2.1 $\text{GL}_{n}$ and $\text{SL}_{n}$ Hitchin fibrations

A standard reference for what follows is [Reference NitsureNit91].

The  $\text{GL}_{n}$  case. Let $\mathscr{M}$ be the moduli space of stable, $D$ -twisted, $\text{GL}_{n}$ Higgs bundles of rank $n$ and degree $e$ on the curve $C$ . Then $\mathscr{M}$ is a nonsingular and quasi-projective variety of pure dimension $n^{2}d+1$ . It parameterizes stable pairs $(E,\unicode[STIX]{x1D719})$ , where: $E$ is a rank $n$ and degree $e$ vector bundle on the curve $C$ , and $\unicode[STIX]{x1D719}:E\rightarrow E(D)$ is a morphism of ${\mathcal{O}}_{C}$ -modules. The notion of stability is the usual one: for every $\unicode[STIX]{x1D719}$ -invariant proper sub-bundle $F\subseteq E$ , the slopes $\unicode[STIX]{x1D707}:=\deg /\text{rk}$ satisfy the inequality $\unicode[STIX]{x1D707}(F)<\unicode[STIX]{x1D707}(E)$ . There is the projective characteristic morphism

sending $(E,\unicode[STIX]{x1D719})$ to the coefficients $(-\text{tr}(\unicode[STIX]{x1D719}),+\text{ tr}(\wedge ^{2}\unicode[STIX]{x1D719}),\ldots ,(-1)^{n}\det (\unicode[STIX]{x1D719}))$ of the characteristic polynomial of $\unicode[STIX]{x1D719}$ . The elements of $\mathscr{A}$ are called characteristics.

The pure-dimensional nonsingular variety $\mathscr{M}$ is connected, hence irreducible. One way to see this, is to couple the fact that the proper characteristic morphism is of pure relative dimension [Reference Chaudouard and LaumonCL16, Corollaire 8.2] with the fact (Remark 2.4.4) that the general fiber, being the Jacobian of a nonsingular and connected spectral curve, is connected. I thank the anonymous referee for bringing this to my attention.

The moduli space $N$ of rank $n$ and degree $e$ vector bundles on $C$ sits naturally in $\mathscr{M}$ (take $\unicode[STIX]{x1D719}:=0$ ). It is well known that $N$ is integral, nonsingular, projective and of dimension $n^{2}(g-1)+1$ . We have inclusions $\mathscr{M}=\overline{T}\supseteq T\supseteq N$ , where $T$ is the total space of the vector bundle of rank $n^{2}[d-(g-1)]$ over $N$ with fiber at $E$ given by $H^{0}(C,\text{End}(E)(D))$ ; see [Reference NitsureNit91, Proposition 7.1 and the formula above it]. Then $T$ is integral, nonsingular, of dimension $n^{2}d+1$ , and it is a Zariski-dense open subvariety of $\mathscr{M}$ ; see [Reference NitsureNit91, pp. 297–298].

The  $\text{GL}_{n}$  traceless case. We need the following simple traceless variant of the $D$ -twisted $\text{GL}_{n}$ moduli space: geometrically, it is the pre-image via the morphism $h:\mathscr{M}\rightarrow \mathscr{A}$ of the locus $\mathscr{A}(0)\subseteq \mathscr{A}$ of traceless characteristics. Let $\mathscr{M}(0)\subseteq \mathscr{M}$ be the moduli space of stable pairs $(E,\unicode[STIX]{x1D719})$ as above, subject to the additional traceless constraint $\text{tr}(\unicode[STIX]{x1D719})=0$ . By repeating the arguments in [Reference NitsureNit91] concerning $\mathscr{M}$ , but with the traceless constraint, we see that $\mathscr{M}(0)$ is a nonsingular and quasi-projective variety, of pure dimension $nd^{2}+1-h^{0}(D)$ . Moreover, we have a natural isomorphism $\mathscr{M}\cong H^{0}(C,D)\times \mathscr{M}(0)$ (see § 4.3, (51)), implying that the nonsingular $\mathscr{M}(0)$ is connected and irreducible.

As above, we have inclusions $\mathscr{M}(0)=\overline{T(0)}\supseteq T(0)\supseteq N$ , with the same properties listed above, except that we take traceless endomorphisms, and the rank of the corresponding vector bundle on $N$ equals $h^{0}(C,\text{End}^{0}(E)(D))=n^{2}[d-(g-1)]-h^{0}(D)$ . We have the projective characteristic morphism

The  $\text{SL}_{n}$  case. Finally, we introduce the moduli space to which this paper is devoted. Fix a line bundle $\unicode[STIX]{x1D716}\in \text{Pic}^{e}(C)$ on $C$ , of degree $e$ . Let $\mathscr{M}(0,\unicode[STIX]{x1D716})\subseteq \mathscr{M}(0)\subseteq \mathscr{M}$ be the moduli space of stable pairs $(E,\unicode[STIX]{x1D719})$ as above, subject to $\text{ tr}(\unicode[STIX]{x1D719})=0$ and to $\det (E)=\unicode[STIX]{x1D716}$ . By repeating the arguments in [Reference NitsureNit91], but with the traceless and fixed-determinant constraints, we see that the variety $\mathscr{M}(0,\unicode[STIX]{x1D716})$ is nonsingular and quasi-projective, of pure dimension $n^{2}d+1-h^{0}(D)-g$ . We have the projective characteristic map

Let $\mathscr{M}(0,\unicode[STIX]{x1D716})_{o}$ be the irreducible (also a connected) component containing the moduli space $N(\unicode[STIX]{x1D716})$ of stable rank $n$ and degree $e$ bundles on $C$ with fixed determinant $\unicode[STIX]{x1D716}\in \text{Pic}^{e}(C)$ . It is well known that the variety $N(\unicode[STIX]{x1D716})$ is integral, nonsingular, projective, and of dimension $(n^{2}-1)(g-1)$ . As above, we have inclusions $\mathscr{M}(0,\unicode[STIX]{x1D716})_{o}=\overline{T(0,\unicode[STIX]{x1D716})}\supseteq T(0,\unicode[STIX]{x1D716})\supseteq N(\unicode[STIX]{x1D716})$ , with the same properties listed above (again, we take traceless endomorphisms).

Note that $\mathscr{M}(0,\unicode[STIX]{x1D716})=\mathscr{M}(0,\unicode[STIX]{x1D716})_{o}$ and that the isomorphism class of $\mathscr{M}(0,\unicode[STIX]{x1D716})_{o}$ is independent of $\unicode[STIX]{x1D716}\in \text{Pic}^{e}(C)$ . This can be seen as in the proof of the following simple lemma.

2.2 Simplified notation for Hitchin fibrations

We want to simplify our notation, while emphasizing the role of the rank $n$ .

