exactly the proof of [Bou89,
III 10, Proposition 18], with the difference that $S\otimes_{R}M$ has a finite
$S$-basis.

To prove the exactness on the left, first consider the following morphism of
$R$-modules:

$\displaystyle
Q=\sum_{i=0}^{r-1}x^{i}\sum_{j=0}^{r-1-i}\overline{x}^{r-1-i-j}\overline{a}_{j}:M$
$\displaystyle\longrightarrow S\otimes_{R}M$ $\displaystyle m$
$\displaystyle\longmapsto\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot
m\right)$

where we put $a_{0}=1$. We prove first that $\Psi\circ Q=0$. Let $m\in M$ be
any element. If we compute $x\cdot Q(m)$, recalling that
$x^{r}=-\sum_{i=0}^{r-1}a_{r-i}x^{i}$, we obtain

$\displaystyle x\cdot Q(m)$
$\displaystyle=x\cdot\left(\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot
m\right)\right)=$
$\displaystyle=\sum_{i=0}^{r-2}x^{i+1}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot
m\right)-\sum_{i=0}^{r-1}\left(a_{r-i}x^{i}\right)\otimes m.$

Changing $i+1$ with $i$ in the first sum and moving the $a_{r-i}$ by
linearity, we obtain

$\displaystyle x\cdot Q(m)$
$\displaystyle=\sum_{i=1}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-i}(x^{r-i-j}a_{j})\cdot
m\right)-\sum_{i=0}^{r-1}x^{i}\otimes(a_{r-i}\cdot m)=$
$\displaystyle=\sum_{i=1}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-i-1}x^{r-i-j}a_{j}+a_{r-i}\right)\cdot
m-1\otimes(a_{r}\cdot m)=$
$\displaystyle=\sum_{i=1}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-i-1}(x^{r-i-j}a_{j})\cdot
m\right)-1\otimes(a_{r}\cdot m).$

Finally, recalling that $-a_{r}=\sum_{j=0}^{r-1}x^{r-j}a_{j}$, we can write

$x\cdot
Q(m)=\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-i-1}(x^{r-i-j}a_{j})\cdot
m\right).$

On the other hand, the computation of $\overline{x}(Q(m))$ gives

$\displaystyle\overline{x}(Q(m))$
$\displaystyle=\overline{x}\left(\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot
m\right)\right)=$
$\displaystyle=\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-i-j}a_{j})\cdot
m\right).$

We conclude that $\Psi(Q(m))=\overline{x}(Q(m))-x\cdot Q(m)=0$ for arbitrary
$m\in M$; then $\Psi\circ Q=0$.

We can now prove that $Q$ is a morphism of $S$-modules. Since $Q$ is
$R$-linear by definition, it remains only to show that $Q(x\cdot m)=x\cdot
Q(m)$; but it is clear by the definition of $\overline{x}$ that $Q(x\cdot
m)=\overline{x}(Q(m))$, which is equal to $x\cdot Q(m)$ by the previous
computation.

Note also that $Q$ is injective. Indeed, after choosing the monomials
$1,x,\dots,x^{r-1}$ as a $R$-basis for $S$, we have a canonical isomorphism
$S\simeq\bigoplus_{i=0}^{r-1}Rx^{i}$ that induces an isomorphism
$S\otimes_{R}M\simeq\bigoplus_{i=0}^{r-1}(x^{i}\otimes M)$. To see that $Q$ is
injective, let $m\in M$ be any element such that $m\neq 0$. Then, the
component of degree $r-1$ of $Q(m)$ is $x^{r-1}\otimes m$, which is different
from $0$; we conclude that $Q(m)\neq 0$.

We are left to prove that $\ker(\Psi)\subseteq\operatorname{Im}(Q)$. Let
$z=\sum_{i=0}^{r-1}x^{i}\otimes m_{i}$ be any element in $S\otimes_{R}M$ such
that $\Psi(z)=0$ or, equivalently,

$\sum_{i=0}^{r-1}x^{i+1}\otimes m_{i}=\sum_{i=0}^{r-1}x^{i}\otimes x\cdot
m_{i}.$

Recalling that $x^{r}=-\sum_{i=0}^{r-1}a_{r-i}x^{i}$, by $R$-linearity we
obtain

(1.1) $\displaystyle\sum_{i=1}^{r-1}x^{i}\otimes(m_{i-1}-a_{r-i}\cdot
m_{r-1})-a_{r}\otimes m_{r-1}=\sum_{i=0}^{r-1}x^{i}\otimes x\cdot m_{i}.$

Let $m:=m_{r-1}$. Looking at the components in degree $r-1$ of Equation 1.1,
we obtain

$-a_{1}\cdot m+m_{r-2}=x\cdot m,$

or, explicitely,

$m_{r-2}=(x+a_{1})\cdot m=\sum_{j=0}^{1}(x^{1-j}a_{j})\cdot m$

Proceeding by induction on $i$ decreasing from $r-2$ to $1$, suppose that the
equality

$m_{i}=\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot m$

is proven. Then, looking at the components in degree $i$ of Equation 1.1, we
obtain:

$m_{i-1}-a_{r-i}\cdot m=x\cdot m_{i},$

hence:

$\displaystyle m_{i-1}$
$\displaystyle=x\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot m+a_{r-i}\cdot m=$
$\displaystyle=\sum_{j=0}^{r-i}(x^{r-i-j}a_{j})\cdot m.$

Hence, we conclude that $m_{i}=\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot m$
for any $i$ from $0$ to $r-1$; in other terms:

$z=\sum_{i=0}^{r-1}x^{i}\otimes\left(\sum_{j=0}^{r-1-i}(x^{r-1-i-j}a_{j})\cdot
m\right)=Q(m)\in\operatorname{Im}(Q).$