Fix $(n,e,\unicode[STIX]{x1D716},D)$ . Denote the characteristic Hitchin morphisms

as follows:

We are denoting the same object ${\check{A}}_{n}=A_{n}(0)$ in two different ways: we prefer to use the notation $A_{n}(0)$ when dealing with $M_{n}(0)$ , and to use ${\check{A}}_{n}$ when dealing with $\check{M}_{n}$ .

The projective morphisms $h_{n}$ and $\check{h}_{n}$ are known as the $D$ -twisted, Hitchin $\text{GL}_{n}$ and $\text{SL}_{n}$ fibrations. The morphism $h_{n}(0)$ plays an important auxiliary role in this paper.

We shall also need to consider two several-variable-variants of these Hitchin fibrations, namely $h_{n_{\bullet }}:M_{n_{\bullet }}\rightarrow A_{\bullet }$ , and $h_{n_{\bullet }m_{\bullet }}(0):M_{n_{\bullet }m_{\bullet }}(0)\rightarrow A_{n_{\bullet }m_{\bullet }}(0)$ (cf. §§ 5.1 and 5.3).

An important locus inside the base of the Hitchin fibration is the elliptic locus. In the case of $\text{GL}_{n}$ and $\text{SL}_{n}$ we define it as follows.

Clearly, ${\check{A}}_{n}^{\text{ell}}=A_{n}^{\text{ell}}\,\cap \,{\check{A}}_{n}$ .

2.3 Spectral covers and the norm map

Let $\unicode[STIX]{x1D70B}:V(D)\rightarrow C$ be the surface total space of the line bundle $D$ on $C$ . Let $t$ be the universal section of $\unicode[STIX]{x1D70B}^{\ast }D$ , with zero set on $V(D)$ given by $C$ , viewed as the zero section on $V(D)$ . Let ${\mathcal{C}}={\mathcal{C}}_{n}\subseteq V(D)\times A_{n}$ be the universal spectral curve, that is the relative curve over $A_{n}$ with fiber ${\mathcal{C}}_{a}$ over a closed point $a=(a(1),\ldots ,a(n))\in A_{n}$ , given by the zero set in $V(D)\times \{a\}$ of the section $P_{a}(t):=t^{n}+\unicode[STIX]{x1D70B}^{\ast }a(1)t^{n-1}+\unicode[STIX]{x1D70B}^{\ast }a(2)t^{n-2}+\cdots +\unicode[STIX]{x1D70B}^{\ast }a(n)$ of the line bundle $\unicode[STIX]{x1D70B}^{\ast }(nD)$ on $V(D)\times \{a\}$ . Note that $A_{n}$ is an affine space inside the projective space given by the linear system $|nC|$ on the standard projective completion $\mathbb{P}_{C}({\mathcal{O}}_{C}\oplus {\mathcal{O}}_{C}(-D))$ of $V(D)$ , where $C$ sits as the zero section. Let $p:{\mathcal{C}}\rightarrow A_{n}$ be the natural ensuing morphism. For $a\in A_{n}$ , the spectral curve ${\mathcal{C}}_{a}$ is geometrically connected and maps $n:1$ onto $C_{a}:=C\otimes k(a)$ via the flat finite morphism $p_{a}:=p_{|{\mathcal{C}}_{a}}:{\mathcal{C}}_{a}\rightarrow C_{a}$ . The total space of the family ${\mathcal{C}}$ is integral and nonsingular, and the natural morphism ${\mathcal{C}}\rightarrow C\times A_{n}$ is finite, flat and of degree $n$ .

When we view each spectral curve ${\mathcal{C}}_{a}$ over a geometric point $a$ of $A_{n}$ , as an effective Cartier divisor on $V(D)\otimes k(a)$ , we may write ${\mathcal{C}}_{a}=\sum _{k=1}^{s}m_{k,a}{\mathcal{C}}_{k,a}$ , where each ${\mathcal{C}}_{k,a}$ is geometrically integral, each integer $m_{k,a}>0$ , and the expression is unique. Each curve ${\mathcal{C}}_{k,a}$ maps finitely onto $C_{a}$ ; denote the corresponding degree by $n_{k,a}$ . Clearly, $n=\sum _{k}m_{k,a}n_{k,a}$ . By considering the coefficients $a(i)$ above as the $i$ th symmetric functions of the $D$ -valued roots of the polynomial equation $P_{a}(t)$ , we obtain the unique factorization $P_{a}(t)=\prod _{k}P_{a_{k}}^{m_{k,a}}(t)$ , where each $a_{k}(i)\in H^{0}(C,iD)$ , $1\leqslant i\leqslant n_{k}$ , is the $i$ th symmetric function of the $D$ -valued roots of $P_{a}(t)$ that lie on ${\mathcal{C}}_{k,a}$ . In particular, we have that $a_{k}$ is a geometric point of $A_{n_{k}}$ (base of the Hitchin fibration for $(n_{k},e,D)$ ), and that ${\mathcal{C}}_{k,a}$ is a spectral curve for the $D$ -twisted $\text{GL}_{n_{k}}$ Hitchin fibration.

Let $a\in A_{n}$ . We need to list the various covers of the curve $C_{a}=C\otimes k(a)$ that arise from the given spectral cover $p_{a}:{\mathcal{C}}_{a}\rightarrow C_{a}$ . In doing so, we also simplify and abuse the notation a little bit. We do not assume the point $a\in A_{n}$ to be a geometric one, so that the intervening integral curves may not be geometrically integral.

We denote the curve ${\mathcal{C}}_{a}=\sum _{k}m_{k}\unicode[STIX]{x1D6E4}_{k}$ , where: each $\unicode[STIX]{x1D6E4}_{k}$ is a spectral curve, zero-set of a section $\mathfrak{s}_{k}$ of the line bundle $\unicode[STIX]{x1D70B}^{\ast }(n_{k}D)$ on the surface $V(D)\otimes k(a)$ ; the $n_{k}>0$ are uniquely-determined positive integers, and we have $n=\sum n_{k}m_{k}$ . Scheme-theoretically, $m_{k}\unicode[STIX]{x1D6E4}_{k}$ is the zero set of the $m_{k}$ th power $\mathfrak{s}^{m_{k}}$ , and ${\mathcal{C}}_{a}=\sum _{k}m_{k}\unicode[STIX]{x1D6E4}_{k}$ is the zero set of the product $\prod _{k}\mathfrak{s}_{k}^{m_{k}}$ . We denote by $\unicode[STIX]{x1D709}_{3,k}:\widetilde{\unicode[STIX]{x1D6E4}_{k}}\rightarrow \unicode[STIX]{x1D6E4}_{k}$ the normalization morphism.