Finally, note that the map
$Q=\sum_{i=0}^{r-1}x^{i}\sum_{j=0}^{r-1-i}\overline{x}^{r-1-i-j}\overline{a}_{j}$
has been defined at the level of $S$-modules, so it defines only a local map
of sheaves on $V$. In order to define a global map of sheaves, note that any
power of $x$ and $\overline{x}$, considered as global maps, introduces a twist
by $\pi_{\underline{a}}^{*}L$ up to the same tensor power, and any
$\overline{a}_{j}$ introduces a twist by $\pi_{\underline{a}}^{*}(L^{\otimes
j})$. The sum of these twistings in the definition of $Q$, for any $i$ and $j$
in the sum, is equal to $i+(r-1-i-j)+j=r-1$. Hence, in order to extend the map
$Q$ globally, one has to consider a map

$\mathcal{Q}:\mathcal{M}\otimes\pi_{\underline{a}}^{*}(L^{1-r})\rightarrow\pi_{\underline{a}}^{*}E$

whose local expression equals the one of $Q$ on any affine open
$V=\pi_{\underline{a}}^{-1}(U)$ such that $L$ is trivial on $U$. Then, such
map fits the global exact sequence:

$0\rightarrow\mathcal{M}\otimes\pi_{\underline{a}}^{*}(L^{1-r})\xrightarrow{\mathcal{Q}}\pi_{\underline{a}}^{*}E\xrightarrow{\pi_{\underline{a}}^{*}(\Phi)-x}\pi_{\underline{a}}^{*}E\otimes\pi_{\underline{a}}^{*}L\xrightarrow{\operatorname{ev}}\mathcal{M}\otimes\pi_{\underline{a}}^{*}L\rightarrow
0.$

∎

######  Remark 1.4.6.

Proposition 1.4.5 suggests that the spectral correspondence stated in [HP12,
Proposition 6.1] is somehow misstated. Indeed, by definition of the spectral
curve, the morphism $\pi_{\underline{a}}^{*}\Phi-x$ has non-trivial kernel at
each point of $X_{\underline{a}}$. In particular, any spectral data
$\mathcal{M}$ fitting in the exact sequence above needs to be supported on the
whole $X_{\underline{a}}$. This is in general not true for torsion-free
sheaves of polarized rank equal to 1, even for connected planar curves, as
showed in [Lop04, §2].

##  Chapter 2 The Norm map on the compactified Jacobian

In this chapter we study how to generalize the Norm map for line bundles to
the compactified Jacobian stack. We look first at the geometric counterpart of
a smaller problem, regarding the direct image of Cartier divisors and
generalized divisors, in order to understand how to proceed in the general
situation of torsion-free sheaves with rank 1.

Let $X\xrightarrow{\pi}Y$ be any finite, flat morphism between embeddable
noetherian schemes of pure dimension 1. In the first part of this chapter
(Sections 2.1, 2.2), we review the classical Norm map and the theory of
generalized divisors on curves, in relation with the objects introduced in
Chapter 1. In the second part (Section 2.3) we propose the definitions of the
direct and inverse image for generalized divisors and generalized line
bundles, and we study their properties. In the third part of the chapter
(Sections 2.4 and 2.5), restricting to the case when $X$ and $Y$ are
projective curves over a field, we consider the same notions for families. On
one hand, families of effective generalized divisors on a curve $X$ are
essentially families of subschemes of finite length, hence they are
parametrized by the Hilbert scheme. On the other hand, families of generalized
line bundles on $X$ are parametrized by the generalized Jacobian
$\operatorname{\textit{G}\mathbb{J}}(X)$, and in this case the direct image
map is called Norm map in analogy with the Norm map defined between the
Jacobians of the curves. In both cases, we show that giving the definitions of
the direct image and the Norm map on such moduli spaces is possible only when
the curve $Y$ is smooth. Finally, we show that the Norm map can be extended
under the same hypothesis to the compactified Jacobian
$\overline{\mathbb{J}}(X)$.

Notation. In the rest of the chapter, in the absence of further
specifications, by curve we refer to a noetherian scheme of pure dimension 1
which is embeddable (i.e. it can be embedded as a closed subscheme of a
regular scheme). This implies that the canonical (or dualizing) sheaf $\omega$
of the curve is well defined.

###  2.1. Review of the Norm map

We resume now the definition and properties of the direct and inverse image
for Cartier divisors and the Norm map for line bundles, associated to a
finite, flat morphism between curves. For a complete treatment, the standard
reference is [Gro67, §21] together with [Gro61, §6.5]. We start with the
definition of the norm at the level of sheaves of algebras.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism between curves of degree
$n$. Since $Y$ is noetherian, this is equivalent to require that $f$ is finite
and locally free, i.e. that $\pi_{*}\mathcal{O}_{X}$ is a locally free
$\mathcal{O}_{Y}$-algebra [Sta19, https://stacks.math.columbia.edu/tag/02K9Tag
02K9].

######  Definition 2.1.1.

The sheaf $\pi_{*}\mathcal{O}_{X}$ is endowed with a homomorphism of
$\mathcal{O}_{Y}$-modules, called the norm and defined on local sections by:

$\displaystyle\mathcal{N}_{Y/X}:\hskip 20.00003pt\pi_{*}\mathcal{O}_{X}$
$\displaystyle\longrightarrow\mathcal{O}_{Y}$ $\displaystyle s$
$\displaystyle\longmapsto\det(\cdot s)$

where $\cdot s:\pi_{*}\mathcal{O}_{X}\rightarrow\pi_{*}\mathcal{O}_{X}$ is the
multiplication map induced by $s$ and $\det(\cdot s)$ is given locally by the
determinant of the matrix with entries in $\mathcal{O}_{Y}$ associated to
$\cdot s$.

By the standard properties of determinants, for local sections $s,s^{\prime}$
of $\pi_{*}\mathcal{O}_{X}$ and any section $\mu$ of $\mathcal{O}_{Y}$, we
have:

(2.1) $\displaystyle\mathcal{N}_{Y/X}(s\cdot
s^{\prime})=\mathcal{N}_{Y/X}(s)\cdot\mathcal{N}_{Y/X}(s^{\prime}),\hskip
30.00005pt\mathcal{N}_{Y/X}(\mu s)=\mu^{n}\mathcal{N}_{Y/X}(s).$

Before giving the definitions of the present section, we need a technical
lemma.

######  Lemma 2.1.2.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism of degree $n$ between
curves and let $\mathcal{L}$ be an invertible $\mathcal{O}_{X}$-module. Then,
$\pi_{*}\mathcal{L}$ is an invertible $\pi_{*}\mathcal{O}_{X}$-module and
there exists an open affine cover $\\{V_{i}\\}_{i\in I}$ of $Y$ s.t.
$\pi_{*}\mathcal{O}_{X}$ is trivial on each $V_{i}$ and $\pi_{*}\mathcal{L}$
is trivial both as a $\pi_{*}\mathcal{O}_{X}$-module and as an
$\mathcal{O}_{Y}$-module on each $V_{i}$.

###### Proof.

By [Sta19, https://stacks.math.columbia.edu/tag/02K9Tag 02K9],
$\pi_{*}\mathcal{O}_{X}$ is a locally free $\mathcal{O}_{Y}$-module; denote
with $\\{W_{\alpha}\\}_{\alpha\in A}$ an open affine cover such that
${\pi_{*}\mathcal{O}_{X}}_{|W_{\alpha}}\simeq{\mathcal{O}_{Y}^{n}}_{|W_{\alpha}}$
for each $\alpha\in A$. By [Gro61, Proposition 6.1.12], $\pi_{*}\mathcal{L}$
is an invertible $\pi_{*}\mathcal{O}_{X}$-module; denote with
$\\{W^{\prime}_{\beta}\\}_{\beta\in B}$ an open affine cover such that
${\pi_{*}\mathcal{L}}_{|W^{\prime}_{\beta}}\simeq{\pi_{*}\mathcal{O}_{X}}_{|W^{\prime}_{\beta}}$
for each $\beta\in B$. Let $\\{V_{\alpha,\beta}=W_{\alpha}\cap
W^{\prime}_{\beta}\\}_{(\alpha,\beta)\in A\times B}$ be a common refinement.
Then, for $I=A\times B$, $\\{V_{i}\\}_{i\in I}$ is an open affine cover of $Y$
such that $\pi_{*}\mathcal{O}_{X}$ is trivial on each open of the cover and
$\pi_{*}\mathcal{L}$ is trivial both as $\pi_{*}\mathcal{O}_{X}$-module and as
$\mathcal{O}_{Y}$-module on each open. ∎

####  2.1.1. Direct and inverse image of Cartier divisors

We recall now the definitions of direct and inverse image for Cartier
divisors. For any curve $X$, denote with $\mathcal{K}_{X}$ the sheaf of total
quotient rings of the curve. Recall that the set of Cartier divisors on $X$ is
the set of global sections of the quotient sheaf of multiplicative groups
$\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*}$:

$\operatorname{CDiv}(X)=\Gamma(X,\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*}).$

Although the group operation on $\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*}$ is
multiplication, the group operation on $\operatorname{CDiv}(X)$ is denoted
additively. The group of Cartier divisors of $X$ contains the subgroup
$\operatorname{Prin}(X)$ of principal divisors defined as the image of the
canonical homomorphism

$\Gamma(X,\mathcal{K}_{X}^{*})\longrightarrow\Gamma(X,\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*}).$

It is well known [Gro67, §21.2] that the set of Cartier divisors is in one-to-
one correspondence with the set of invertible fractional ideals, i.e. the set
of subsheaves $\mathcal{I}\subseteq\mathcal{K}_{X}$ that are also invertible
$\mathcal{O}_{X}$-modules [Gro67, Proposition 21.2.6]. If
$D\in\Gamma(X,\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*})$ is represented by an
open cover $\\{U_{i}\\}_{i\in I}$ and a collection of sections
$f_{i}\in\Gamma(U_{i},\mathcal{K}_{X}^{*})$ such that
$f_{i}/f_{j}\in\Gamma(U_{i}\cap U_{j},\mathcal{O}_{X}^{*})$, the corresponding
fractional ideal $\mathcal{I}_{D}$ is the sub $\mathcal{O}_{X}$-module of
$\mathcal{K}_{X}$ equal to $\mathcal{O}_{X|U_{i}}\cdot f_{i}$ on any
$U_{i}$.111Differently from Grothendieck’s EGA, we pick $f_{i}$ instead of
$f_{i}^{-1}$. This does not affect the other results. Under this
correspondence, the sum of Cartier divisors corresponds to the multiplication
of fractional ideals.

######  Definition/Lemma 2.1.3.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism between curves and let
$\pi^{\sharp}:\mathcal{O}_{Y}\rightarrow\pi_{*}\mathcal{O}_{X}$ be the
associated canonical map of sheaves of modules.

Let $D\in\operatorname{CDiv}(X)$ be a Cartier divisor on $X$ corresponding to
the invertible fractional ideal $\mathcal{I}$ and let $\\{V_{i}\\}_{i\in I}$
be an affine cover as in Lemma 2.1.2. Then, on each $V_{i}$,
$\pi_{*}\mathcal{I}_{|V_{i}}$ is equal to the subsheaf
$h_{i}\cdot(\pi_{*}\mathcal{O}_{X})_{|V_{i}}$ of
$(\pi_{*}\mathcal{K}_{X})_{|V_{i}}$ generated by a meromorphic regular section
$h_{i}=f_{i}/g_{i}$, with
$f_{i},g_{i}\in\Gamma(V_{i},\pi_{*}\mathcal{O}_{X}^{*})$. The direct image of
$D$, denoted $\pi_{*}(D)$, is the Cartier divisor on $Y$ corresponding to the
fractional ideal generated on any $V_{i}$ by the meromorphic regular section
$\mathcal{N}_{Y/X}(f_{i})/\mathcal{N}_{Y/X}(g_{i})\in\Gamma(V_{i},\mathcal{K}_{Y})$.