We have the following commutative diagram of finite surjective morphisms of curves.

Each of the morphisms to $C_{a}$ in diagram (2) comes with an associated norm morphism into $\text{Pic}(C_{a})$ , and with an associated pull-back morphism from $\text{Pic}(C_{a})$ . Similarly, if we replace $\text{Pic}$ with $\text{Pic}^{0}$ . For the definition and properties of the norm morphism, see [Reference GrothendieckEGAII, § 6.5] and [Reference GrothendieckEGAIV.4, § 21.5]. For a quick reference for the facts we use in this paper, see also [Reference Hausel and PaulyHP12, § 3]. See also Fact 2.4.5. We have the norm morphism

We also have the norm morphisms $N_{\widetilde{p}},N_{p^{\prime }}$ and $N_{p^{\prime \prime }}$ , as well as the pull-back morphisms $\widetilde{p}^{\ast },{p^{\prime }}^{\ast }$ and ${p^{\prime \prime }}^{\ast }$ ; similarly, for each of their $k$ th components.

We end this section with the following consideration that will play a role later.

2.4 The fibers of the Hitchin fibrations

Let $a\in A_{n}$ and let $p:{\mathcal{C}}_{a}=:\unicode[STIX]{x1D6E4}=\sum _{k}m_{k}\unicode[STIX]{x1D6E4}_{k}\rightarrow C_{a}$ be the corresponding spectral cover, with $n=n_{\unicode[STIX]{x1D6E4}}=\deg (p)=\sum _{k}n_{k}m_{k}=\text{rk}_{C}(p_{\ast }{\mathcal{O}}_{\unicode[STIX]{x1D6E4}})$ ; see (2). Let $j_{k}:\unicode[STIX]{x1D702}_{k}\rightarrow \unicode[STIX]{x1D6E4}$ be the finitely many generic points in $\unicode[STIX]{x1D6E4}$ , one for each irreducible component $m_{k}\unicode[STIX]{x1D6E4}_{k}$ . A coherent sheaf ${\mathcal{E}}$ on $\unicode[STIX]{x1D6E4}$ is torsion free if and only if the natural map ${\mathcal{E}}\rightarrow \prod _{k}{\mathcal{E}}_{k}$ is injective, where ${\mathcal{E}}_{k}={j_{k}}_{\ast }j_{k}^{\ast }{\mathcal{E}}$ ; see [Reference SchaubSch98, Definition 1.1. and Proposition 1.1]. A torsion-free ${\mathcal{E}}$ is said to have $\text{Rk}_{\unicode[STIX]{x1D6E4}}({\mathcal{E}})=r$ if its lengths at the generic points satisfy $l_{k}({\mathcal{E}}):=l_{{\mathcal{O}}_{\unicode[STIX]{x1D702}_{k}}}({\mathcal{E}}_{\unicode[STIX]{x1D702}_{k}})=r\,m_{k}$ , for every $k$ ; such a rank is then a nonnegative rational number, which is zero if and only if ${\mathcal{E}}=0$ . A torsion free ${\mathcal{E}}$ may fail to have a well-defined $\text{Rk}_{\unicode[STIX]{x1D6E4}}({\mathcal{E}})$ . When this rank is well defined, one defines the degree by setting $\text{Deg}_{\unicode[STIX]{x1D6E4}}({\mathcal{E}}):=\unicode[STIX]{x1D712}({\mathcal{E}})-\text{Rk}_{\unicode[STIX]{x1D6E4}}({\mathcal{E}})\unicode[STIX]{x1D712}({\mathcal{O}}_{\unicode[STIX]{x1D6E4}})$ .

Let $P_{\unicode[STIX]{x1D6E4}}=\prod _{k}P_{\unicode[STIX]{x1D6E4}_{k}}^{m_{k}}$ be the characteristic equation defining $\unicode[STIX]{x1D6E4}$ . A torsion-free coherent sheaf ${\mathcal{E}}$ on $\unicode[STIX]{x1D6E4}$ corresponds, via $p_{\ast }$ , to a pair $(E,\unicode[STIX]{x1D719}:E\rightarrow E(D))$ on $C$ , where: $E=p_{\ast }{\mathcal{E}}$ is locally free of rank $\text{rk}_{C}(E)=\sum _{k}n_{k}l_{k}$ ; $\unicode[STIX]{x1D719}$ is the twisted endomorphism corresponding to multiplication by $t$ on ${\mathcal{E}}$ . Then $\unicode[STIX]{x1D719}$ has characteristic polynomial $P_{\unicode[STIX]{x1D719}}=\prod _{k}P_{\unicode[STIX]{x1D6E4}_{k}}^{l_{k}}$ . It follows that $P_{\unicode[STIX]{x1D719}}=P_{\unicode[STIX]{x1D6E4}}$ if and only if $\text{Rk}_{\unicode[STIX]{x1D6E4}}({\mathcal{E}})$ is well defined and equals $1$ (this is the content of [Reference SchaubSch98, Proposition 2.1]).

Note that [Reference Hausel and PaulyHP12, § 3.3] introduces, via the Riemann–Roch theorem, a different notion of rank and degree for every coherent ${\mathcal{O}}_{\unicode[STIX]{x1D6E4}}$ -module, even for those torsion-free ones for which the notion of degree given above is not well defined. In this paper, we use the notion of rank and degree given above [Reference SchaubSch98], not the one in [Reference Hausel and PaulyHP12]. The forthcoming modular description of the fibers of the Hitchin fibration is given in terms of the notions employed in this paper, and the torsion-free sheaves on spectral curves that arise are, by necessity, the ones for which the rank is well defined and it has value one.

It is easy to show that in the context of torsion-free and $\text{ Rk}_{\unicode[STIX]{x1D6E4}}(-)=r$ coherent sheaves on $\unicode[STIX]{x1D6E4}$ , the notion of slope in [Reference SimpsonSim94, p. 55] and [Reference SimpsonSim95, Corollary 6.9] and the notion of slope $\text{Deg}_{\unicode[STIX]{x1D6E4}}/\text{Rk}_{\unicode[STIX]{x1D6E4}}$ yield coinciding notions of slope stability. In turn, this coincides with the notion of slope-stable Higgs pair $(p_{\ast }{\mathcal{E}},\unicode[STIX]{x1D719})$ , with slopes defined by taking $\deg _{C}/\text{rk}_{C}$ . By working with quotients, instead of with subobjects, the stability condition takes the form (6) below. Define