Let $M\in\operatorname{CDiv}(Y)$ be a Cartier divisor on $Y$ corresponding to
the invertible fractional ideal $\mathcal{J}\subseteq\mathcal{K}_{Y}$, and let
$\\{V_{i}\\}_{i\in I}$ be an affine cover of $Y$ such that, on each $V_{i}$,
$\mathcal{J}_{|V_{i}}$ is equal to the fractional ideal of
$\mathcal{O}_{Y|V_{i}}$-modules generated by a meromorphic regular section
$u_{i}=s_{i}/t_{i}$ with $s_{i},t_{i}\in\Gamma(V_{i},\mathcal{O}_{Y}^{*})$.
The inverse image of $M$, denoted $\pi^{*}(M)$, is the Cartier divisor on $X$
corresponding to the fractional ideal generated on any $U_{i}=\pi^{-1}(V_{i})$
by the meromorphic regular section
$\pi^{\sharp}(s_{i})/\pi^{\sharp}(t_{i})\in\Gamma(U_{i},\mathcal{K}_{X})$.

###### Proof.

The definition of direct image is a reformulation of the one given in [Gro67,
§21.5.5]. The fact that $\mathcal{N}_{Y/X}(f_{i})/\mathcal{N}_{Y/X}(g_{i})$ is
a regular meromorphic section follows from the discussion in [Gro67, §21.5.3];
the definition is independent of the choice of the $h_{i}$’s since the norm of
sheaves is multiplicative.

The definition of inverse image is a reformulation of [Gro67, Definition
21.4.2], obtained as a consequence of [Gro67, §21.4.3], together with the
result of [Gro67, Proposition 21.4.5] that ensures that
$\pi^{\sharp}(s_{i})/\pi^{\sharp}(t_{i})$ is regular. ∎

######  Proposition 2.1.4.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism of curves of degree $n$.
The direct and inverse image for Cartier divisors induce homorphisms of
groups:

$\displaystyle\pi_{*}:\operatorname{CDiv}(X)$
$\displaystyle\longrightarrow\operatorname{CDiv}(Y)$
$\displaystyle\pi^{*}:\operatorname{CDiv}(Y)$
$\displaystyle\longrightarrow\operatorname{CDiv}(X),$

such that $\pi_{*}\circ\pi^{*}$ is the multiplication map by $n$.

###### Proof.

The direct image is a homomorphism thanks to [Gro67, §21.5.5.1], while the
inverse image is a homomorphism as a consequence of [Gro67, Definition
21.5.5.1]. The result on $\pi_{*}\circ\pi^{*}$ is stated in [Gro67,
Proposition 21.5.6] ∎

####  2.1.2. Norm of line bundles

We come now to the definition of the Norm map for invertible
$\mathcal{O}_{X}$-modules.

######  Definition/Lemma 2.1.5.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism between curves and let
$\mathcal{L}$ be an invertible $\mathcal{O}_{X}$-module. Let
$\\{V_{i}\\}_{i\in I}$ be an affine cover of $Y$ as in Lemma 2.1.2. In
particular, there is for any $i\in I$ an isomorphism
$\lambda_{i}:(\pi_{*}\mathcal{L})_{|V_{i}}\rightarrow(\pi_{*}\mathcal{O}_{X})_{|V_{i}}$.
For any $i,j\in I$, the isomorphism
$\omega_{ij}:=\lambda_{i}\circ\lambda_{j}^{-1}$ can be interpreted as a
section of $\pi_{*}\mathcal{O}_{X}$ over $V_{i}\cap V_{j}$. The collection of
norms $\\{\mathcal{N}_{Y/X}(\omega_{ij})\\}_{i,j\in I}$ is a 1-cocycle with
values in $\mathcal{O}_{Y}^{*}$.

The cocyle $\\{\mathcal{N}_{Y/X}(\omega_{ij})\\}_{i,j\in I}$ defines up to
isomorphism an invertible $\mathcal{O}_{Y}$-module, which is called the Norm
of $\mathcal{L}$ relative to $\pi$ and is denoted as
$\operatorname{Nm}_{\pi}(\mathcal{L})$ or
$\operatorname{Nm}_{Y/X}(\mathcal{L})$.

###### Proof.

If $\mathcal{L}^{\prime}$ is an invertible $\mathcal{O}_{X}$-module isomorphic
to $\mathcal{L}$ through an isomorphism
$h:\mathcal{L}^{\prime}\rightarrow\mathcal{L}$, then a local trivialization
for $\pi_{*}\mathcal{L}^{\prime}$ over $V_{i}$ is given by
$\lambda_{i}\circ(\pi_{*}h)$. Running over all $i\in I$, the resulting
1-cocycle
$\\{\mathcal{N}_{Y/X}(\lambda_{i}\circ\pi_{*}h\circ\pi_{*}h^{-1}\circ\lambda_{j}^{-1})\\}_{i,j\in
I}$ is the same as for $\mathcal{L}$. ∎

Recall that, for any curve $X$, the Picard group of $X$ is the set
$\operatorname{Pic}(X)$ of isomorphism classes of invertible
$\mathcal{O}_{X}$-modules, endowed with the operation of tensor product.

######  Proposition 2.1.6.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism of curves of degree $n$.
The Norm and the inverse image map for line bundles induce homomorphism of
groups:

$\displaystyle\operatorname{Nm}_{\pi}:\operatorname{Pic}(X)$
$\displaystyle\longrightarrow\operatorname{Pic}(Y)$
$\displaystyle\pi^{*}:\operatorname{Pic}(Y)$
$\displaystyle\longrightarrow\operatorname{Pic}(X),$

such that $\operatorname{Nm}_{\pi}\circ\pi^{*}$ is the $n$-th tensor power.