Modular description of the Hitchin fiber $M_{n,a}:=h_{n}^{-1}(a)$ , $a\in A_{n}$ . The discussion that follows does not require that one first proves that $M_{n}$ is irreducible; in particular, it can be used in order to establish this fact, as it has been done in § 2.1. The Hitchin fiber $M_{n,a}:=h_{n}^{-1}(a)$ , i.e. the moduli space of stable $D$ -twisted Higgs pairs with rank $n$ and degree $e$ and with characteristic $a\in A_{n}$ , is isomorphic to the moduli space of torsion-free sheaves ${\mathcal{E}}$ on the spectral curve ${\mathcal{C}}_{a}$ with $\text{ Rk}_{{\mathcal{C}}_{a}}({\mathcal{E}})=1$ (and hence with associated characteristic polynomial $P_{\unicode[STIX]{x1D719}}=P_{{\mathcal{C}}_{a}}$ ) and $\text{Deg}_{{\mathcal{C}}_{a}}({\mathcal{E}})=e^{\prime }$ , subject to the following stability condition: for every closed subscheme $i_{Z}:Z\rightarrow {\mathcal{C}}_{a}$ of pure dimension one, for every torsion-free quotient ${\mathcal{O}}_{Z}$ -module  with $\text{Rk}_{Z}({\mathcal{E}}_{Z})=1$ , we have

The isomorphism is given by the push-forward morphism ${p_{a}}_{\ast }$ on coherent sheaves under the finite, flat, degree $n$ , spectral cover morphism $p_{a}:{\mathcal{C}}_{a}\rightarrow C_{a}=C\otimes k(a)$ .

Let us record the properties of the norm map that we need.

Modular description of the Hitchin fiber $\check{h}_{n}^{-1}(a)$ , $a\in {\check{A}}_{n}$ . The description in question is the same as the modular description given above, except for the added constraint on the determinant $\det ({p_{a}}_{\ast }{\mathcal{E}})=\unicode[STIX]{x1D716}$ , where $\unicode[STIX]{x1D716}\in \text{Pic}^{e}(C)$ is the fixed line bundle involved in the definition (1) of $\check{M}_{n}$ .

In general, the Prym variety $\text{Prym}_{a}$ is a disconnected group scheme with finitely many components; see [Reference Hausel and PaulyHP12] for a description of these components at geometric points of $A_{n}$ . We also call Prym variety the corresponding identity connected component. In a given context, we shall make it clear which Prym variety we are using.

If $a\in A_{n}$ is general, then $\text{Prym}_{a}$ is geometrically connected (Fact 2.4.5(3)).

2.5 Endoscopy loci of the Hitchin $\text{SL}_{n}$ fibration

Let $a$ be a geometric point of $A_{n}^{\text{ell}}$ , so that the spectral curve ${\mathcal{C}}_{a}$ is (geometrically) integral. The $D$ -twisted, $\text{GL}_{n}$ Hitchin fiber $M_{n,a}$ is also integral: it is isomorphic to the compactified Jacobian of the integral locally planar spectral curve, parameterizing rank one and degree $e^{\prime }$ torsion-free coherent sheaves on it. In particular, the regular part $\text{Pic}^{e^{\prime }}({\mathcal{C}}_{a})\cong M_{n,a}^{\text{reg}}\subseteq M_{n,a}$ of this fiber is integral, Zariski open and dense in the whole fiber, and it is a $\text{ Pic}^{0}({\mathcal{C}}_{a})$ -torsor.

Let $a$ be a geometric point of ${\check{A}}_{n}$ . Then the $D$ -twisted, $\text{SL}_{n}$ Hitchin fiber $\check{M}_{n,a}=\mathfrak{p}_{a}^{-1}(\unicode[STIX]{x1D716})$ (cf. (10)), and it is (geometrically) connected. Since the morphism $\check{h}_{n}$ is flat and ${\check{A}}_{n}$ is nonsingular, every fiber of $\check{h}_{n}$ is a local complete intersection (l.c.i).

Assume, in addition, that $a$ is a geometric point of ${\check{A}}_{n}^{\text{ell}}$ . By the $\text{Pic}^{0}(C)$ -equivariance of $\mathfrak{p}_{a}$ , the regular part of $\check{M}_{n,a}$ satisfies $\check{M}_{n,a}^{\text{reg}}=M_{n,a}^{\text{reg}}\,\cap \,\check{M}_{n,a}$ , and it is Zariski open and dense. Since the fiber $\check{M}_{n,a}$ is a l.c.i., we have that, being smooth on a Zariski-dense open subset, it is also reduced. The regular part $\check{M}_{n,a}^{\text{reg}}$ is made of line bundles ${\mathcal{E}}$ on the spectral curve with $\mathfrak{p}_{a}({\mathcal{E}})=\unicode[STIX]{x1D716}$ . It is clear that $\check{M}_{n,a}^{\text{reg}}$ is then a $\text{Prym}_{a}$ -torsor.

For every $a\in A_{n}$ , the group of connected components $\unicode[STIX]{x1D70B}_{0}(\text{ Prim}_{a})$ is described in [Reference Hausel and PaulyHP12, Theorem 1.1]. The locus ${\check{A}}_{n,\text{endo}}\subseteq {\check{A}}_{n}$ over which $\text{ Prym}_{a}$ is disconnected is called the endoscopic locus of the $\text{SL}_{n}$ Hitchin fibration and it is described in [Reference Hausel and PaulyHP12, § 5, especially Lemmas 5.1; 7.1]:

where $\unicode[STIX]{x1D6E4}$ ranges over the finite set of cyclic subgroups of $\text{Pic}^{0}(C)[n]$ of prime number order. Each ${\check{A}}_{n,\unicode[STIX]{x1D6E4}}\subseteq {\check{A}}_{n}$ is a geometrically integral subvariety. The codimension of each ${\check{A}}_{n,\unicode[STIX]{x1D6E4}}$ can be computed in the same way as in the proof of [Reference Hausel and PaulyHP12, Lemma 7.1], whose proof in the case $D=K_{C}$ , remains valid for $D$ : we need the knowledge of $d_{{\check{A}}_{n}}$ (78), obtained by the Riemann–Roch theorem, and the formula directly above [Reference Hausel and PaulyHP12, Lemma 5.1]. The resulting value

is strictly positive in view, for example, of our assumption $d>2(g-1)$ .

The subvarieties ${\check{A}}_{n,\unicode[STIX]{x1D6E4}}^{\text{ell}}:={\check{A}}_{n,\unicode[STIX]{x1D6E4}}\,\cap \,{\check{A}}_{n}^{\text{ell}}\subseteq {\check{A}}_{n}^{\text{ell}}$ are nonsingular and mutually disjoint [Reference NgôNgô06, Proposition 10.3]. By construction, the number

of connected components of $\text{Prym}_{a}$ is independent of $a\in {\check{A}}_{n,\unicode[STIX]{x1D6E4}}^{\text{ell}}$ .