###### Proof.

The inverse image for line bundles is a homomorphism since it commutes
obviously with tensor products. The other results follow from (2.1), using the
fact that tensor product of line bundles corresponds to multiplication at the
level of defining cocycles. ∎

######  Proposition 2.1.7.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism between curves. For any
invertible $\mathcal{O}_{X}$-modules $\mathcal{L}$, we have

(2.2)
$\displaystyle\operatorname{Nm}_{\pi}(\mathcal{L})\simeq\det(\pi_{*}\mathcal{L})\otimes\det(\pi_{*}\mathcal{O}_{X})^{-1}.$

###### Proof.

See [HP12, Corollary 3.12]. ∎

As we show in the next proposition, the direct image for Cartier divisors and
the Norm map for line bundles are related, as well as the inverse image maps.
Recall that, on any curve $X$, the Picard group is canonically isomorphic to
the group of Cartier divisors modulo the subgroup $\operatorname{Prin}(X)$ of
principal divisors. This gives rise to a canonical quotient of groups:

$\displaystyle q_{X}:\operatorname{CDiv}(X)$
$\displaystyle\longrightarrow\operatorname{CDiv}(X)/\operatorname{Prin}(X)=\operatorname{Pic}(X)$
$\displaystyle D$ $\displaystyle\longmapsto\mathcal{I}_{D}.$

######  Proposition 2.1.8.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism between curves. The direct
and inverse image for Cartier divisors are compatible via the quotient maps
$q_{X}$ and $q_{Y}$ respectively with the Norm and the inverse image map for
line bundles; i.e. the following diagrams of groups are commutative:

${\operatorname{CDiv}(X)}$${\operatorname{CDiv}(Y)}$${\operatorname{Pic}(X)}$${\operatorname{Pic}(Y)}$$\scriptstyle{\pi_{*}}$$\scriptstyle{q_{X}}$$\scriptstyle{q_{Y}}$$\scriptstyle{\operatorname{Nm}_{\pi}}$
${\operatorname{CDiv}(Y)}$${\operatorname{CDiv}(X)}$${\operatorname{Pic}(Y)}$${\operatorname{Pic}(X).}$$\scriptstyle{\pi^{*}}$$\scriptstyle{q_{Y}}$$\scriptstyle{q_{X}}$$\scriptstyle{\pi^{*}}$

###### Proof.

The commutativity of the first diagram follows from correspondence between $D$
and $\mathcal{I}_{D}$ and the definitions of $\pi_{*}$ and
$\operatorname{Nm}_{\pi}$. The second diagram is commutative as consequence of
[Gro67, Picture 21.4.2.1] together with [Gro67, Proposition 21.4.5]. ∎

We are finally interested in reviewing the Norm map for families of line
bundles.222For the sake of completeness, we note that a complete discussion
regarding families of Cartier divisors and their direct and inverse images is
done in [Gro67, §21.15].  To study families of line bundles, we assume that
$X$ and $Y$ are projective curves over a base field $k$ and that
$\pi:X\rightarrow Y$ is a finite, flat morphism of degree $n$ defined over
$k$. Families of line bundles are parametrized by the Jacobian schemes of $X$
and $Y$ (denoted $J(X)$ and $J(Y)$, see Chapter 1, Section 1.2).

######  Definition/Lemma 2.1.9.

Let $\pi:X\rightarrow Y$ be a finite, flat morphism of projective curves over
a base field $k$. For any $k$-scheme $T$, the Norm map for line bundles
associated to $\pi$ is defined on the $T$-valued points of the Jacobian of $X$
as:

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

where $\pi_{T}:X\times_{k}T\rightarrow Y\times_{k}T$ is induced by pullback
from $\pi$.

###### Proof.

We need to check that the map is well defined. Since $\mathcal{L}$ is a line
bundle on $X\times_{k}T$, its pushforward $\pi_{T,*}(\mathcal{L})$ is a
locally free sheaf of rank $n$ on $Y\times_{k}T$, as for
$\pi_{T,*}\mathcal{O}_{X\times_{k}T}$. Taking determinants produces line
bundles, so
$\det\left(\pi_{T,*}(\mathcal{L})\right)\otimes\det\left(\pi_{T,*}\mathcal{O}_{X\times_{k}T}\right)^{-1}$
is a well-defined line bundle on $Y\times_{k}T$. The whole construction is
functorial, so the above definition gives a well-defined morphism of schemes

$\operatorname{Nm}_{\pi}:\hskip 20.00003ptJ(X)\longrightarrow J(Y).$

∎

###  2.2. Review of generalized divisors

In this section, we review known facts about generalized divisors and
generalized line bundles on curves. Moreover, we compare their moduli spaces
with the compactified Jacobian.

######  Remark 2.2.1.

The theory of generalized divisors is developed by Hartshorne in his papers
[Har86], [Har94] and [Har07], in order to generalize the notion of Cartier
divisor on schemes satisfying condition $S_{2}$ of Serre. Since we are dealing
with schemes of dimension 1, the condition $S_{2}$ of Serre coincides with the
condition $S_{1}$, which in turn coincides with the fact of not having
embedded components (i.e. embedded points). In particular, any scheme of pure
dimension 1 is also $S_{1}$, and hence $S_{2}$. Similarly, a coherent sheaf on
a curve satisfies condition $S_{2}$ if and only if it is torsion-free.

####  2.2.1. Generalized divisors

######  Definition 2.2.2.

Let $X$ be a curve and let $\mathcal{K}_{X}$ be the sheaf of total quotient
rings on $X$. A generalized divisor on $X$ is a nondegenerate fractional ideal
of $\mathcal{O}_{X}$-modules, i.e. a subsheaf
$\mathcal{I}\subseteq\mathcal{K}_{X}$ that is a coherent sheaf of
$\mathcal{O}_{X}$-modules and such that
$\mathcal{I}_{\eta}=\mathcal{K}_{X,\eta}$ for any generic point $\eta\in X$.
It is effective if $\mathcal{I}\subseteq\mathcal{O}_{X}$. It is Cartier if
$\mathcal{I}$ is an invertible $\mathcal{O}_{X}$-module, or equivalently
locally principal. It is principal if $\mathcal{I}=\mathcal{O}_{X}\cdot f$
(also denoted $(f)$) for some global section
$f\in\Gamma(X,\mathcal{K}^{*}_{X})$.

The set of generalized divisors on $X$ is denoted with
$\operatorname{GDiv}(X)$, the subset of Cartier divisors with
$\operatorname{CDiv}(X)$ and the set of principal divisors with
$\operatorname{Prin}(X)$.

Using the usual notion of subscheme associated to a sheaf of ideals, the set
$\operatorname{GDiv}^{+}(X)$ of effective generalized divisors on $X$ is in
one-to-one correspondence with the set of closed subschemes $D\subset X$ of
pure codimension one (i.e. of dimension zero). With a slight abuse of
notation, also for non-effective divisors, we denote with $D$ the generalized
divisor and we refer to $\mathcal{I}$ (or $\mathcal{I}_{D}$) as the fractional
ideal of $D$ (also called defining ideal of $D$, or ideal sheaf of $D$ if $D$
is effective).

Let $D_{1},D_{2}$ be generalized divisors on $X$, with fractional ideals
$\mathcal{I}_{1},\mathcal{I}_{2}$. The sum $D_{1}+D_{2}$ is the generalized
divisor associated to the fractional ideal
$\mathcal{I}_{1}\cdot\mathcal{I}_{2}\subseteq\mathcal{K}_{X}$. The sum is
commutative, associative, with neutral element $0$ defined by the trivial
ideal $\mathcal{O}_{X}$. The inverse of a generalized divisor $D$ associated
to $\mathcal{I}$ is the generalized divisor $-D$ associated to the fractional
ideal $\mathcal{I}^{-1}$, i.e. the sheaf which on any open subset $U$ of $X$
is defined as $\\{f\in\Gamma(U,\mathcal{K}_{X})\hskip 3.00003pt|\hskip
3.00003pt\mathcal{I}(U)\cdot f\subseteq\mathcal{O}_{X}(U)\\}$.

The inverse operation behaves well with the sum only for Cartier divisors. For
any pair of generalized divisors $D,E$ on $X$ with $E$ Cartier,
$-(D+E)=(-D)+(-E)$, but $D+(-D)=0$ if and only if $D$ is a Cartier divisor. As
a consequence, the set $\operatorname{GDiv}(X)$ of all generalized divisors
over $X$ endowed with the sum operation is not a group, but the subset
$\operatorname{CDiv}(X)$ of Cartier divisors is. The set
$\operatorname{Prin}(X)$ of principal divisors is a subgroup of
$\operatorname{CDiv}(X)$ and both the groups act by addition on the set
$\operatorname{GDiv}(X)$.

Tthe following Lemma, inspired by [Har94, Proposition 2.11], shows that any
generalized divisor is equal to the sum of an effective generalized divisor
and the inverse of an effective Cartier divisor, as

######  Lemma 2.2.3.

Let $X$ be a curve and let $D\in\operatorname{GDiv}(X)$ be any generalized
divisor on $X$. Then, there exist an effective generalized divisor
$D^{\prime}$ and an effective Cartier divisor $E$ on $X$ such that
$D=D^{\prime}+(-E)$.

###### Proof.

Cover $X$ by open affines $U_{i}=\operatorname{Spec}(A_{i})$, $i=1,\dots,r$.
For each $i$, denote with $I_{i}$ the fractional ideal of $D$ restricted to
$U_{i}$. This is a finitely generated $A_{i}$-module, so there exists a
nonzero-divisor $f_{i}\in A_{i}$ such that $f\cdot I_{i}\subseteq A$. Let
$Y_{i}\subset U_{i}$ be the closed subscheme defined by $f_{i}$, which is an
effective Cartier divisor of $U_{i}$; after the composition with
$U_{i}\hookrightarrow X$, $Y_{i}$ becomes an effective Cartier divisor of $X$.
Now, the sum of divisors $D+\sum_{i=1}^{r}Y_{i}$ is effective as it is sum of
effective divisors on each $U_{i}$. By putting $E=\sum_{i=1}^{r}Y_{i}$ and
$D^{\prime}=D+E$ we get the result. ∎

Two generalized divisors $D_{1},D_{2}$ over $X$ are linearly equivalent
(written $D_{1}\sim D_{2}$) if there is a divisor
$(f)\in\operatorname{Prin}(X)$ such that $D_{1}+(f)=D_{2}$. We define the
generalized Picard of X as the set of divisors on $X$ modulo linear
equivalence:

$\operatorname{GPic}(X)=\operatorname{GDiv}(X)/\operatorname{Prin}(X).$

We have the following commutative diagram of sets, where the vertical maps are
quotients by $\operatorname{Prin}(X)$:

${\operatorname{CDiv}(X)}$${\subseteq}$${\operatorname{GDiv}(X)}$${\operatorname{Pic}(X)}$${\subseteq}$${\operatorname{GPic}(X)}$

Taking inverse and sums preserve linear equivalence, so the two operations are
well defined also on $\operatorname{GPic}(X)$ and the subset
$\operatorname{Pic}(X)\subseteq\operatorname{GPic}(X)$ is a group that acts on
$\operatorname{GPic}(X)$ by addition. For a curve $X$, the condition
$\operatorname{GDiv}(X)=\operatorname{CDiv}(X)$ is also equivalent to
$\operatorname{GPic}(X)=\operatorname{Pic}(X)$, which is also equivalent to
the curve $X$ being smooth.