A point $a\in {\check{A}}_{n,\unicode[STIX]{x1D6E4}}^{\text{ell}}$ if and only if the spectral cover $p_{a}:{\mathcal{C}}_{a}\rightarrow C$ has the property that the induced morphism from the normalization of the integral spectral curve $\widetilde{p_{a}}=\widetilde{{\mathcal{C}}_{a}}\rightarrow C$ factors through the étale cyclic cover of $C$ associated with $\unicode[STIX]{x1D6E4}$ (cf. [Reference Hausel and PaulyHP12, Proof of Theorem 5.3]).

The locus

is the $G=\text{SL}_{n}$ endoscopic locus introduced by Ngô in [Reference NgôNgô06, § 10] for $D$ -twisted, $G$ Hitchin fibrations ( $G$ reductive). It determines the socle $\text{Socle}({R\check{h}_{n}}_{\ast }\overline{\mathbb{Q}}_{\ell })\,\cap \,{\check{A}}_{n}^{\text{ell}}$ over the elliptic locus; see § 4.9.

2.6 Weak abelian fibrations and $\unicode[STIX]{x1D6FF}$ -regularity

The notion of $\unicode[STIX]{x1D6FF}$ -regular weak abelian fibration has been introduced in [Reference NgôNgô10] as an encapsulation of some important features of the Hitchin fibration over the elliptic locus: presence of the action of a commutative smooth group scheme with affine stabilizers, polarizability of the associated Tate module, and $\unicode[STIX]{x1D6FF}$ -regularity of the group scheme. See also [Reference NgôNgô11] for an introduction to this circle of ideas.

In this section, let $g:J\rightarrow A$ be a smooth commutative group scheme over an irreducible variety $A$ such that $g$ has geometrically connected fibers.

Chevalley devissage. References for what follows are, for example, [Reference MilneMil, Theorem 10.25, Propositions 10.24, 10.5 (and its proof), Proposition 10.3] and [Reference ConradCon02].

Let $\overline{a}$ be a geometric point on $A$ with underlying point a Zariski point $a\in A$ . Let $J_{\overline{a}}$ be the fiber of $J$ at $\overline{a}$ . There is a canonical short exact sequence of commutative connected group schemes over the residue field of $\overline{a}$ :

where $J_{\overline{a}}^{\text{aff}}\subseteq J_{\overline{a}}$ is the maximal connected affine linear subgroup of $J_{\overline{a}}$ , and $J_{\overline{a}}^{\text{ ab}}$ is an abelian variety. The dimensions of these varieties depend only on the Zariski point $a\in A$ , and are denoted by $d_{a}^{\text{aff}}(J)$ and $d_{a}^{\text{ab}}(J)$ , respectively. Clearly,

The notion of  $\unicode[STIX]{x1D6FF}$ -regularity. The function

is upper semicontinuous (jumps up on closed subsets); see [Reference Grothendieck and DemazureSGA3.II, X, Remark 8.7]. We have the disjoint union decomposition

of $A$ into locally closed subvarieties of $A$ . We call $S_{\unicode[STIX]{x1D6FF}}$ the $\unicode[STIX]{x1D6FF}$ -locus of $J/A$ .

The following lemma is an immediate consequence of the upper-semicontinuity of the function $\unicode[STIX]{x1D6FF}$ and of the identity (16).

The Tate module  $T_{\overline{\mathbb{Q}}_{\ell }}(J)$  and the notion of its polarizability. Let $g:J\rightarrow A$ be as above and let $d_{g}:=\dim (J)-\dim (A)$ be the pure relative dimension of $g$ . The Tate module of $J$ is the $\overline{\mathbb{Q}}_{\ell }$ -adic sheaf [Reference NgôNgô10, § 4.12]

Its stalk at any geometric point $\overline{a}$ of $A$ is given by the Tate module $T_{\overline{\mathbb{Q}}_{\ell }}(J_{\overline{a}})$ , i.e. the inverse limit, with respect to $i\in \mathbb{N}$ , of the $\ell ^{i}$ -torsion points on $J_{\overline{a}}$ , tensored with $\overline{\mathbb{Q}}_{\ell }$ over $\mathbb{Z}_{\ell }$ . The Chevalley devissage at the stalks yields the natural short exact sequence

The Tate module $T_{\overline{\mathbb{Q}}_{\ell }}(J)$ is said to be polarizable if it admits a polarization, i.e. an alternating bilinear pairing

such that, for every geometric point $\overline{a}$ of $A$ , we have that the kernel of $\unicode[STIX]{x1D713}_{\overline{a}}$ is exactly $T_{\overline{\mathbb{Q}}_{\ell }}(J_{\overline{a}}^{\text{ aff}})$ . In this case, the pairings $\unicode[STIX]{x1D713}_{\overline{a}}$ descend to nondegenerate, alternating, bilinear parings on the $T_{\overline{\mathbb{Q}}_{\ell }}(J_{\overline{a}}^{\text{ab}})$ .

Note that by general principles (cf. [Reference Grothendieck, Raynaud and RimSGA7.I, VIII, Corollary 4.10]), the alternating bilinear pairings we consider in this paper are automatically trivial on the ‘affine’ part, and do descend to the ‘abelian’ part. We do verify this fact along the way to proving the key fact that, in the cases we deal with, they in fact descend to nondegenerate pairings.

Affine stabilizers. Let $h:M\rightarrow A$ be a morphism of varieties and let $J\rightarrow A$ be a group scheme acting on $M/A$ . We say that the action has affine stabilizers if for every geometric point $m$ of $M$ , we have that the stabilizer subgroup $\text{St}_{\text{m}}\subseteq J_{h(m)}$ is affine.

$\unicode[STIX]{x1D6FF}$ -regular weak abelian fibrations. See [Reference NgôNgô10, Reference NgôNgô11]. Let $h:M\rightarrow A\leftarrow J:g$ be a pair of morphisms of varieties, where $g$ is as in the beginning of this section (smooth commutative group scheme, with geometrically connected fibers over an irreducible $A$ ), $h$ is proper, and $J/A$ acts on $M/A$ . We denote this situation simply by $(M,A,J)$ ; the context will make it clear which morphisms $h,g$ are being used.

Ngô support inequality. The following is a remarkable, and remarkably useful, topological restriction on the dimensions of the supports appearing in the context of weak abelian fibrations. If $a\in A$ , then $d_{a}:=\dim \overline{\{a\}}$ is the dimension of the closed subvariety of $A$ with generic point $a$ . For the notion of socle, see § 1. The celebrated Ngô support theorem [Reference NgôNgô10, Theorem 7.2.1] is a more refined restriction on the geometry of the supports, and it is proved also by using the support inequality.

Given that we are assuming $d_{h}=d_{g}$ , we may re-formulate (24) as follows via (16):

3.1 The action of the Jacobi group scheme $J_{n}$

For what follows, see [Reference Chaudouard and LaumonCL16, § 5]. Let $J_{n}\rightarrow A_{n}$ be the identity connected component of the degree zero component of the relative Picard stack $\text{Pic}_{{\mathcal{C}}/A_{n}}$ . This is a connected, smooth, commutative group scheme over $A_{n}$ , whose fiber $J_{n,a}$ over a point $a\in A_{n}$ is $\text{Pic}^{0}({\mathcal{C}}_{a})$ ; see Fact 2.3.1 for a description of this group. In particular, the structural morphism $g_{n}:J_{n}\rightarrow A_{n}$ is of pure relative dimension, call it $d_{g_{n}}$ , the arithmetic genus of the spectral curves, which coincides with the pure relative dimension $d_{h_{n}}$ (7) of $h_{n}:M_{n}\rightarrow A_{n}$ , i.e. we have

The group scheme $J_{n}/A_{n}$ acts on the Hitchin fibration $M_{n}/A_{n}$ ; see Proposition 2.4.9.

3.2 Affine stabilizers for the action of the Jacobi group scheme

3.3 The Tate module of the Jacobi group scheme is polarizable

We refer to § 2.6 for the terminology. Let $g_{n}:J_{n}\rightarrow A_{n}$ be the structural morphism for Picard. The Tate module is the $\overline{\mathbb{Q}}_{\ell }$ -adic sheaf (22) $T_{\overline{\mathbb{Q}}_{\ell }}(J_{n}):=R^{2d_{h_{n}}-1}{g_{n}}_{!}\overline{\mathbb{Q}}_{\ell }(d_{h_{n}})$ . If $a$ is a geometric point of $A_{n}$ , then the Chevalley devissage yields the natural short exact sequences

Note that: (i) $\dim _{\overline{\mathbb{Q}}_{\ell }}T_{\overline{\mathbb{Q}}_{\ell }}(J_{n,a}^{\text{ab}})=2\dim J_{n,a}^{\text{ab}}$ ; (ii) $\dim _{\overline{\mathbb{Q}}_{\ell }}T_{\overline{\mathbb{Q}}_{\ell }}(J_{n,a}^{\text{aff}})\leqslant \dim J_{n,a}^{\text{aff}}$ , and that the strict inequality can occur: this is due to the fact that the affine part $J_{n,a}^{\text{aff}}$ is an iterated extension of the additive and of the multiplicative group $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 9], and only the latter contribute to the Tate module.

The goal of this section is to prove the following polarizability result, which has been proved over the elliptic locus $A_{n}^{\text{ell}}$ in [Reference NgôNgô10], and is stated implicitly over the whole base $A_{n}$ and then used in [Reference Chaudouard and LaumonCL16, § 9].

3.4 $\unicode[STIX]{x1D6FF}$ -regularity of the action of the Jacobi group scheme over the elliptic locus

4.1 The action of the Prym group scheme $\check{J}_{n}$

Let $p:{\mathcal{C}}\rightarrow A_{n}$ be the family of spectral curves as in § 2.3. The norm morphism (3) defines a morphism of group schemes over $A_{n}$ (cf. [Reference Hausel and PaulyHP12, Corollary 3.12], for example)

The $A_{n}$ -morphism $p:{\mathcal{C}}\rightarrow A_{n}$ induces the morphism $p^{\ast }:\text{Pic}(C)\times A\rightarrow J_{n}$ of group schemes over $A_{n}$ . One verifies that $N_{p}(p^{\ast }(-))=(-)^{\otimes n}$ ; see Fact 2.4.5(1). In particular, the morphism $N_{p}$ is surjective. The differential of the composition $N_{p}\circ p^{\ast }$ along the identity section is multiplication by $n$ , so that the morphism $N_{p}$ is smooth. The kernel $\text{Ker}(N_{p})$ of $N_{p}$ is a closed subgroup scheme that is smooth over $A_{n}$ . We call it the Prym group scheme. Its fibers are precisely the Prym varieties (8). Then, by [Reference Grothendieck and DemazureSGA3.I, Exp VI-B, Theorem 3.10], there is the open subgroup scheme over $A_{n}$

of the kernel, which (set-theoretically) is the union of the identity connected components of the fibers of this kernel group scheme over $A_{n}$ . Since this whole construction is compatible with arbitrary base change, the fiber $J_{n,a}^{\prime }$ over $a\in A_{n}$ is precisely the identity connected component of the kernel of the norm morphism associated with the spectral cover ${\mathcal{C}}_{a}\rightarrow C_{a}=C\otimes k(a)$ .

We restrict this whole picture to the $\text{SL}_{n}$ Hitchin base ${\check{A}}_{n}=A_{n}(0)\subseteq A_{n}$ and set

which we also call the Prym group scheme.

Then $\check{J}_{n}/{\check{A}}_{n}$ is a smooth connected group scheme with connected fibers over ${\check{A}}_{n}$ that acts on $M_{n}(0)/A_{n}(0)$ (trace zero) preserving $\check{M}_{n}/{\check{A}}_{n}$ (trace zero and fixed determinant $\unicode[STIX]{x1D716}$ ); see Fact 2.4.5(3) and the proof of Proposition 2.4.9. It follows that $\check{J}_{n}/{\check{A}}_{n}$ acts on $\check{M}_{n}/{\check{A}}_{n}$ .

According to Proposition 2.4.9, on each fiber $\check{M}_{n,a}$ , this action is free on the open part given by those rank one torsion-free sheaves which are locally free. The Hitchin fibers $\check{M}_{n,a}$ corresponding to nonsingular spectral curves are $\check{J}_{n,a}$ -torsors via this action.

4.2 The abelian variety parts

Let $a$ be a geometric point of $A_{n}$ ( ${\check{A}}_{n}$ , respectively). Recalling the Chevalley devissage § 2.6 for $J_{n,a}$ ( $\check{J}_{n,a}$ , respectively), we set, by taking dimensions as varieties over the algebraically closed residue field of $a$ ,

these dimensions depend only on the Zariski point underlying $a$ .

In fact, as Proposition 4.2.2 below shows, the inequality of Lemma 4.2.1 is an equality.

4.3 Product structures

4.4 Product structures, re-mixed

In analogy with Lemma 4.3.1, and keeping in mind the construction of spectral curves § 2.3 as the universal divisor inside of $V(D)\times A_{n}$ , we have the cartesian square diagram with $q,q^{\prime \prime }$ isomorphisms

where $(\unicode[STIX]{x1D70E},u_{\bullet }=(u_{2},\ldots ,u_{n}))\in H^{0}(C,D)\times A(0)$ and $(x,v)\in V(D)_{x}$ is the line fiber of $V(D)$ over a point $x\in C$ . For every fixed $(\unicode[STIX]{x1D70E},u_{\bullet })$ , the resulting morphism $q^{\prime \prime }:V(D)\stackrel{{\sim}}{\rightarrow }V(D)$ is, fiber-by-fiber, the translation in the line direction by the amount $\unicode[STIX]{x1D70E}$ (linear change of coordinates $t\mapsto t+\unicode[STIX]{x1D70E}$ ) (cf. [Reference Hausel and PaulyHP12, Remark 2.5]).

Consider the spectral curve family ${\mathcal{C}}\subseteq A_{n}\times V(D)$ and the pre-image ${\mathcal{C}}(0)$ of $A_{n}(0)$ . Then, by restricting $q^{\prime \prime }$ to ${\mathcal{C}}$ , we obtain a cartesian square diagram

with $q,q^{\prime \prime \prime }$ isomorphisms. For every fixed $(\unicode[STIX]{x1D70E},u_{\bullet })\in H^{0}(C,D)\times A_{n}(0)$ , we have the spectral curve $(\text{Id}\times p(0))^{-1}(\unicode[STIX]{x1D70E},u_{\bullet })=p(0)^{-1}(u_{\bullet })={\mathcal{C}}_{0,u_{\bullet }}$ . The morphism $q^{\prime \prime }$ maps ${\mathcal{C}}_{0,u_{\bullet }}$ isomorphically onto ${\mathcal{C}}_{q(\unicode[STIX]{x1D70E},u_{\bullet })}$ , via the fiber-by-fiber translation by the amount $\unicode[STIX]{x1D70E}$ .

By recalling that $J_{n}(0)={J_{n}}_{|A_{n}(0)}$ , and by setting $q^{iv}:=((q^{\prime \prime \prime })^{-1})^{\ast }$ , we obtain a cartesian square diagram with $q,q^{iv}$ isomorphisms.

4.5 $\unicode[STIX]{x1D6FF}$ -regularity of Prym over the elliptic locus

Recall that the elliptic locus $A_{n}^{\text{ell}}\subseteq A_{n}$ is the locus of characteristics $a\in A_{n}$ yielding geometrically integral spectral curves ${\mathcal{C}}_{a}$ . We denote by $A_{n}^{\text{ell}}(0)$ and by ${\check{A}}_{n}^{\text{ell}}$ the restriction of the elliptic locus to $A_{n}(0)={\check{A}}_{n}$ . Recall Definition 2.6.1 ( $\unicode[STIX]{x1D6FF}$ -regularity).

4.6 The norm morphism $N_{p}^{\text{ab}}$

Fix a geometric point $a$ of ${\check{A}}_{n}$ . Recall the diagram (2) of finite morphisms of curves and let us focus on $\unicode[STIX]{x1D709},p,\widetilde{p}$ . We have the surjection (36) $\unicode[STIX]{x1D709}^{\ast }:J_{n,a}=\text{Pic}^{0}({\mathcal{C}}_{a}=:\sum _{k}m_{k}\unicode[STIX]{x1D6E4}_{k})\rightarrow \text{Pic}^{0}(\widetilde{{\mathcal{C}}_{a,\text{red}}}=\coprod _{k}\widetilde{\unicode[STIX]{x1D6E4}_{k}})$ . Keeping in mind the Chevalley devissage, we have the following commutative diagram of short exact sequences completing the right-hand square in (50) (recall that $C_{a}:=C\otimes k(a)$ )

where $N_{p}^{\text{ab}}$ is the arrow induced by $N_{p}$ , in view of the fact that, since $N_{p}$ has target an abelian variety, it must be trivial when restricted to the connected and affine $\text{Ker}\,\unicode[STIX]{x1D709}^{\ast }=J_{n,a}^{\text{aff}}\subseteq J_{n,a}$ .

The arrow $N_{p}^{\text{ab}}$ is not the norm $N_{\widetilde{p}}$ associated with the morphism $\widetilde{p}$ . In fact, we have the following lemma.

4.7 The Tate module of Prym is polarizable

4.8 Recap for the $\text{SL}_{n}$ weak abelian fibration

Theorem 3.4.1 tells us that in the $D$ -twisted, $\text{GL}_{n}$ case, the triple $(M_{n},A_{n},J_{n})$ is a weak abelian fibration that is $\unicode[STIX]{x1D6FF}$ -regular over the elliptic locus.

Proposition 4.5.1 implies that the analogous conclusion holds for ( $M_{n}(0),A_{n}(0),J_{n}(0)$ ). In fact, the polarizability of the Tate module is automatic when restricting from $A_{n}$ to $A_{n}(0)$ : because $J_{n}(0)={J_{n}}_{|A_{n}(0)}$ , and the Tate module is the restriction of the Tate module. Similarly, the stabilizers are the same and they are thus affine. Even though the Chevalley devissages are un-effected when passing from $A_{n}$ to $A_{n}(0)$ , it is not a priori evident that the $\unicode[STIX]{x1D6FF}$ -regularity should be preserved (intersecting may spoil codimensions), and this is precisely what Proposition 4.5.1 ensures.

The $D$ -twisted $\text{SL}_{n}$ case, i.e.  $(\check{M}_{n},{\check{A}}_{n},\check{J}_{n})$ , is slightly trickier because, in addition to the discussion in the previous paragraph, the polarizability Theorem 4.7.2 for the Tate module $T_{\overline{\mathbb{Q}}_{\ell }}(\check{J}_{n})$ did not follow immediately from the $\text{GL}_{n}$ analogous Theorem 3.3.1.

We record for future use the following result.

4.9 Endoscopy and the $\text{SL}_{n}$ socle over the elliptic locus

We employ the notation and results in § 2.5, especially Fact 2.5.1.

According to [Reference NgôNgô10, Proposition 6.5.1], we have

In view of Theorem 4.8.1, the triple $(\check{M}_{n},{\check{A}}_{n},\check{J}_{n})$ is a weak abelian fibration that is $\unicode[STIX]{x1D6FF}$ -regular over ${\check{A}}_{n}^{\text{ell}}$ , so that we may use Ngô support theorem [Reference NgôNgô10, Theorem 7.2.1], to the effect that the supports over the elliptic locus must also be the supports appearing in (65), and conclude that

5.1 The weak abelian fibration $(M_{n_{\bullet }},A_{n_{\bullet }},J_{n_{\bullet }})$

Let $n_{\bullet }=(n_{1},\ldots ,n_{s})$ be a finite sequence of positive integers. Define

A geometric point of $A_{n_{\bullet }}^{\text{ell}}$ correspond to an ordered $s$ -tuple of geometrically integral spectral curves $(\unicode[STIX]{x1D6E4}_{1},\ldots ,\unicode[STIX]{x1D6E4}_{s})$ of respective spectral degrees $(n_{1},\ldots ,n_{s})$ .

The requirements of Definition 2.6.3 (same pure relative dimensions, affine stabilizers, polarizability of Tate modules, $\unicode[STIX]{x1D6FF}$ -regularity on the elliptic locus) are met on each factor separately by virtue of Theorem 3.4.1. (In verifying $\unicode[STIX]{x1D6FF}$ -regularity, one needs a simple application of Lemma 2.6.2(2) to each factor: let $a\in A_{n_{\bullet }}^{\text{ell}}$ ; let $x_{\bullet }$ be a closed general point in $\overline{\{a\}}$ ; let $a_{k}$ be the projection of $a$ to the $k$ th factor; then $a_{k}\in A_{n_{k}}^{\text{ell}}$ , $x_{k}$ is a closed general point of $\overline{\{a_{k}\}}$ , and $\sum _{k}d_{a_{k}}\geqslant d_{a}$ (because $\overline{\{a\}}\subseteq \prod _{k}\overline{\{a_{k}\}}$ ); we have $d_{a}^{\text{ab}}(J_{n_{\bullet }})=d_{x_{\bullet }}^{\text{ab}}(J_{n_{\bullet }})=\sum _{k}d_{x_{k}}^{\text{ab}}(J_{n_{k}})=\sum _{k}d_{a_{k}}^{\text{ab}}(J_{n_{k}})\geqslant \sum _{k}(d_{a_{k}}(J_{n_{k}})-d_{A_{n_{k}}}+d_{a_{k}})=\sum _{k}d_{a_{k}}(J_{n_{k}})-d_{A_{n_{\bullet }}}+\sum _{k}d_{a_{k}}\geqslant \sum _{k}d_{a_{k}}(J_{n_{k}})-d_{A_{n_{\bullet }}}+d_{a}=\sum _{k}d_{x_{k}}(J_{n_{k}})-d_{A_{n_{\bullet }}}+d_{a}=d_{a}(J_{n_{\bullet }})-d_{A_{n_{\bullet }}}+d_{a}$ .) It follows immediately that they are met on the product, so that (67) is a weak abelian fibration which is $\unicode[STIX]{x1D6FF}$ -regular over $A_{n_{\bullet }}^{\text{ell}}$ .

5.2 Stratification by type of the $\text{GL}_{n}$ Hitchin base $A_{n}$

Let $n\in \mathbb{Z}^{{\geqslant}1}$ and let $s\in \mathbb{Z}^{{\geqslant}1}$ with $1\leqslant s\leqslant n$ . We consider the set $NM(s)$ of pairs $(n_{\bullet },m_{\bullet })$ subject to the following requirements: (1) $n_{1}\geqslant \ldots \geqslant n_{s}$ ; (2) $m_{k}\geqslant m_{k+1}$ whenever $n_{k}=n_{k+1}$ ; (3) $\sum _{k=1}^{s}m_{k}n_{k}=n$ . There is the partition of the integral variety

into the locally closed integral subvarieties

where $\overline{a}\rightarrow a$ is given by an algebraic closure $k(a)\subseteq \overline{k(a)}$ , and each spectral curve ${\mathcal{C}}_{k,\overline{a}}$ is irreducible of spectral curve degree $n_{k}$ . The closure $\overline{S_{n_{\bullet },m_{\bullet }}}\subseteq A_{n}$ is the image of the finite morphism [Reference Chaudouard and LaumonCL16, § 9]

which on closed points is defined as follows: $(a_{1},\ldots ,a_{s})\mapsto a$ , where we view $a_{k}$ as a characteristic polynomial $P_{a_{k}}$ of degree $n_{k}$ , we consider the degree $n$ polynomial $\prod _{k=1}^{s}P_{a_{k}}^{m_{k}}$ , and we take $a$ to be the corresponding closed point on $A_{n}$ . The stratum $S_{n_{\bullet }m_{\bullet }}$ is the image of a suitable Zariski-dense open subvariety inside the Zariski-dense open subvariety $\prod _{k=1}^{s}A_{n_{k}}^{\text{ell}}\subseteq A_{n_{\bullet }}$ . Given a point $a\in A_{n}$ , we have $a\in S_{n_{\bullet }m_{\bullet }}$ for a unique triple $(s,(n_{\bullet },m_{\bullet }))$ , with $1\leqslant s\leqslant n$ the number of irreducible components of ${\mathcal{C}}_{\overline{a}}$ , and with $(n_{\bullet },m_{\bullet })\in NM(s)$ , which we call the type of $a\in A_{n}$ . Since the spectral curve ${\mathcal{C}}_{a}$ may have a strictly smaller number of components than ${\mathcal{C}}_{\overline{a}}$ , the type of $a$ is observed on ${\mathcal{C}}_{\overline{a}}$ .

Geometrically, we may think of the morphism $\unicode[STIX]{x1D706}_{n_{\bullet }m_{\bullet }}$ as sending an ordered $s$ -tuple of integral curves $(\unicode[STIX]{x1D6E4}_{1},\ldots ,\unicode[STIX]{x1D6E4}_{s})$ , to the spectral curve denoted (§ 2.3) by $\sum _{k}m_{k}\unicode[STIX]{x1D6E4}_{k}$ . As it is already clear in the case $s=2$ , with $(n_{1},n_{2};m_{1},m_{2})=(1,1;1,1)$ , in general, the finite morphisms $\unicode[STIX]{x1D706}_{n_{\bullet }m_{\bullet }}$ are not birational.

The morphisms (71) are introduced in [Reference Chaudouard and LaumonCL16, § 9] in order to exploit the $\text{GL}_{n}$ $\unicode[STIX]{x1D6FF}$ -regularity inequalities for each $J_{n_{k}}^{\text{ell}}/A_{n_{k}}^{\text{ell}}$ , $k=1,\ldots ,s$ (however, see Remark 5.4.3).

The resulting inequalities are of no use to us for the $\text{SL}_{n}$ case: they are too weak. One may be tempted to replace them by taking the multi-variable counterpart to the $\text{SL}_{n}$ $\unicode[STIX]{x1D6FF}$ -regularity inequality (64). As it turns out, these $\text{SL}_{n}$ inequalities are also of no use to us towards the proof of Theorem 1.0.2 on the $\text{SL}_{n}$ socle: they are not relevant in the proof given in § 6.2 of Theorem 1.0.2 (the $\text{SL}_{n}$ support inequality (63) plays a crucial role, though).

The multi-variable $\unicode[STIX]{x1D6FF}$ -regularity inequalities that we need for the proof of Theorem 1.0.2 on the $\text{SL}_{n}$ socle are given by Corollary 5.4.4(76), and are to be extracted from the constructions of § 5.3.