The set $\operatorname{GPic}(X)$ has an alternative description in terms of
generalized line bundles. Let $D$ be a generalized divisor on $X$; then, its
fractional ideal $\mathcal{I}$ is a generalized line bundle. If $D^{\prime}$
is another generalized line bundle, it is linearly equivalent to $D$ if and
only if its fractional ideal $\mathcal{I}^{\prime}$ is a generalized line
bundle isomorphic to $X$ as an $\mathcal{O}_{X}$-module. Viceversa, any
generalized line bundle $\mathcal{F}$ on $X$ is isomorphic to the fractional
ideal of some generalized divisor $D$ [Har07, Proposition 2.4]. Then,
$\operatorname{GPic}(X)$ can be also defined as the set of generalized line
bundles of $X$, up to isomorphism. Under this description, classes of Cartier
divisors correspond to isomorphism classes of line bundles, and the operations
of sum by a Cartier divisor and inverse of a divisor are replaced with tensor
product and taking dual of the corresponding sheaves.

####  2.2.2. Degree of generalized divisors

We now assume that $X$ is a projective curve over a base field $k$. In this
case, the notion of degree of a divisor can be introduced.

######  Definition/Lemma 2.2.4.

Let $D\in\operatorname{GDiv}^{+}(X)$ be an effective generalized divisor on
$X$ with ideal sheaf $\mathcal{I}_{D}\subseteq\mathcal{O}_{X}$, and let $x\in
X$ be any point of $X$ in codimension 1. We define the degree of $D$ at $x$ as
the non-negative integer:

$\deg_{x}(D)=\ell_{\mathcal{O}_{X,x}}\big{(}\mathcal{O}_{X,x}/\mathcal{I}_{D,x}\big{)}[\kappa(x):k].$

The degree of any generalized divisor $D\in\operatorname{GDiv}(X)$ at $x$ is
defined as $\deg_{x}(D)=\deg_{x}(E)-\deg_{x}(F)$, where $D=E-F$ with $E,F$
effective generalized divisors and $F$ Cartier by Lemma 2.2.3.

The degree of $D$ on $X$ (also denoted as $\deg(D)$ when there is no
ambiguity) is equal to the sum of the degrees of $D$ at all points of $X$ in
codimension 1:

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

###### Proof.

First, note that the local ring $\mathcal{O}_{X,x}/\mathcal{I}_{D,x}$ in
nonzero if and only if $x$ is in the support of $D$. In such case the local
ring has Krull dimension 0, so it is Artinian and hence has finite length.

The definition of degree at a point of any generalized divisor is well-given
thanks to [Har94, Lemma 2.17], and $\deg_{x}(-D)=-\deg_{x}(D)$.

Finally, the supporty of $D$ contains only finitely many points, hence the
degree of $D$ on $X$ is a well-defined integer. ∎

The degree of a generalized divisor is strictly related to the degree of its
fractional ideal, as the following proposition shows.

######  Proposition 2.2.5.

Let $D$ be a generalized divisor on $X$ with fractional ideal
$\mathcal{I}_{D}\subseteq\mathcal{K}_{X}$. Then, the degree of
$\mathcal{I}_{D}$ as a torsion-free sheaf is well defined and equal to
$-\deg_{X}(D)$. Moreover, linearly equivalent divisors have the same degree on
$X$.

###### Proof.

The fractional ideal $\mathcal{I}_{D}$ is a generalized line bundle, hence it
has rank $1$ on $X$ and well-defined degree as a torsion-free sheaf. In order
to compute $\deg(\mathcal{I}_{D})$ consider the short exact sequence

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

By definition of degree,
$\deg(\mathcal{I}_{D})=\chi(\mathcal{I}_{D})-\chi(\mathcal{O}_{X})$. Then, by
additivity of Euler characteristic with respect to exact sequences, we deduce
that $\deg(\mathcal{I}_{D})=-\chi(\mathcal{O}_{X}/\mathcal{I}_{D})$. Since
$\mathcal{O}_{X}/\mathcal{I}_{D}$ has dimension 0, its Euler characteristic is
equal to
$h^{0}(\mathcal{O}_{X}/\mathcal{I}_{D}):=\dim_{k}H^{0}(X,\mathcal{O}_{X}/\mathcal{I}_{D})$.
The support of $\mathcal{O}_{D}=\mathcal{O}_{X}/\mathcal{I}_{D}$ is by
definition the support of $D$ and
$H^{0}(X,\mathcal{O}_{X}/\mathcal{I}_{D})=\oplus_{x\in\operatorname{Supp}D}\mathcal{O}_{D,x}$.
Then we compute:

$\displaystyle\deg(\mathcal{I}_{D})$
$\displaystyle=-\chi(\mathcal{O}_{X}/\mathcal{I}_{D})=-\sum_{x\in\operatorname{Supp}D}\dim_{k}\mathcal{O}_{D,x}=$
$\displaystyle=-\sum_{x\in\operatorname{Supp}D}\dim_{\kappa(x)}\big{(}\mathcal{O}_{D,x}\big{)}[\kappa(x):k]=$
$\displaystyle=-\sum_{x\in\operatorname{Supp}D}\ell_{\mathcal{O}_{X},x}\big{(}\mathcal{O}_{D,x}\big{)}[\kappa(x):k]=$
$\displaystyle=-\deg_{X}(D).$

Consider now a generalized divisor $D$ with fractional ideal $\mathcal{I}_{D}$
and let $D=E-F$ with $E,F$ effective and $F$ Cartier by Lemma 2.2.3. Since $F$
is effective, consider first the short exact sequence:

(2.3) $\displaystyle
0\rightarrow\mathcal{I}_{F}\rightarrow\mathcal{O}_{X}\rightarrow T\rightarrow
0$

where $T$ is a sheaf of dimension 0. Since $F$ is Cartier, we can tensor by
$\mathcal{I}_{F}^{-1}$ to obtain:

(2.4) $\displaystyle
0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{I}_{F}^{-1}\rightarrow\mathcal{I}_{F}^{-1}\otimes
T\rightarrow 0.$

Using addivity of Euler characteristics in the exact sequences 2.3 and 2.4 and
the fact that $\chi(\mathcal{I}_{F}^{-1}\otimes T)=\chi(T)$, we see that:

$\chi(\mathcal{I}_{F}^{-1})-\chi(\mathcal{O}_{X})=\chi(\mathcal{O}_{X})-\chi(\mathcal{I}_{F}).$

Also $E$ is effective, hence there is a short exact sequence:

(2.5) $\displaystyle
0\rightarrow\mathcal{I}_{E}\rightarrow\mathcal{O}_{X}\rightarrow Q\rightarrow
0$

where $Q$ is a sheaf of dimension 0. Tensoring again by $\mathcal{I}_{F}^{-1}$
we obtain:

(2.6) $\displaystyle
0\rightarrow\mathcal{I}_{D}\rightarrow\mathcal{I}_{F}^{-1}\rightarrow\mathcal{I}_{F}^{-1}\otimes
Q\rightarrow 0$

where $\mathcal{I}_{D}\simeq\mathcal{I}_{E}\otimes\mathcal{I}_{F}^{-1}$ since
$D=E-F$ and $F$ is Cartier. Using addivity of Euler characteristics in the
exact sequences 2.5 and 2.6 and the fact that
$\chi(\mathcal{I}_{F}^{-1}\otimes Q)=\chi(Q)$, we can compute the degree of
$\mathcal{I}_{D}$:

$\displaystyle\deg(\mathcal{I}_{D})$
$\displaystyle=\chi(\mathcal{I}_{D})-\chi(\mathcal{O}_{X})=$
$\displaystyle=\chi(\mathcal{I}_{F}^{-1})-\chi(\mathcal{I}_{F}^{-1}\otimes
Q)-\chi(\mathcal{O}_{X})=$
$\displaystyle=\chi(\mathcal{I}_{F}^{-1})-\chi(Q)-\chi(\mathcal{O}_{X})=$
$\displaystyle=\chi(\mathcal{I}_{F}^{-1})-2\chi(\mathcal{O}_{X})+\chi(\mathcal{I}_{E})=$
$\displaystyle=\chi(\mathcal{O}_{X})-\chi(\mathcal{I}_{F})+\chi(\mathcal{I}_{E})-\chi(\mathcal{O}_{X}).$

Since $E$ and $F$ are effective, we conclude that:

$\deg(\mathcal{I}_{D})=-(\deg_{X}(E)-\deg_{X}(F))=-\deg_{X}(D).$

To prove the last statement, let $D$ and $D^{\prime}$ be linearly equivalent
divisors; then, their fractional ideal $\mathcal{I}_{D}$ and
$\mathcal{I}_{D^{\prime}}$ are isomorphic. Hence,

$\deg_{X}(D)=-\deg(\mathcal{I}_{D})=-\deg(\mathcal{I}_{D}^{\prime})=\deg_{X}(D^{\prime}).$

∎

####  2.2.3. Moduli of generalized divisors and the Abel Map

We start with considering the moduli space for generalized divisors. Since we
are dealing with families of sheaves and related moduli problems, we assume
that $X$ is a projective curve over a base field $k$.

######  Remark 2.2.6.

In general $\operatorname{GDiv}(X)$ cannot be represented by a geometric
object of finite type, even for fixed degrees. Also, it is not easy to give a
correct definition for flat families of non-effective generalized divisors.
Hence, only families of effective divisors are considered, using the Hilbert
scheme.

######  Definition 2.2.7.

The Hilbert scheme of effective generalized divisors of degree $d$ on $X$ is
the Hilbert scheme $\operatorname{Hilb}_{X}^{d}$ parametrizing 0-dimensional
subschemes of $X$, with Hilbert polynomial equal to a costant integer $d$.
Recall that, given a $k$-scheme $T$, a $T$-valued point of
$\operatorname{Hilb}_{X}^{d}$ is a $T$-flat subscheme $D\subset X\times_{k}T$
such that $D$ restricted to the fiber over any $t\in T$ is a 0-dimensional
subscheme of $X\times_{k}\\{t\\}\simeq X$, of degree $d$. In other words, the
corresponding ideal sheaf $\mathcal{I}_{D}\subset\mathcal{O}_{X\times_{k}T}$
restricted to any fiber over $T$ defines a generalized divisor $D_{t}$ on $X$.

The Hilbert scheme of effective Cartier divisors of degree $d$ is the open
subscheme
$\operatorname{{}^{\ell}Hilb}_{X}^{d}\subseteq\operatorname{Hilb}_{X}^{d}$
parametrizing subschemes of $X$ of degree $d$ whose ideal sheaf is locally
principal.

Note that $\operatorname{{}^{\ell}Hilb}_{X}^{d}=\operatorname{Hilb}^{d}_{X}$
when $X$ is smooth.

Families of effective generalized divisors can be added with families of
Cartier divisors, giving rise to a morphism:

$\displaystyle\operatorname{Hilb}_{X}^{d}\times\operatorname{{}^{\ell}Hilb}_{X}^{e}$
$\displaystyle\longrightarrow\operatorname{Hilb}_{X}^{d+e}$
$\displaystyle(D,E)\hskip 20.00003pt$
$\displaystyle\longmapsto\mathcal{Z}(\mathcal{I}_{D}\cdot\mathcal{I}_{E}).$

Effective generalized divisors on $X$ up to linear equivalence are generalized
line bundles, hence they are parametrized by the generalized Jacobian
$\operatorname{\textit{G}\mathbb{J}}(X)$.

By Remark 1.2.9, the Jacobian scheme $J(X)$ of $X$ is contained in
$\operatorname{\textit{G}\mathbb{J}}(X)$ as an open subscheme. The operations
of tensor products and taking inverse are well defined on $J(X)$ and make
$J(X)$ into an algebraic group, acting on
$\operatorname{\textit{G}\mathbb{J}}(X)$ via tensor product: