Choose $a=(A,t)\in{\overline{C}}$ with a lifting $\widehat{a}=(A,\tilde{t})$
and look at the fiber over $\widehat{a}$. Here, the subvarieties have
dimension $2$ and we want to show (for general $a$) that they do not share any
$2$-dimensional components.

In view of the decomposition of the fiber $\widehat{{\operatorname{{\sf
Residual}}}}_{0}(a)$ into $16$ pieces that move around under the monodromy
operation when we move $a$, whereas on the other hand
$\widehat{{\operatorname{{\sf Residual}}}}_{1}(a)$ can’t have more than $4$
irreducible components since it is a $4$-sheeted covering of the irreducible
${\rm Wob}_{1}$, we see that $\widehat{{\operatorname{{\sf Residual}}}}_{0}$
and $\widehat{{\operatorname{{\sf Residual}}}}_{1}$ can not share irreducible
components. Thus, we are reduced to the question of showing that
$\widehat{{\operatorname{{\sf Residual}}}}_{1}$ and $\widehat{K}$ do not share
any components.

Now, by Lemma 7.11, the image of ${\widehat{\mathcal{H}}}^{\rm ab}(\tilde{a})$
in ${\overline{\mathcal{H}}}(a)$ only intersects the Hecke fibers over the
Kummer $\mathsf{Kum}$, in the intersection points of the two lines. But this
collection of points maps to the Kummer K3 surface inside $X_{1}$ that isn’t
the same as ${\rm Wob}_{1}$. Thus, the intersection
$\widehat{{\operatorname{{\sf Residual}}}}_{1}\cap\widehat{K}$ does not have
any $2$-dimensional pieces. This completes the proof of the theorem. ∎

######  Corollary 7.18.

Inside the big Hecke correspondence ${\overline{\mathcal{H}}}$, the
intersection of the two pullbacks of the wobbly loci with the abelianized
Hecke is, as for its $3$-dimensional pieces:

$p^{-1}({\rm Wob}_{1})\cap d^{-1}({\rm Wob}_{0})\cap
g({\widehat{\mathcal{H}}}^{\rm
ab})=g(\boldsymbol{\mathit{Exc}}_{0}\cup\boldsymbol{\mathit{Exc}}_{1}).$

###### Proof.

This is the image by $g:{\widehat{\mathcal{H}}}^{\rm
ab}\rightarrow{\overline{\mathcal{H}}}$ of the statement in the previous
theorem. ∎

Recall that for each $a\in{\widehat{C}}$, the piece
$g(\boldsymbol{\mathit{Exc}}_{0})(a)$ in the fiber
${\overline{\mathcal{H}}}(a)$ contracts to a lower-dimensional subvariety (in
this case the union of $16$ lines in the wobbly locus) under the projection to
$X_{1}$. And similarly, the piece $g(\boldsymbol{\mathit{Exc}}_{1})(a)$
contracts to a lower-dimensional subvariety (a birational copy of the curve
${\overline{C}}$ in $\mathsf{Kum}$) under the projection to $X_{0}$.

The piece $g(\boldsymbol{\mathit{Exc}}_{0})(a)$ mapping to $X_{0}$ consists of
$16$ planes over the trope planes, which are the ramification of the
horizontal divisor in the Hecke correspondence that contribute to the
singularities of the Hecke transformed system from $X_{1}$ to $X_{0}$. The
singularities over the Kummer surface come from the degenerations of Hecke
conics to pairs of lines.

On the other hand, the piece $g(\boldsymbol{\mathit{Exc}}_{1})(a)$ mapping to
$X_{1}$ is the ramification of the inverse image of the Kummer surface in the
Hecke correspondence, giving the singularities of the Hecke transformed system
from $X_{0}$ to $X_{1}$.

In the next chapters we will be testing the Hecke transforms by restricting
over lines in the target spaces, so the ramification locations contributing to
singularities will show up as intersected with the inverse images of the lines
in the Hecke correspondence.

##  8 Hecke transformation from $X_{0}$ to $X_{1}$

We recall the standard diagrams. The big Hecke correspondence fits into a
diagram of the form

$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{b}$$\textstyle{X_{0}}$$\textstyle{X_{1}\times{\overline{C}}.}$

Fix a point $a=(A,t)\in{\overline{C}}$ where $A^{\otimes
2}={\mathcal{O}}_{C}(t-\boldsymbol{\mathsf{p}})$ and consider the Hecke
correspondence ${\overline{\mathcal{H}}}(\overline{a})$ fitting into the
diagram

$\textstyle{{\overline{\mathcal{H}}}(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{b}$$\textstyle{X_{0}}$$\textstyle{X_{1}.}$

The objective is to pull-back the constructed Higgs bundle from $X_{0}$ and
take the higher direct image along $b$ to $X_{1}$.

In order to simplify the measurement of the result, fix a general line
$\ell\subset X_{1}$. Let ${\overline{\mathcal{H}}}(a)_{\ell}$ be the inverse
image of $\ell$ in the Hecke variety ${\overline{\mathcal{H}}}(a)$. This is a
Hirzebruch surface $\mathbb{F}_{1}$, which is a ${\mathbb{P}}^{1}$-bundle over
$\ell\cong{\mathbb{P}}^{1}$.

Let $P\subset X_{0}$ be the image of the map from
${\overline{\mathcal{H}}}(a)_{\ell}$ to $X_{0}$. This is a plane
$P\cong{\mathbb{P}}^{2}\subset{\mathbb{P}}^{3}$. The map
$d:{\overline{\mathcal{H}}}(a)_{\ell}\rightarrow P$ is blowing up a point
$\boldsymbol{\mathfrak{p}}\in P$. This point is a general point contained in
the Kummer $\mathsf{Kum}\subset X_{0}$. There is also a point
$\boldsymbol{\mathfrak{q}}\in P\cap\mathsf{Kum}$ where $P$ is tangent to
$\mathsf{Kum}$. In particular $\boldsymbol{\mathfrak{p}}$ and
$\boldsymbol{\mathfrak{q}}$ are not on the trope planes.

Let $\boldsymbol{\mathcal{T}}\subset{\overline{\mathcal{H}}}(a)_{\ell}$ be the
strict transform of $\mathsf{Kum}\cap P$, and let
$\boldsymbol{\mathcal{N}}\subset{\overline{\mathcal{H}}}(a)_{\ell}$ be the
exceptional divisor. It is the unique section of the $\mathbb{F}_{1}$ surface
that has self-intersection $-1$.

We have that
$\boldsymbol{\mathcal{T}}\cap\boldsymbol{\mathcal{N}}=\boldsymbol{\mathfrak{p}}^{\prime}$
is a single point in ${\overline{\mathcal{H}}}(a)_{\ell}$. The point
$\boldsymbol{\mathfrak{q}}$ corresponds to a unique point
$\boldsymbol{\mathfrak{q}}^{\prime}\in{\overline{\mathcal{H}}}(a)_{\ell}$
where the curve $\boldsymbol{\mathcal{T}}$ has a node. The distinction in
notation here is that $\boldsymbol{\mathfrak{p}}\in P$ is a point of the plane
inside $X_{0}$, while $\boldsymbol{\mathfrak{p}}^{\prime}$ is one of the
points in the exceptional divisor above $\boldsymbol{\mathfrak{p}}$, namely
the intersection point of $\boldsymbol{\mathcal{T}}$ and
$\boldsymbol{\mathcal{N}}$; it also corresponds to the tangent direction of
the Hecke line at $\boldsymbol{\mathfrak{p}}$ or equivalently the tangent
direction of $P\cap\mathsf{Kum}$ at $\boldsymbol{\mathfrak{p}}$. The points
$\boldsymbol{\mathfrak{q}}$ and $\boldsymbol{\mathfrak{q}}^{\prime}$ are
basically identical since there is no blowing-up, we just use the notation
$\boldsymbol{\mathfrak{q}}^{\prime}$ rather than $\boldsymbol{\mathfrak{q}}$
for uniformity.

Figure 1: The Hirzebruch surface ${\overline{\mathcal{H}}}(a)_{\ell}$

Let $Y_{1,\ell}\subset Y_{1}$ be the inverse image of $\ell$. This is a smooth
curve in view of the moving properties of the family of lines $\ell$, indeed,
locally this family looks like the family of complete intersections of two
moving divisors in $X_{1}$.

###  8.1 The abelianized Hecke as a critical locus

The big abelianized Hecke fits into a diagram of the form

$\textstyle{{\widehat{\mathcal{H}}}^{\rm
ab}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{\rm
ab}}$$\scriptstyle{b^{\rm
ab}}$$\textstyle{Y_{0}}$$\textstyle{Y_{1}\times{\widehat{C}}.}$

The inverse image in ${\widehat{C}}$ of $a\in{\overline{C}}$ consists of two
points that we will note $\widehat{a}$ and $\tau\widehat{a}$. We obtain two
abelianized Hecke varieties ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$
and ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})$. Denote their disjoint
union by

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a}):={\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})\sqcup{\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a}).$

The map ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})\rightarrow Y_{1}$ is
the blow-up along $16$ lines contained in the wobbly locus, as was discussed
in Section 7.2.

A general line $\ell\subset X_{1}$ does not meet those lines. Thus, if we let
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ denote the inverse
image of $\ell$ in ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$, the
projection induces an isomorphism

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}Y_{1,\ell}.$

The same holds for the other piece ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$.

The curve ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ has degree
$8$ over $\ell$. The full curve of degree $16$ over $\ell$ is

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}={\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}\sqcup{\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}.$

The $16$ exceptional divisors in ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ contract to the $16$ exceptional planes in $Y_{0}$.
It follows that the image of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ in $Y_{0}$ does not meet those.

Let $Y_{0,P}:=Y_{0}\times_{X_{0}}P$, and set

$\Sigma:=Y_{0,P}\times_{P}{\overline{\mathcal{H}}}(a)_{\ell}.$

This is the spectral variety for the Higgs bundle
$(\mathscr{F}_{0,\bullet},\Phi_{0})_{{\overline{\mathcal{H}}}(a)_{\ell}}$
which is $(\mathscr{F}_{0,\bullet},\Phi_{0})$ pulled back to
${\overline{\mathcal{H}}}(a)_{\ell}$ via the map
$d:{\overline{\mathcal{H}}}(a)_{\ell}\to X_{0}$. In particular $\Sigma$ has
degree $8$ over ${\overline{\mathcal{H}}}(a)_{\ell}$. We have a lifting

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\rightarrow\Sigma.$

######  Proposition 8.1.

The map ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\rightarrow\Sigma$ identifies the
abelianized Hecke as the upper critical locus (see Proposition 12.1)

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\cong\widetilde{{\rm
Crit}}\left({\overline{\mathcal{H}}}(a)_{\ell}/\ell,(\mathscr{F}_{0,\bullet},\Phi_{0})_{{\overline{\mathcal{H}}}(a)_{\ell}}\right)$

away from the points $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$.

###### Proof.

Suppose we are given a point of $\Sigma$. It corresponds to a point of $Y_{0}$
together with a Hecke operation leading to a Hecke transformed bundle which is
a point of $\ell\subset X_{1}$.

If the point of $Y_{0}$ is on an exceptional divisor, it means that the point
of $\Sigma$ is a ramification point of $\Sigma$ over the horizontal divisor in
${\overline{\mathcal{H}}}(a)_{\ell}$. The residues of the Higgs field
$\Phi_{0}$ are nilpotent, so their Jordan types must be constant along the
parts of the horizontal divisor that are etale over $\ell$, and this implies
that ramification points of $\Sigma$ over this subset of the horizontal
divisor can not be in the relative critical locus (we will see this argument
again in the next subsection). The points $\boldsymbol{\mathfrak{p}}^{\prime}$
and $\boldsymbol{\mathfrak{q}}^{\prime}$, as well as the ramification points
of the horizontal divisor, are contained in the image of
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$ and
also in the relative critical locus.

We claim that the upper critical locus is smooth at the ramification points of
the horizontal divisor, i.e. at points where $\boldsymbol{\mathcal{T}}$
ramifies over $\ell$. In local coordinates $x,t$ where the map to $\ell$ is
given by $t$, we can write $\boldsymbol{\mathcal{T}}$ as $x^{2}-t=0$, and
$\Sigma$ is $x^{2}-t=w^{2}$. Thus $x,w$ are coordinates on $\Sigma$ and
$t=x^{2}-w^{2}$. Write $\alpha$ as $adx+bdw$. The vertical direction over the
ramification point is the $w$-axis, and the fact that the Jordan form of the
residue of the Higgs field is constant along smooth points of ${\rm Wob}_{1}$
(our ramification is such a smooth point) implies that $\alpha$ is nonzero on
the vertical direction, as was pointed out in Lemma 8.3. This says that
$b(0)\neq 0$. Now, the relative differentials are obtained by setting $dt=0$
so $xdx=wdw$. Thus $\alpha^{\rm rel}$ is written as

$\alpha^{\rm rel}=(a+(x/w)b)dx$

and the condition $\alpha^{\rm rel}=0$ may be written as $wa+xb=0$. As
$b(0)\neq 0$, the linear term $wa(0)+xb(0)$ is nonzero, so this defines a
smooth curve which is the local branch of the relative critical locus at this
point of $\Sigma$.

This shows that the relative critical locus is smooth at ramification points
of $\boldsymbol{\mathcal{T}}/\ell$, so a set-theoretical identification
between the two subvarieties (neither of which has embedded points) is an
isomorphism.

Over the points $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$, we will see that the relative critical
locus has two branches in each of the four nodes of $\Sigma$ over one of these
points, and the abelianized Hecke is isomorphic to the normalization.

In view of the previous paragraphs, we can now assume that the point of
$Y_{0}$ is not on an exceptional divisor, so it corresponds to a line bundle
$L$ of degree $2$ on ${\widetilde{C}}$ whose norm to $C$ is $\omega_{C}$. The
corresponding vector bundle on $C$ is $E=\pi_{*}L$. Our Hecke transformation
is given by a pair $(A,x)$ where $x\in C$ and $A^{\otimes
2}={\mathcal{O}}_{C}(x-\boldsymbol{\mathsf{p}})$. The Hecke transformation is
centered at a line $V\subset E_{x}$ and we let $E^{\prime}$ be the kernel of
the map $E\rightarrow E_{x}/V$. This is a bundle of degree $-1$ with
determinant ${\mathcal{O}}_{C}(-x)$. Then $E^{\prime}\otimes
A(\boldsymbol{\mathsf{p}})$ has degree $1$ and determinant
${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$ so it is a point in $X_{1}$.

In turn we have a line $V^{\prime}\subset E^{\prime}(x)_{x}$ and the kernel of
$E^{\prime}\rightarrow E^{\prime}(x)_{x}/V^{\prime}$ is $E$. The line
$V^{\prime}$ corresponds to a line in $E^{\prime}\otimes
A(\boldsymbol{\mathsf{p}})$ and the opposite Hecke transform involving
tensoring again with a square-root, gets us back to $E$. The Hecke line
through the point $[E]$ consists of all the bundles obtained as kernels of
$E^{\prime}\rightarrow E^{\prime}(x)/W$ where $W$ is another line $W\subset
E^{\prime}(x)_{x}$. Letting $W$ be an infinitesimal deformation of
$V^{\prime}$ we get a tangent vector to $X_{0}$ at the point $[E]$,
represented by a class $\eta\in H^{1}(\text{End}_{0}(E))$. We’ll give an
expression for this class using an exact sequence, below.

The tautological $1$-form on $Y_{0}$ is a section of
$f_{0}^{*}T_{X_{0}}^{\vee}$ At the point $[L]\in Y_{0}$ corresponding to the
line bundle $L$ that projects to $[E]\in X_{0}$,
$\alpha([L])\in\left(f_{0}^{*}T^{\vee}_{X_{0}}\right)_{[L]}=T^{\vee}_{X_{0},[E]}$
can be paired with a tangent vector of the form $\eta$. The condition that we
are at a point of the upper critical locus means that the value of the pairing
is zero.

Our point on $Y_{0}$ actually represents a Higgs bundle over $C$, because
$Y_{0}$ was the blow-up of ${\mathcal{P}}_{2}\subset\mathsf{Higgs}_{0}$. The
underlying bundle is $E$ (since we are assuming that we are not over the
wobbly locus), and the Higgs field $\theta:E\to E\otimes\omega_{C}$ is induced
by the tautological $1$-form of ${\widetilde{C}}$. In these terms, the pairing
is just the pairing between $\theta\in
H^{0}(\text{End}_{0}(E)\otimes\omega_{C})$ and $\eta\in
H^{1}(\text{End}_{0}(E))$. The upper critical locus condition is that this
pairing is $0$.

Let us now look more closely at the deformation class $\eta$. The deformation
of $V^{\prime}$ is given by an element of
$\text{Hom}(V^{\prime},E^{\prime}(x)_{x}/V^{\prime})$. Tensor

$0\rightarrow E\rightarrow E^{\prime}(x)\rightarrow
E^{\prime}(x)_{x}/V^{\prime}\rightarrow 0$

with $E^{\vee}$ to get

$0\rightarrow\operatorname{End}(E)\rightarrow E^{\vee}\otimes
E^{\prime}(x)\rightarrow E_{x}^{\vee}\otimes
E^{\prime}(x)_{x}/V^{\prime}\rightarrow 0.$

Note that $E_{x}\rightarrow V^{\prime}$ so $(V^{\prime})^{\vee}\rightarrow
E_{x}^{\vee}$. Compose with the connecting map for the previous exact sequence
as follows:

$(V^{\prime})^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime})\rightarrow
E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime})\rightarrow
H^{1}(\operatorname{End}(E)).$

The image of an element of
$\operatorname{Hom}(V^{\prime},(E(x)_{x}/V^{\prime}))$ is the deformation
class of $E$ generated by doing the Hecke operation back using the
infinitesimally close $W$. The determinant of the new bundle is the same as
that of $E$, so the deformation class lies in the trace-free part
$H^{1}(\operatorname{End}_{0}(E))$.

The exact sequence for the fiber of the Hecke transformation is

$0\rightarrow E^{\prime}_{x}/V^{\prime}(-x)\rightarrow E_{x}\rightarrow
V^{\prime}\rightarrow 0.$

The exact sequence

$0\rightarrow\operatorname{End}(E)\rightarrow E^{*}\otimes
E^{\prime}(x)\rightarrow
E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime})\rightarrow 0.$

has as dual, taking $\operatorname{Ext}^{1}(-,\omega_{C})$:

$0\rightarrow
E\otimes(E^{\prime})^{\vee}(-x)\otimes\omega_{C}\rightarrow\operatorname{End}(E)\otimes\omega_{C}\rightarrow\operatorname{Ext}^{1}(E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime}),\omega_{C})\rightarrow
0.$

We have

$\operatorname{Ext}^{1}(E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime}),\omega_{C})=\operatorname{Hom}({\mathcal{O}}_{C},E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime}))^{\vee}=\operatorname{Hom}((E^{\prime}(x)_{x}/V^{\prime}),E_{x}).$

This maps to $\operatorname{Hom}(E^{\prime}(x)_{x}/V^{\prime},V^{\prime})$.
Use the residue identification $\omega_{C}(x)_{x}\cong{\mathbb{C}}$ and hence
$V^{\prime}\otimes\omega_{C}(x)\cong V^{\prime}$, to say that

$\operatorname{Hom}(E^{\prime}(x)_{x}/V^{\prime},V^{\prime})\cong\operatorname{Hom}(E^{\prime}(x)_{x}/V^{\prime},V^{\prime}\otimes\omega_{C}(x))\cong\operatorname{Hom}(E^{\prime}_{x}/V^{\prime}(-x),V^{\prime}\otimes\omega_{C}).$

The previous map now becomes

$\operatorname{Ext}^{1}(E_{x}^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime}),\omega_{C})\rightarrow\operatorname{Hom}(E^{\prime}_{x}/V^{\prime}(-x),V^{\prime}\otimes\omega_{C}).$

Altogether, the second map in the dual exact sequence becomes

$\operatorname{End}(E)\otimes\omega_{C}\rightarrow\operatorname{Hom}(E^{\prime}_{x}/V^{\prime}(-x),V^{\prime}\otimes\omega_{C}),$

and this is just the evaluation map evaluating a Higgs field on the subspace
$E^{\prime}_{x}/V^{\prime}(-x)\subset E_{x}$ and projecting the answer to the
quotent $V^{\prime}$ tensored with $\omega_{C}$.

The Serre dual of the map
$(V^{\prime})^{\vee}\otimes(E^{\prime}(x)_{x}/V^{\prime})\rightarrow
H^{1}(\operatorname{End}(E))$ is the action of the previous map on global
sections:

$H^{0}(\operatorname{End}(E)\otimes\omega_{C})\rightarrow\operatorname{Hom}(E^{\prime}_{x}/V^{\prime}(-x),V^{\prime}\otimes\omega_{C}),$

To say that this vanishes for a Higgs field $\theta:E\rightarrow
E\otimes\omega_{C}$ is equivalent to saying that the filtration with subspace
$E^{\prime}_{x}/V^{\prime}(-x)\subset E_{x}$ and quotient $E_{x}\rightarrow
V^{\prime}$ is respected by the Higgs field.

This is now the same thing as saying that our original Hecke transformation
was in the abelianized Hecke. ∎

######  Lemma 8.2.

For a general $\ell$, ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ does not intersect the inverse images
in ${\overline{\mathcal{H}}}(a)_{\ell}$ of the $32$ points in $P$ given by the
intersection of $P$ with the trope conics.

###### Proof.

Recall that ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}\rightarrow
Y_{0}$ is the blow-up along a curve isomorphic to ${\widehat{C}}$, this curve
intersecting the planes of the exceptional locus in $Y_{0}$ in a finite set.
The trope conics are conics in these planes, so their inverse image in
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ has dimension $1$. This
projects to a dimension $1$ subset of $X_{1}$, so a general line $\ell$ does
not intersect that. The same holds for the other piece. So,
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})$ intersected
with the pullbacks of the trope conics, intersected with the pullback of
$\ell$, will be empty. ∎

###  8.2 Description of the configuration

We look more closely at the details of the configuration inside and above
${\overline{\mathcal{H}}}(a)_{\ell}$. Inside $\Sigma$, there are $16$ lines
$L_{\kappa,\Sigma}\subset\Sigma$ lying over the (strict transforms in
${\overline{\mathcal{H}}}(a)_{\ell}$ of the) intersections
$P\cap\mathsf{Trope}_{\kappa}$ of $P$ with the trope planes, on which
$\Sigma/{\overline{\mathcal{H}}}(a)_{\ell}$ has a simple ramification. Lemma
8.2 says that the curve ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ does not meet these
$L_{\kappa,\Sigma}$.

Over general points of
$\boldsymbol{\mathcal{T}}\cup\boldsymbol{\mathcal{N}}\subset{\overline{\mathcal{H}}}(a)_{\ell}$
the ramification of $\Sigma$ consists of four sheets each of which is a simple
ramification. We also know that $\Sigma$ has four ordinary double points over
each of $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$. There are probably other singularities
for example over the intersection of $P$ with the trope conics, however the
curve ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$
does not touch those.

There is a spectral line bundle $\mathscr{L}_{0}$ on $Y_{0}$, which pulls back
to a line bundle $\mathscr{L}_{\Sigma}$ on $\Sigma$, such that
${\mathcal{E}}_{{\overline{\mathcal{H}}}(a)_{\ell}}$ is the pushforward of
$\mathscr{L}_{\Sigma}$ from $\Sigma$ to ${\overline{\mathcal{H}}}(a)_{\ell}$.
The Higgs field of ${\mathcal{E}}_{{\overline{\mathcal{H}}}(a)_{\ell}}$ is
given by a holomorphic $1$-form $\alpha$ on the smooth locus of $\Sigma$. It
projects to a section of
$\Omega^{1}_{{\overline{\mathcal{H}}}(a)_{\ell}/\ell}(\log D)|_{\Sigma}$ where
the divisor $D$ is the union of $\boldsymbol{\mathcal{T}}$,
$\boldsymbol{\mathcal{N}}$ and the strict transforms in
${\overline{\mathcal{H}}}(a)_{\ell}$ of the $16$ trope lines
$P\cap\mathsf{Trope}_{\kappa}$.

Along a point where $\boldsymbol{\mathcal{T}}\cup\boldsymbol{\mathcal{N}}$ is
horizontal over $\ell$, the residue of the Higgs field consists of a sum of
four nonzero nilpotent transformations, since the Jordan type of the residue
has to stay fixed along the divisor because it corresponds to monodromy.
Furthermore, over $\boldsymbol{\mathcal{N}}$ at least, the inverse image of
$\boldsymbol{\mathcal{N}}$ in $\Sigma$ is a union of four lines along which
the map has simple ramification. Thus, the form $\alpha$ is nonvanishing in
the transverse direction to these lines so as a section of
$\Omega^{1}_{{\overline{\mathcal{H}}}(a)_{\ell}/\ell}(\log D)|_{\Sigma}$ it is
nonvanishing along these lines except at the points over
$\boldsymbol{\mathfrak{p}}^{\prime}$. On the other hand, the curve
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$ is the
vanishing locus of the section of
$\Omega^{1}_{{\overline{\mathcal{H}}}(a)_{\ell}/\ell}(\log D)|_{\Sigma}$. It
follows that ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\subset\Sigma$ does not intersect the
four lines over $\boldsymbol{\mathcal{N}}$ except at points over
$\boldsymbol{\mathfrak{p}}^{\prime}$. In sum, the image of
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$ in
${\overline{\mathcal{H}}}(a)_{\ell}$ only intersects
$\boldsymbol{\mathcal{N}}$ at $\boldsymbol{\mathfrak{p}}^{\prime}$. Similar
considerations show that this also holds for the horizontal part of
$\boldsymbol{\mathcal{T}}-\\{\boldsymbol{\mathfrak{p}}^{\prime},\boldsymbol{\mathfrak{q}}^{\prime}\\}$.

We would like to know how many branches of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ pass through the point
$\boldsymbol{\mathfrak{p}}^{\prime}$. We will see by the local calculations
below that each double point of $\Sigma$ over
$\boldsymbol{\mathfrak{p}}^{\prime}$ corresponds to two branches of the polar
curve of zeros of the relative Higgs field. This says that the full curve
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$ has $2$
branches in each of the four double points, thus it has $8$ branches over
$\boldsymbol{\mathfrak{p}}^{\prime}$. A monodromy argument says that these
have to be evenly distributed, so ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ has $4$ branches passing through
$\boldsymbol{\mathfrak{p}}^{\prime}$. The local argument will also show that
they intersect $\boldsymbol{\mathcal{N}}$ transversally. Thus, if we denote by
$[{\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}]$ the image of this
curve in ${\overline{\mathcal{H}}}(a)_{\ell}$, we have

$[{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}]\cdot\boldsymbol{\mathcal{N}}=4.$

We will discuss later the distribution of these $4$ branches among the $4$
double points of $\Sigma$.

We next consider intersection numbers inside the $\mathbb{F}_{1}$ surface
${\overline{\mathcal{H}}}(a)_{\ell}$. Let $\mathsf{fib}$ denote a general
fiber of $b:{\overline{\mathcal{H}}}(a)_{\ell}\rightarrow\ell$. Then the
divisors $\mathsf{fib}$ and $\boldsymbol{\mathcal{N}}$ generate the Picard
group of ${\overline{\mathcal{H}}}(a)_{\ell}$, with $\mathsf{fib}^{2}=0$,
$\boldsymbol{\mathcal{N}}^{2}=-1$ and
$\mathsf{fib}\cdot\boldsymbol{\mathcal{N}}=1$.

The curve ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ has degree
$8$ over $\ell$, so

$[{\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}]\cdot\mathsf{fib}=8.$

The intersection numbers with $\boldsymbol{\mathcal{N}}$ and $\mathsf{fib}$
uniquely determine the class of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ inside ${\overline{\mathcal{H}}}(a)_{\ell}$:

$[{\widehat{\mathcal{H}}}^{\rm ab}(\tilde{a})_{\ell}]\sim
8\boldsymbol{\mathcal{N}}+12\mathsf{fib}.$

On the other hand, we also have that $\boldsymbol{\mathcal{T}}$ has degree $3$
over $\ell$ since the intersection with a fiber is all the points in that line
intersected the Kummer, except the point $\boldsymbol{\mathfrak{p}}^{\prime}$
which corresponds to the point of $\boldsymbol{\mathcal{N}}$ intersected that
fiber. We have $\boldsymbol{\mathcal{T}}\cdot\mathsf{fib}=3$ and
$\boldsymbol{\mathcal{T}}\cdot\boldsymbol{\mathcal{N}}=1$ so

$\boldsymbol{\mathcal{T}}\sim 3\boldsymbol{\mathcal{N}}+4\mathsf{fib}.$

Combining these we get

$[{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}]\cdot\boldsymbol{\mathcal{T}}=(8\boldsymbol{\mathcal{N}}+12\mathsf{fib})(3\boldsymbol{\mathcal{N}}+4\mathsf{fib})=36+32-24=44.$

We already know the intersections over the point
$\boldsymbol{\mathfrak{p}}^{\prime}$, there are $4$ branches. Also over
$\boldsymbol{\mathfrak{q}}^{\prime}$ there will similarly be $4$ branches, but
$\boldsymbol{\mathfrak{q}}^{\prime}$ is a double point of
$\boldsymbol{\mathcal{T}}$ (whereas the double point of $D$ at
$\boldsymbol{\mathfrak{p}}^{\prime}$ involved both $\boldsymbol{\mathcal{T}}$
and $\boldsymbol{\mathcal{N}}$). Thus, the intersection counts for $8$ points
over $\boldsymbol{\mathfrak{q}}^{\prime}$. This leaves $32$ intersection
points over the ramification points of $\boldsymbol{\mathcal{T}}/\ell$. We
note that there are $8$ such points, mapping to the $8$ points in $\ell$ where
$\ell\subset X_{1}$ intersects ${\rm Wob}_{1}$. We get an intersection number
of $4$ at each ramification point (a monodromy argument shows that they need
to be evenly distributed). This will come from two tacnodes between the curve
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ and the divisor
$\boldsymbol{\mathcal{T}}$, in accordance with our calculations for the local
contribution of the direct image at a simple ramification point of the
horizontal divisor.

The curve ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ has to meet
the ramified locus of $Y_{0}$ over $\mathsf{Kum}$ in two of the four sheets,
as each one will contribute a tacnode and from the above discussion there
should not be more than $2$ tacnodes. The curve ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$ meets the ramified locus in the other two sheets.
On the other hand, for the nodal points we’ll see that
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ has one branch in each
sheet and ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})_{\ell}$ has the
other branch.

For each of the direct summands of the direct image Higgs bundle, we will get
two ramification points of the spectral variety over each point of
$\ell\cap{\rm Wob}_{1}$. This agrees with the ramification pattern of
$Y_{1}\rightarrow X_{1}$ over ${\rm Wob}_{1}$.

We need to verify the part of Hypothesis 3.16 about points of type 3.11.1(d),
namely the ramification points of $\boldsymbol{\mathcal{T}}$ over $\ell$.

######  Lemma 8.3.

For a general line $\ell$, the value of the spectral $1$-form $\alpha$ on the
vertical direction in the tangent space of $Y_{0}$ at any of the ramification
points of $\boldsymbol{\mathcal{T}}/\ell$, is nonzero.

###### Proof.

As $\ell$ varies, each of these ramification points varies and constitutes a
general point of $\mathsf{Kum}$. So it suffices to note that the value of
$\alpha$ on the vertical direction is nonzero. This is because the nilpotent
residue of the Higgs field is nonzero along $\mathsf{Kum}$, which in turn is a
consequence of the Bogomolov-Gieseker inequality as explained in the proof of
Lemma 8.9 below. ∎

###  8.3 Apparent singularities

The discriminant of
$({\overline{\mathcal{H}}}(a)_{\ell},\boldsymbol{\mathcal{T}}\cup\boldsymbol{\mathcal{N}})$
relative to $\ell$ consists of the following kinds of points:

  * •

images of simple ramification points of $\boldsymbol{\mathcal{T}}$ over
$\ell$, these are the points of $\ell\cap{\rm Wob}_{1}$ and we expect
singularities of the local system at these points;

  * •

images of points where the Hecke line goes through a trope conic— by Lemma 8.2
these points are not contained in the lower critical locus, so by Proposition
12.2 they do not contribute singularities of the higher direct image local
system; and

  * •

images of the points $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$ where
$\boldsymbol{\mathcal{T}}\cup\boldsymbol{\mathcal{N}}$ has a node—we’ll see
for topological reasons below that these do not contribute singularities, and
indeed the Dolbeault higher direct image consideration will also show that.

The following lemma gives the topological proof for the third part.

######  Lemma 8.4.

Suppose given a point where the horizontal divisor has a simple normal
crossing with both branches étale over the base, and suppose that the
monodromy transformations around each branch are direct sums of identical size
$2$ unipotent Jordan blocks. Then such a point does not contribute to the
monodromy of the higher direct image local system.

###### Proof.

This description characterizes the nontrivial local rank $2$ pieces of the
local system upstairs. However, we also obtain the same type of situation if
we start with a unipotent rank $2$ local system having singularities along a
horizontal divisor that is simply ramified over the base, then pull back to a
simply ramified double cover of the base. We will see in subsection 12.4.3
that the monodromy for a simply ramified horizontal divisor with unipotent
local system, is a transformation of order $2$. Therefore, its pullback by a
ramified double cover has trivial monodromy transformation. ∎

Even though the topological monodromy transformation for such a point is
trivial, we still need to use the description of [DPS16] to get the
description of the spectral line bundle for the higher direct image at such a
point, since the map is singular.

######  Corollary 8.5.

The singularities of the higher direct image local system over $\ell$ are
located at the points of $\ell\cap{\rm Wob}_{1}$.

######  Remark 8.6.

In the situation of Lemma 8.4, if we had monodromy over the nodal horizontal
divisor decomposing into size $2$ blocks with a reflection instead of a
nilpotent monodromy transformations, then the higher direct image would have
the square of a nilpotent transformation (subsection 12.4.1), so again a
nontrivial nilpotent transformation. Applied to the Hecke situation, this
means that if we do the Hecke transformation starting with a local system with
parabolic weights $1/2$ over the Kummer, we will get singularities at the
images of $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$ in $\ell$. These are probably the
intersection of $\ell$ with a singular K3 surface inside $X_{1}$, the image of
the K3 inside the Hecke variety, and which is the intersection of $X_{1}$ with
another quadric in ${\mathbb{P}}^{5}$.

####  8.3.1 General position arguments

Let us now look locally at the point $\boldsymbol{\mathfrak{p}}^{\prime}$. The
horizontal divisor $\boldsymbol{\mathcal{T}}\cup\boldsymbol{\mathcal{N}}$ has
a node there, and the spectral variety $\Sigma$ decomposes as a union of four
ordinary double points.

The point $\boldsymbol{\mathfrak{q}}^{\prime}$ is a place where
$\boldsymbol{\mathcal{T}}$ has a node, with both branches being étale over the
base $\ell$. This corresponds to a point where the plane $P$ is tangent to
$\mathsf{Kum}$. Again, $\Sigma$ decomposes into $4$ pieces, and each piece
restricted to the tangent plane gives an ordinary double point.

In order to apply the construction of Theorems 3.17 and 12.12, we need the
following result.

######  Theorem 8.7.

The points $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$ are points of type 3.11.1(e) in the
classification of Subsection 3.11.

###### Proof.

The geometric picture shows that the inverse image of
$\boldsymbol{\mathfrak{p}}^{\prime}$ resp.
$\boldsymbol{\mathfrak{q}}^{\prime}$ in $\Sigma$ decomposes into four ordinary
double points. Recall that over a general point of $\mathsf{Kum}$, the
covering $Y_{0}\rightarrow X_{0}$ decomposes into four local pieces each of
which is a double cover simply ramified over the local piece of
$\mathsf{Kum}$. Each of these four pieces leads to a simple double point in
the fiber of $\Sigma$, either over $\boldsymbol{\mathfrak{p}}^{\prime}$ or
over $\boldsymbol{\mathfrak{q}}^{\prime}$.

Over $\boldsymbol{\mathfrak{p}}^{\prime}$, we are restricting to a transverse
plane where it is again a smooth double cover ramified over a smooth curve,
then blowing up the origin. The point $\boldsymbol{\mathfrak{p}}^{\prime}$
corresponds to the place where the exceptional divisor
$\boldsymbol{\mathcal{N}}$ meets the strict transform
$\boldsymbol{\mathcal{T}}$ of $\mathsf{Kum}$, and the double cover gives there
an ordinary double point.

Over $\boldsymbol{\mathfrak{q}}^{\prime}$, we have a plane that is tangent to
$\mathsf{Kum}$ with nondegenerate second fundamental form, such that the two
branches of the intersection between the plane and $\mathsf{Kum}$ correspond
to the two branches of $\boldsymbol{\mathcal{T}}$ at
$\boldsymbol{\mathfrak{q}}^{\prime}$. The restriction of the local double
cover to the tangent plane is again an ordinary double point.

To complete the verification of the conditions required for the classification
of our points in the category “type 3.11.1(e)”, we need to show that the
relative critical locus has two branches at each node of $\Sigma$, with the
branches having distinct tangent vectors in the tangent cone of $\Sigma$.

A degree and monodromy calculation shows that the relative critical locus,
that we have identified also with ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$, has two branches at each node of
$\Sigma$. We need to show that they are smooth with distinct tangent vectors.
This statement amounts to a general position argument concerning the
relationship between the local geometry at a general point of $\mathsf{Kum}$,
and the tautological $1$-form on the covering $Y_{0}/X_{0}$.

The cases of $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$ are different although similar. The basic
idea is to find three (or more) directions in a plane that come from the
geometry of the moduli spaces, independent of the point in the Hitchin base;
then the tautological $1$-form is a fourth direction. If the cross-ratio
between these directions did not move, it means that the data coming from the
tautological $1$-form would be constant, and we try to get a contradiction.
That will show that the cross-ratio is general for general points, which in
turn gives the required general position property.

The proofs for $\boldsymbol{\mathfrak{p}}^{\prime}$ and
$\boldsymbol{\mathfrak{q}}^{\prime}$ will be completed in detail in the
following subsections. ∎

Let $Y_{\mathsf{Kum}}\subset Y_{0}$ be the reduced inverse image of
$\mathsf{Kum}$ in $Y_{0}$. We saw in Proposition 5.12 that
$Y_{\mathsf{Kum}}\rightarrow\mathsf{Kum}$ is a $4$-sheeted covering with an
explicit description.

Let $H_{\mathsf{Kum}}$ be the set of points in the Hecke correspondence
${\overline{\mathcal{H}}}(a)$ which are intersection points of two lines in
the Hecke fiber over points of $\mathsf{Kum}$. Thus, $H_{\mathsf{Kum}}$ is the
Kummer K3-surface obtained by blowing up the $16$ nodes of $\mathsf{Kum}$,
with a map $H_{\mathsf{Kum}}\rightarrow\mathsf{Kum}$ that is an isomorphism
outside of the nodes. We obtain a map $H_{\mathsf{Kum}}\rightarrow X_{1}$ that
we’ll view as a rational map $\nu:\mathsf{Kum}\dashrightarrow X_{1}$, whose
image is the Kummer K3 surface in $X_{1}$ [Bea96, Dol20, GH94, Hud05, Keu97].

######  Lemma 8.8.

Suppose $x\in\mathsf{Kum}$, not a node. Let $z:=\nu(x)\in X_{1}$ be the image
of the point of $H_{\mathsf{Kum}}$ lying over $x$. Let $V_{x}\subset X_{0}$ be
the Hecke line over $z$. Then $V_{x}$ is tangent to $\mathsf{Kum}$ at $x$.

###### Proof.

Let $d^{-1}(z)\subset{\overline{\mathcal{H}}}(\widehat{a})$ be the inverse
image, which is a ${\mathbb{P}}^{1}$ mapping isomorphically to the Hecke line
$V_{x}$. If $x^{\prime}\in
H_{\mathsf{Kum}}\subset{\overline{\mathcal{H}}}(\widehat{a})$ is the point
over $x$, so that $z=d(x^{\prime})$, then $d^{-1}(z)$ passes through
$x^{\prime}$. The divisor
$b^{-1}(\mathsf{Kum})\subset{\overline{\mathcal{H}}}(\widehat{a})$ is, locally
near $x^{\prime}$, a union of two branches each of which is a part of the
${\mathbb{P}}^{1}$-bundle over the Jacobian of $C$. The curve $d^{-1}(z)$
meets each of the branches, so its intersection with the divisor
$b^{-1}(\mathsf{Kum})$ has multiplicity $\geq 2$. This intersection number is
the same as the local intersection number of $V_{x}$ with $\mathsf{Kum}$ at
$x$. The fact that it is $\geq 2$ means that the line $V_{x}$ is tangent to
$\mathsf{Kum}$ at $x$. ∎

The inverse image of $\mathsf{Kum}$ in $Y_{0}$ is $2Y_{\mathsf{Kum}}$ as a
divisor, since the map $Y_{0}\rightarrow X_{0}$ is fully simply ramified along
$Y_{\mathsf{Kum}}$. At points where $Y_{\mathsf{Kum}}/\mathsf{Kum}$ is etale,
there is a specified tangent direction to $Y$, normal to $Y_{\mathsf{Kum}}$,
namely the directions that map to zero in the tangent space of $X_{0}$. Call
this subsheaf

$N^{v}\hookrightarrow T(Y_{0})|_{Y_{\mathsf{Kum}}}.$

######  Lemma 8.9.

The restriction of the spectral $1$-form $\alpha$ of $Y_{0}$ to $N^{v}$ is
nonzero.

###### Proof.

If it were zero everywhere, this would imply that the Higgs field on the Higgs
bundle $\pi_{*}(\mathscr{L}_{0})$ does not have singularities over
$\mathsf{Kum}$. Recall from Corollary 5.11 that for an appropriate choice of
$\mathscr{L}_{0}$, the second Chern character violates the Bogomolov-Gieseker
inequality. But, if there were no singularities of the Higgs field along
$\mathsf{Kum}$, then the Higgs field would be logarithmic along just the trope
part of the wobbly divisor. From Corollary 5.5, the trope part of the wobbly
divisor has normal crossings up to codimension $2$. Since the spectral variety
$Y$ is irreducible, the Higgs bundle is stable. This would contradict
Mochizuki’s Bogomolov-Gieseker inequality [Moc06]. Therefore, the restriction
of $\alpha$ to $N^{v}$ is nonzero.

In fact, it is nonzero at all the points where it is well-defined over the
smooth locus of $\mathsf{Kum}$ minus the trope conics, because the residue of
the Higgs field has to have constant Jordan type along the smooth points of
the wobbly divisor. ∎

######  Lemma 8.10.

Consider the subspace $\ker(\alpha)\cap T(Y_{\mathsf{Kum}})$ varying as a
function of the point in $Y_{\mathsf{Kum}}$. Over a general point
$x\in\mathsf{Kum}$, then the projections of these subspaces to
$T_{x}(\mathsf{Kum})$ vary as a function of the point in the Hitchin base.

###### Proof.

If the subspaces depend only on $C$ and $x\in\mathsf{Kum}$ then, in
particular, their values on different sheets of $Y_{\mathsf{Kum}}$ would be
the same. We’ll see that this is not the case.

Points of $Y_{\mathsf{Kum}}$ are represented by $(L,\tilde{u}+\tilde{v})$
where $\tilde{u},\tilde{v}\in{\widetilde{C}}$ are points lying over their
images denoted $u,v\in C$, and $L^{\otimes
2}={\mathcal{O}}_{C}(u+v-2\boldsymbol{\mathsf{p}})$. The restriction of
$\alpha$ to $Y_{\mathsf{Kum}}$ is given by adding the values of the
tautological form on ${\widetilde{C}}$ evaluated on $\tilde{u}$ and
$\tilde{v}$. Specialize near a point where $\tilde{u}$ is not a ramification
of ${\widetilde{C}}/C$ but $\tilde{v}$ is a ramification point. As $v$ moves
around the branch point, $\tilde{v}$ changes branches and the tautological
form $\alpha$ changes sign when viewed as a dual element of $T_{v}(C)$. Then,
$\alpha$ is the sum of a fixed part namely the tautological form at
$\tilde{u}$, plus a part that changes sign namely the value at $\tilde{v}$.
Thus, the directions of $\ker(\alpha)$ over these two points in different
sheets of $Y_{\mathsf{Kum}}$ are different. ∎

####  8.3.2 Arguments for $\boldsymbol{\mathfrak{p}}^{\prime}$

Look first at the point $\boldsymbol{\mathfrak{p}}^{\prime}$. View it as a
general point $\boldsymbol{\mathfrak{p}}\in\mathsf{Kum}\subset X_{0}$. The
Hecke fiber over $x$ has two lines that meet in a point, and the image of this
point in $X_{1}$ is a point $z$ where the two lines meet. The line $\ell$ in
our picture is one of these two. Going back, the line $\ell$ corresponds to a
plane $P\subset X_{0}$, transverse to $\mathsf{Kum}$ at
$\boldsymbol{\mathfrak{p}}$ and tangent to it at $\boldsymbol{\mathfrak{q}}$.
On the other hand, the Hecke fiber over $z$ is a line in $X_{0}$, and this
line is tangent to $P\cap\mathsf{Kum}$ at $\boldsymbol{\mathfrak{p}}$. Indeed,
after blowing up the Hecke line and the strict transform of $\mathsf{Kum}$
both meet the exceptional divisor $\boldsymbol{\mathcal{N}}$ at the same
point, meaning that the Hecke line was tangent to $\mathsf{Kum}$ before
blowing up.

The double cover $Y_{P}\rightarrow P$ ramified along $\mathsf{Kum}$ has a
point $y$ over $\boldsymbol{\mathfrak{p}}$, and its tangent space at $y$ is a
$2$-dimensional space containing the following subspaces: the vertical space
$N^{v}(y)$ of the ramification; the tangent space $T_{y}(Y_{P,\mathsf{Kum}})$
of the ramification divisor $Y_{P,\mathsf{Kum}}:=Y_{P}\cap Y_{\mathsf{Kum}}$;
and the two directions of the pullback of the Hecke line
$V_{\boldsymbol{\mathfrak{p}}}$ which pulls back to a pair of crossed lines in
the double cover since it is tangent to $\mathsf{Kum}$ (Lemma 8.8). The cross-
ratio of these four points is fixed, because the $4$ directions are symmetric
for the involution of the double cover. Furthermore, Lemma 8.9 implies that
the restriction of the spectral $1$-form to $Y_{P}$ is nonzero. It therefore
defines a $1$-dimensional subspace $A\subset T_{y}(Y_{P})$.

######  Proposition 8.11.

For general global parameters $C,{\widetilde{C}}$, as the point
$\boldsymbol{\mathfrak{p}}$ moves around in $\mathsf{Kum}$, the subspace $A$
moves with respect to the framing of $T_{y}(Y_{P})$ given by the four
previously discussed directions.

###### Proof.

Proceed by contradiction: suppose that the subspace $A$ has a fixed direction
with respect to the framing. We first show that it must then be the direction
$T_{y}(Y_{P,\mathsf{Kum}}$. By Lemma 8.9, $A$ is not the direction $N^{v}(y)$.
It seems likely that the two directions coming from
$V_{\boldsymbol{\mathfrak{p}}}$ should interchange under a global symmetry or
monodromy operation, and if $A$ were in a fixed direction it would have to be
invariant under that operation, which would imply that it is the direction
$T_{y}(Y_{P,\mathsf{Kum}}$. However, we have not been able to specify such an
interchange operation. So, instead, let us calculate at a special point.

Recall from Proposition 5.12 that the points of $Y_{\mathsf{Kum}}$ are
represented by $(L,\tilde{u}+\tilde{v})$ where
$\tilde{u},\tilde{v}\in{\widetilde{C}}$ are points lying over their images
denoted $u,v\in C$, and $L^{\otimes
2}={\mathcal{O}}_{C}(u+v-2\boldsymbol{\mathsf{p}})$. Specialize near a point
on $\mathsf{Kum}$ where both $u$ and $v$ are (different) branch points of
${\widetilde{C}}/C$. This corresponds to a place where two other pieces of the
movable ramification locus of $Y/X$ meet the ramification over $\mathsf{Kum}$.
Both $\tilde{u}$ and $\tilde{v}$ then branch. We can write this in local
coordinates $y_{1},y_{2},y_{3}$ for $Y$ over coordinates $x_{1},x_{2},x_{3}$
for $X$, such that $x_{1}=0$ is the equation of $\mathsf{Kum}$, $y_{1}=0$ is
the equation of $Y_{\mathsf{Kum}}$, and the map is given by $x_{i}=y_{i}^{2}$.
The spectral $1$-form along $Y_{\mathsf{Kum}}$ is obtained from the
tautological $1$-form on ${\widetilde{C}}$ by adding the values at $\tilde{u}$
and $\tilde{v}$. The tautological $1$-form on ${\widetilde{C}}$ looks locally
like $z^{2}dz$ if $z$ is the coordinate on ${\widetilde{C}}$. We can therefore
write the leading term as

$\alpha=a(y)d(y_{1})+y_{2}^{2}d(y_{2})+y_{3}^{2}d(y_{3}).$

Since $\alpha$ comes from a linear form on the abelian variety, it does not
vanish at any point, thus $a(0)$ has to be nonzero.

Consider a point $x=(x_{1},x_{2},x_{3})$ near to
$\boldsymbol{\mathfrak{p}}=(0,0,0)$. The Hecke line $V_{x}$ becomes, in these
coordinates, a curve tangent to $(x_{1}=0)$. Let us write it (again looking at
the highest order term) as

$(x_{1},x_{2},x_{3})=(b_{1}t^{2},\epsilon_{2}+b_{2}t,\epsilon_{3}+b_{3}t)$

for some coefficients $b_{1}\neq 0$ and $(b_{2},b_{3})\neq(0,0)$, and small
$(\epsilon_{2},\epsilon_{3})$. Note that as
$(\epsilon_{2},\epsilon_{3})\rightarrow(0,0)$, the second order term
$b_{1}t^{2}$ approaches a nonzero limit ($(0,0,0)$ being itself a point of
$\mathsf{Kum}$ that is general with respect to $C$ although not necessarily
with respect to ${\widetilde{C}}$).

Specifying a lifting of $(0,\epsilon_{2},\epsilon_{3})$ in $Y$ i.e. extracting
$\epsilon_{2}^{1/2}$ and $\epsilon_{3}^{1/2}$, the curve lifts into $Y$ in two
branches corresponding to $\pm b_{1}^{1/2}$. One of these branches is

$(y_{1},y_{2},y_{3})=\left(b_{1}^{1/2}t,\epsilon_{2}^{1/2}(1+b_{2}t/2\epsilon_{2}+\ldots),\epsilon_{3}^{1/2}(1+b_{3}t/2\epsilon_{3}+\ldots)\right).$

Its tangent vector is
$(b_{1}^{1/2},b_{2}\epsilon_{2}^{-1/2}/2,b_{3}\epsilon_{3}^{-1/2}/2)$.

The vertical tangent vector is $(1,0,0)$. We use the middle direction to
normalize the horizontal tangent vector with respect to the vertical one, in
other words the middle vector should be the sum of the same multiple of the
vertical and normalized horizontal vectors. The multiple is $b_{1}^{1/2}$, so
the normalized horizontal vector is

$\left(0,b_{1}^{-1/2}b_{2}\epsilon_{2}^{-1/2}/2,b_{1}^{-1/2}b_{3}\epsilon_{3}^{-1/2}/2\right).$

The values of $\alpha$ on the vertical tangent vector is $a(0)$. On the
normalized horizontal vector, evaluating at the point
$(y_{1},y_{2},y_{3})=(0,\epsilon_{2}^{1/2},\epsilon_{2}^{1/2})$, it is

$\alpha(0,\epsilon_{2}^{1/2},\epsilon_{2}^{1/2})\cdot\left(0,b_{1}^{-1/2}b_{2}\epsilon_{2}^{-1/2}/2,b_{1}^{-1/2}b_{3}\epsilon_{3}^{-1/2}/2\right)=b_{1}^{-1/2}b_{2}\epsilon_{2}^{1/2}/2+b_{1}^{-1/2}b_{3}\epsilon_{3}^{1/2}/2.$

This approaches $0$ as $(\epsilon_{2},\epsilon_{3})\rightarrow(0,0)$. This
tells us that the kernel line $A$ of $\alpha$ on $T(Y_{P})$ approaches the
horizontal direction as we approach the special point.

This shows that if $A$ is some fixed direction with respect to the framing, it
must be the horizontal direction. Let’s now take up that possibility. In that
case, it means that the line $T_{y}(Y_{P,\mathsf{Kum}})$ is always in the
kernel of $\alpha$. In particular, the restriction of $\alpha$ to
$Y_{\mathsf{Kum}}$ has this fixed foliation as a kernel. But Lemma 8.10 says
that this does not happen, completing the contradiction. ∎

The local piece of $\Sigma$ corresponds to a locally defined rank $2$ Higgs
bundle on $X_{0}$. It comes originally from a rank $2$ piece of
$\mathscr{F}_{0}$ whose spectral variety is a double cover of $X_{0}$ simply
ramified at the point of $\mathsf{Kum}$. We’ll use the Higgs bundle to
calculate the lower critical locus, knowing that it is the image of the upper
critical locus so if we separate the two branches of the lower critical locus
that will separate them upstairs too.

Use a coordinate system $(x,t)$ on $P\subset X_{0}$ with $P\cap\mathsf{Kum}$
given by $t=0$. The double cover looks locally like $t=w^{2}$. We can write
the form $\alpha$ as

$\alpha=(a^{+}(x,t)+wa^{-}(x,t))dx+(b^{+}(x,t)+wb^{-}(x,t))dw$
$=(a^{+}(x,t)+wa^{-}(x,t))dx+(wb^{+}(x,t)+tb^{-}(x,t))(dt/t).$

Locally we assume that the line bundle is trivial
$\mathscr{L}_{0}={\mathcal{O}}_{P}$. Then the direct image down to $P$, which
is the rank $2$ piece of our Higgs bundle $\mathscr{F}_{0}|_{P}$, has basis
$1,w$. In these terms, the form $\alpha$ leads to a Higgs field in matrix
form, with coefficients being functions of $(x,t)$:

$\varphi=\left(\begin{array}[]{cc}a^{+}dx+b^{-}dt&ta^{-}dx+b^{+}dt\\\
a^{-}dx+b^{+}(dt/t)&a^{+}dx+tb^{-}dt\end{array}\right).$

We next blow up the point $(0,0)$ with coordinates $(x,v)$ with $t=xv$. This
gives $dt=vdx+xdv$. The Higgs field becomes

$\varphi=\left(\begin{array}[]{cc}(a^{+}+vb^{-})dx+xb^{-}dv&(xva^{-}+vb^{+})dx+xb^{+}dv\\\
a^{-}dx+b^{+}(dx/x+dv/v)&(a^{+}+vb^{-})dx+x^{2}vb^{-}dv\end{array}\right).$

This is the formula for the Higgs field on
${\overline{\mathcal{H}}}(a)_{\ell}$. We may assume that the map
${\overline{\mathcal{H}}}(a)_{\ell}\rightarrow\ell$ is given by the function
$u-v$, so for the relative differentials it induces the relation $dx=dv$. The
relative Higgs field becomes

$\varphi_{{\overline{\mathcal{H}}}(a)_{\ell}/\ell}=\left(\begin{array}[]{cc}(a^{+}+vb^{-}+xb^{-})dv&(xva^{-}+vb^{+}+xb^{+})dv\\\
(a^{-}+b^{+}/x+b^{+}/v)dv&(a^{+}+vb^{-}+x^{2}vb^{-})dv\end{array}\right).$

The support of the cokernel of this matrix is given by its determinant

${\rm
det}(\varphi_{{\overline{\mathcal{H}}}(\overline{a})_{\ell}/\ell})=(a^{+}+vb^{-}+xb^{-})(a^{+}+vb^{-}+x^{2}vb^{-})-(xva^{-}+(x+v)b^{+})(a^{-}+((x+v)/xv)b^{+})$
$=(xv)^{-1}\left[xv(a^{+})^{2}-(x+v)^{2}(b^{+})^{2}+\ldots\right]$

where the next terms have degree $\geq 3$ in $x,v$. The vanishing locus, away
from the axes, is therefore given to first order by the equation

$(b^{+})^{2}x^{2}+(2(b^{+})^{2}-(a^{+})^{2})xv+(b^{+})^{2}v^{2}=0.$

Dividing by $(b^{+})^{2}$ and setting $c:=(a^{+}/b^{+})^{2}$ this becomes

$x^{2}+(2-c)xv+v^{2}.$

Its discriminant is $(2-c)^{2}-4=c^{2}-4c$ so if $c\neq 0,4$, in other words
the quotient is not $0$, $-2$ or $2$, then there are two branches.

######  Corollary 8.12.

If $\boldsymbol{\mathfrak{p}}$ was a general point on $\mathsf{Kum}$ then the
quotient $a^{+}(0,0)/b^{+}(0,0)$ is general.

###### Proof.

This restates the result of Proposition 8.11. ∎

Therefore, at a general point on $\mathsf{Kum}$, the upper critical locus has
two branches meeting the exceptional divisor at two distinct points. This
completes the proof of Theorem 8.7 for the point
$\boldsymbol{\mathfrak{p}}^{\prime}$.

####  8.3.3 Arguments for $\boldsymbol{\mathfrak{q}}^{\prime}$

At the point $\boldsymbol{\mathfrak{q}}=\boldsymbol{\mathfrak{q}}^{\prime}$
the plane $P$ is tangent to $\mathsf{Kum}$. Introduce coordinates $x,y$ for
the plane and $z$ in the transverse direction so that $\mathsf{Kum}$ is given
by $z=0$ and the plane is given by $z=xy$. The covering $Y$ has coordinates
$x,y,w$ with $w^{2}=z$, and $Y_{\mathsf{Kum}}$ has equation $w=0$.

We may scale the coordinates such that the map from the plane to the line
$\ell$ is given by $t=x-y$. Notice that the fibers of the map are not tangent
to the two principal directions of $P\cap\mathsf{Kum}$, as we have seen above
that the two branches of $\boldsymbol{\mathcal{T}}$ at the point
$\boldsymbol{\mathfrak{q}}^{\prime}$ are etale over $\ell$.

Write the spectral differential form on $Y$ as

$\alpha=a(x,y,w)dx+b(x,y,w)dy+c(x,y,w)dw.$

We have $c(0,0,0)\neq 0$ by Lemma 8.9. Also, $\alpha$ does not vanish
identically on $Y_{\mathsf{Kum}}$, by Lemma 8.10, and the kernel of $\alpha$
is not generically given by any construction depending only on $C$ (such as
the lines $x=0$, $y=0$ or $x=y$ in our coordinate system). Thus, at our
general point we can assume that $a(0),b(0),a(0)+b(0)$ are nonzero.

Proceed now with the calculation of the directions of the upper critical
locus. The covering $\Sigma$ has equation $w^{2}=xy$ mapping to the plane $P$
with coordinates $x,y$. Blow up the origin in the $x,y$ plane, so we introduce
a coordinate $v$ with $y=xv$. The exceptional divisor is $x=0$ and $v$ is the
coordinate along it. Now the form is $\alpha=adx+b(xdv+vdx)+cdw$ and $\Sigma$
has equation $w^{2}=x^{2}v$. This normalizes to two branches $w=xv^{1/2}$ and
$w=-xv^{1/2}$. Let $u=v^{1/2}$ be the coordinate at a general point of the
covering of the exceptional divisor, so we have $w=xu$ (there are two branches
as $u,-u$ go to the same point $v$). We have $v=u^{2}$ so $dv=2udu$. We have
$y=xu^{2}$.

This normalization is smooth over $u\neq 0$ (actually everywhere) and the form
becomes

$\alpha=adx+b(2xudu+u^{2}dx)+c(xdu+udx).$

The mapping function from the plane to the line is $t=x-y$ so $dt=dx-dy$.
Relative differentials are obtained by working modulo $dt$, identifying

$dx\sim dy=2xudu+u^{2}dx.$

Modulo $dt$ we get

$dx=\frac{2xu}{1-u^{2}}du.$

Thus

$\frac{\alpha^{\rm
rel}}{du}=a\frac{2xu}{1-u^{2}}+b\left(2xu+\frac{2xu^{3}}{1-u^{2}}\right)+c\left(x+\frac{2xu^{2}}{1-u^{2}}\right).$

The zeros of $\alpha^{\rm rel}$ are given by the equation

$a\cdot 2xu+b\cdot(2xu-2xu^{3}+2xu^{3})+c\cdot(x-xu^{2}+2xu^{2})$
$=(a+b)(2xu)+c(x+xu^{2}).$

One can factor $x$ out of this expression.

We notice that $a(0)+b(0)$ is the value of $\alpha$ on the tangent direction
of the Hecke line $t=0$, and $c(0)$ is the value of $\alpha$ in the vertical
direction. In our current coordinate system, the unit vectors of these
directions are normalized to be related by the lines described in the previous
subsection: the Hecke line is $y=x,z=x^{2}$ and its two lifts are $y=x,w=\pm
x$.

Therefore, curiously enough, we are now in the same general position setting
as in the previous subsection. Proposition 8.11 implies that the ratio
$(a(0)+b(0))/c(0)$ moves as a function of the point, so we may assume that it
is general. The factored expression for the zeros of the relative $1$-form
$\alpha^{\rm rel}/xdu$ along the exceptional curve becomes

$1+\frac{a(0)+b(0)}{c(0)}u+u^{2}=0.$

There are two distinct points $u$ in the exceptional divisor corresponding to
limits of zeros of $\alpha^{\rm rel}$. This gives two branches of the upper
critical locus with distinct tangent vectors in the nodal point of our local
piece of $\Sigma$.

This completes the proof of the $\boldsymbol{\mathfrak{q}}^{\prime}$ part of
Theorem 8.7.

###  8.4 Spectral line bundle for the higher direct image

Recall the _spectral line bundle_ , on $Y_{0}$ or $Y_{1}$, is the line bundle
$\mathscr{L}_{0}$ or $\mathscr{L}_{1}$ whose direct image down to $X_{0}$ or
$X_{1}$ respectively is the level $0$ piece of the parabolic Higgs bundles
$(\mathscr{F}_{0,\bullet},\Phi_{0})$ or $(\mathscr{F}_{1,\bullet},\Phi_{1})$
respectively. For $X_{0}$ the parabolic Higgs bundle has trivial parabolic
structure (away from the codimension $2$ tacnode points). On $X_{1}$ the
parabolic structure has levels $0$ and $-1/2$, so this definition of spectral
line bundle involves a choice.

We would like to calculate the spectral line bundle ${\mathcal{U}}$ on
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$
corresponding to the higher direct image of the pullback Higgs bundle
$\mathscr{F}_{0,{\overline{\mathcal{H}}}(a_{\ell}}$ down to $\ell$, in terms
of the spectral line bundle $\mathscr{L}_{0}$ over $Y_{0}$ and its restriction
$\mathscr{L}_{\ell}$ to ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$.

For this, we use the statement of Theorem 3.17, in particular the part proven
in Theorem 12.12. The previous discussion shows that the points
$\boldsymbol{\mathfrak{p}}^{\prime}$ and $\boldsymbol{\mathfrak{q}}^{\prime}$
are of type 3.11.1(e), and the remaining points are covered by the other parts
of the classification in Subsection 3.11. To use the theorem in the presence
of the points $\boldsymbol{\mathfrak{p}}^{\prime}$,
$\boldsymbol{\mathfrak{q}}^{\prime}$, we should take the answer

${\mathcal{U}}=\mathscr{L}_{\ell}\otimes\omega_{{\overline{\mathcal{H}}}(\overline{a})_{\ell}/\ell}.$

The normalization of the relative critical locus separates the two branches at
each of the nodes. By Theorem 7.12 and Corollary 7.13, the two components of
the abelianized Hecke are disjoint away from
$\boldsymbol{\mathfrak{p}}^{\prime},\boldsymbol{\mathfrak{q}}^{\prime}$, so
the normalization that was denoted by $G$ in Subsection 3.11 is the same as
the disjoint union

$G={\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}={\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}\sqcup{\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}.$

The prescription of Theorem 3.17 gives a parabolic sheaf $F^{\prime}_{\cdot}$
such that

$F^{\prime}_{0}=f_{*}({\mathcal{U}}|_{G})$

and $F^{\prime}_{-1/2}$ is the standard subsheaf coming from the ramification
points of $f:G\rightarrow\ell$ over $\ell\cap{\rm Wob}_{1}$. Theorem 3.17 says
that $F^{\prime}_{\cdot}$ is a parabolic subsheaf of the parabolic Higgs
bundle $F_{\cdot}$ we are looking for. On the other hand, we know from general
principles that $F_{\cdot}$ has parabolic degree $0$. Also, $F$ and
$F^{\prime}$ agree except possibly over the points
$f(\boldsymbol{\mathfrak{p}}^{\prime}),f(\boldsymbol{\mathfrak{q}}^{\prime})\in\ell$
and these are different from the points where there is a parabolic structure.

So, if we can show that the parabolic degree of $F^{\prime}_{\cdot}$ is zero,
this will imply that $F_{\cdot}=F^{\prime}_{\cdot}$ and we get the computation
of $F_{\cdot}$. The degree will be calculated in the next subsection.

###  8.5 Degree calculation

Recall our notation that inside $Y_{0}$ that $\boldsymbol{\mathit{E}}_{0}$
denotes the exceptional divisor of the blow-up
$\boldsymbol{\varepsilon}_{0}:Y_{0}\to{\mathcal{P}}_{2}$, and
$\boldsymbol{\mathit{F}}_{0}$ denotes the inverse image of the hyperplane
class of $X_{0}$.

######  Lemma 8.13.

For the relative canonical class over the degree $1$ moduli space, we have

$\omega_{{{\overline{\mathcal{H}}}(a)}/X_{1}}={\mathcal{O}}_{X_{0}}(-2)|_{{\overline{\mathcal{H}}}(a)}\otimes{\mathcal{O}}_{X_{1}}(1)|_{{\overline{\mathcal{H}}}(a)}.$

If $\Xi$ denotes the theta divisor on the Prym, or its pullbacks to $Y_{0}$,
$Y_{1}$ or ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$, we have
$\boldsymbol{\mathit{F}}_{0}=\Xi-\boldsymbol{\mathit{E}}_{0}$ and
$\boldsymbol{\mathit{F}}_{1}=2\Xi-\boldsymbol{\mathit{E}}_{1}$. We note that
$\Xi^{3}=24$, concording with the calculations in Propositions 4.8 and 5.8.

###### Proof.

The canonical bundle $\omega_{{\overline{\mathcal{H}}}(a)}$ restricts to
${\mathcal{O}}_{{\mathbb{P}}^{1}}(-2)$ on the fibers of the projection to
$X_{1}$ and to ${\mathcal{O}}_{{\mathbb{P}}^{1}}(2)$ on the fibers of the
projection to $X_{0}$. It means that it restricts to the pullback of
${\mathcal{O}}_{X_{0}}(2)$ on the Hecke lines over points of $X_{1}$, and to
the pullback of ${\mathcal{O}}_{X_{1}}(1)$ on the Hecke conics over points of
$X_{0}$. The Picard group of ${{\overline{\mathcal{H}}}(a)}$ is generated by
these two things, and the pullbacks restrict to trivial bundles on their own
Hecke fibers, so we conclude

$\omega_{{\overline{\mathcal{H}}}(a)}={\mathcal{O}}_{X_{0}}(-2)|_{{\overline{\mathcal{H}}}(a)}\otimes{\mathcal{O}}_{X_{1}}(-1)|_{{\overline{\mathcal{H}}}(a)}.$

On the other hand, as $X_{1}$ is an intersection of two quadrics in
${\mathbb{P}}^{5}$, its canonical class is
${\mathcal{O}}_{X_{1}}(-6+2+2)={\mathcal{O}}_{X_{1}}(-2)$. We conclude the
stated formula.

For the last parts, we note that the linear system $|\Xi|$ that produces the
rational map ${\mathcal{P}}_{2}\dashrightarrow{\mathbb{P}}^{3}$ has base
points on the $16$ points that we blow-up to get $Y_{0}$ so on the resulting
map on the blow-up, we subtract $\boldsymbol{\mathit{E}}_{0}$. The
$6$-dimensional subsystem (anti-invariant part) of $|2\Xi|$ that provides the
rational map ${\mathcal{P}}_{3}\dashrightarrow{\mathbb{P}}^{5}$ has
${\widehat{C}}$ as base locus so we subtract $\boldsymbol{\mathit{E}}_{1}$.
For the verification, the formulas of Propositions 5.8 and 4.8 give

$(\boldsymbol{\mathit{F}}_{0}+\boldsymbol{\mathit{E}}_{0})^{3}=24\;\;\;\mbox{
and }\;(\boldsymbol{\mathit{F}}_{1}+\boldsymbol{\mathit{E}}_{1})^{3}=192$

compatible with $\Xi^{3}=24$. ∎

Recall that $\boldsymbol{\mathit{Exc}}_{0}$ and
$\boldsymbol{\mathit{Exc}}_{1}$ are the exceptional divisors in
${\widehat{\mathcal{H}}}^{\rm ab}$. We could denote their restrictions over
the point $\widehat{a}=(A,\tilde{t})\in{\widehat{C}}$ by
$\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$ and
$\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$. Those are thus the divisors in
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ that are strict
transforms of the divisors $\boldsymbol{\mathit{E}}_{0}$ and
$\boldsymbol{\mathit{E}}_{1}$.

We get that

$\boldsymbol{\mathit{E}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}=\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$

whereas

$\boldsymbol{\mathit{E}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}=\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})+\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$

because starting from $Y_{1}$ we blow up lines contained in
$\boldsymbol{\mathit{E}}_{1}$ to get ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$. This gives

$\boldsymbol{\mathit{F}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}=2\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$

and

$\boldsymbol{\mathit{F}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}=4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\tilde{a})-\boldsymbol{\mathit{Exc}}_{1}(\tilde{a}).$

This gives for the relative differentials

$\omega_{{{\overline{\mathcal{H}}}(a)}/X_{1}}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}={\mathcal{O}}(-2\boldsymbol{\mathit{F}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}+\boldsymbol{\mathit{F}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}})$
$={\mathcal{O}}(\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})).$

Recall that—up to tensoring with degree $0$ line bundles
$(\boldsymbol{\varepsilon}_{0}^{*}\mathfrak{L}_{0})$ resp.
$(\boldsymbol{\varepsilon}_{1}^{*}\mathfrak{L}_{1})$—the spectral line bundle
on $Y_{0}$ is
$\mathscr{L}_{0}={\mathcal{O}}_{Y_{0}}(2\boldsymbol{\mathit{F}}_{0}+\boldsymbol{\mathit{E}}_{0})$
and the spectral line bundle on $Y_{1}$ is
$\mathscr{L}_{1}={\mathcal{O}}_{Y_{1}}(\boldsymbol{\mathit{F}}_{1})$. It
follows that up to numerical equivalence

$\mathscr{L}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}={\mathcal{O}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))$

and

$\mathscr{L}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}={\mathcal{O}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})).$

######  Proposition 8.14.

Define the line bundle ${\mathcal{U}}$ over ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ as the pullback of the spectral line
bundle $\mathscr{L}_{0}$ on $Y_{0}$, tensored with
$\omega_{{\overline{\mathcal{H}}}(a)_{\ell}/\ell}$. Then ${\mathcal{U}}$ is
the spectral line bundle for the $L^{2}$ Dolbeault higher direct image Higgs
bundle on $\ell$. Over each of the two pieces of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$, ${\mathcal{U}}$ is isomorphic to the
restriction of the spectral line bundle that we construct for $X_{1}$, from
$Y_{1}$ to ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$
(respectively ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})_{\ell}$).

###### Proof.

It will be convenient to prove the second part first. Let us use the
expressions for the spectral line bundles that we are constructing on $Y_{0}$
and $Y_{1}$, as calculated prior to the statement of the proposition. We get

$\displaystyle{\mathcal{U}}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}$
$\displaystyle=\mathscr{L}_{0}\otimes\omega_{{\overline{\mathcal{H}}}(\widehat{a})_{\ell}/\ell}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}$
$\displaystyle={\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}((4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))+(\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})))$
$\displaystyle={\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(4\Xi-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})).$

We would like to compare this with the spectral line bundle coming from
$Y_{1}$ which is, as we have seen above,

$\mathscr{L}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}={\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})).$

These two differ by $\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$. However,
remember that we are looking at the restriction over a line $\ell\subset
X_{1}$. The divisor components of $\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$
are obtained by blowing up $16$ disjoint ${\mathbb{P}}^{1}$’s in $Y_{1}$.
These ${\mathbb{P}}^{1}$’s are sixteen fibers of
$\boldsymbol{\mathit{E}}_{1}={\widehat{C}}\times{\mathbb{P}}^{1}$, so they map
to $16$ lines in the wobbly locus ${\rm Wob}_{1}\subset X_{1}$. A general line
$\ell$ will miss these. Thus, as restricted over $\ell$, the two expressions
are the same.

The same discussion holds on ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$. Thus,
$\mathscr{L}_{1}|_{G}\cong{\mathcal{U}}|_{G}$.

Let $F_{\cdot}$ denote the $L^{2}$ Dolbeault pushforward (on $\ell$) with its
parabolic structure, and let $F^{\prime}_{\cdot}$ be the parabolic bundle on
$\ell$ whose spectral line bundle is ${\mathcal{U}}$ over $G$. The parabolic
degree of $F_{\cdot}$ is zero since it corresponds to a harmonic bundle.

The spectral line bundle $\mathscr{L}_{1}$ pushes forward to a parabolic Higgs
bundle with first parabolic Chern class equal to $0$, over $X_{1}$ and hence
over a line $\ell\subset X_{1}$. The parabolic structure here is also the
standard one coming from the ramification over ${\rm Wob}_{1}$. As we have
identified the two line bundles $\mathscr{L}_{1}|_{G}$ and
${\mathcal{U}}|_{G}$, and since the parabolic structures are standard coming
from ramification points of $G$ over $\ell\cap{\rm Wob}_{1}$, it follows that
the parabolic degree of $F^{\prime}_{\cdot}$ is zero. Thus,
$F_{\cdot}=F^{\prime}_{\cdot}$. We conclude that ${\mathcal{U}}$ is the
spectral line bundle for $F_{\cdot}$. ∎

###  8.6 Restriction to a line

We have been considering the local system corresponding to the parabolic Higgs
bundle $\left(\mathscr{F}_{1,\bullet},\Phi_{1}\right)$ via the non-abelian
Hodge and Riemann-Hilbert correspondences. So far we focused on the
restriction of this local system to a general line $\ell\subset X_{1}$. We
show in this subsection that knowing this restriction suffices in order to
identify the local system on $X_{1}-{\rm Wob}_{1}$. Throughout the subsection,
$\ell$ denotes a general line in $X_{1}$. Let
$\ell^{\circ}:=\ell-(\ell\cap{\rm Wob}_{1})$.

######  Lemma 8.15.

The local system on $\ell^{\circ}$ constructed in Section 4, is irreducible.

###### Proof.

It suffices to show that the spectral covering is irreducible. Recall from
Lemma 4.15 that the spectral covering has the following description. The line
$\ell$ has a natural trigonal covering $k:C\rightarrow\ell$ whose fiber over
$x\in\ell$ is identified with the set of three lines through $x$ that are
different from $\ell$. The $8$ branch points are the points of $\ell\cap{\rm
Wob}_{1}$.

We have the spectral cover ${\widetilde{C}}/C$. For a point $x\in\ell$ we can
make the following set with $8$ elements: it is the set of liftings of the
subset $k^{-1}(x)$ to a subset of three elements of ${\widetilde{C}}$. This
family determines a covering of $\ell$ of degree $8$, and that is the same as
$Y_{1}\times_{X_{1}}\ell\rightarrow\ell$.

For $\ell$ general, the set of $4$ branch points of ${\widetilde{C}}/C$ maps
to a subset of $4$ distinct points in $\ell$. Indeed, the set of four branch
points is the inverse image of a general pair of points in ${\mathbb{P}}^{1}$
under the hyperelliptic map $\text{\sf h}_{C}:C\to{\mathbb{P}}^{1}$, and a
general trigonal map $C\rightarrow{\mathbb{P}}^{1}$ does not identify opposite
points under the hyperelliptic involution on $C$, so the images of the four
branch points are distinct. Thus, as $x$ moves around in $\ell^{\circ}$, we
can change individually the parity of any one of the liftings of the three
points. This shows that the monodromy action on the $8$ points is transitive,
so $Y_{1}\times_{X_{1}}\ell$ is irreducible. That was the spectral covering of
the Higgs bundle on $\ell$ corresponding to the restricted local system, so
the local system is irreducible. ∎

######  Proposition 8.16.

Suppose $V$ and $V^{\prime}$ are two local systems on $X_{1}-{\rm Wob}_{1}$,
such that

$V|_{\ell^{\circ}}\cong V^{\prime}|_{\ell^{\circ}}$

and such that this local system on $\ell^{\circ}$ is irreducible. Then $V\cong
V^{\prime}$.

###### Proof.

Let $\rho:\pi_{1}(X_{1}-{\rm Wob}_{1},o)\to GL_{N}({\mathbb{C}})$ and
$\rho^{\prime}\pi_{1}(X_{1}-{\rm Wob}_{1},o)\to GL_{N}({\mathbb{C}})$ denote
the monodromy representations of $V$ and $V^{\prime}$ with respect to a
basepoint $o\in\ell^{\circ}$ chosen in $\ell^{\circ}$. Let
$\zeta:\pi_{1}(\ell^{circ},o)\to GL_{N}({\mathbb{C}})$ denote the monodromy
representation of $V|_{\ell^{\circ}}$ and $V^{\prime}|_{\ell^{\circ}}$,
assuming we choose framings making these representations the same.

Suppose $a\in\pi_{1}(X_{1}-{\rm Wob}_{1},o)$. Then $\rho(a)$ and
$\rho^{\prime}(a)$ are morphisms of representations of
$\pi_{1}(\ell^{\circ},o)$ between $\zeta$ and its conjugate $\zeta^{a}$. Since
$\zeta$ is irreducible, these two morphisms differ by a scalar. This gives a
rank $1$ character $\chi:\pi_{1}(X_{1}-{\rm
Wob}_{1},o)\to{\mathbb{C}}^{\times}$ such that
$\rho\otimes\chi\cong\rho^{\prime}$, and $\chi$ is trivial on
$\pi_{1}(\ell^{\circ},o)$.

To finish the proof, we need to note that $\chi$ is trivial. This is because
the map

$H_{1}(\ell^{\circ},{\mathbb{Z}})\rightarrow H_{1}(X_{1}-{\rm
Wob}_{1},{\mathbb{Z}})$ (56)

is surjective. Indeed, this follows from the following

######  Claim 8.17.

The fist homology group $H_{1}(X_{1}-{\rm Wob}_{1},{\mathbb{Z}})$ is cyclic
and generated by the linking loop in $X_{1}$ going around some smooth point of
${\rm Wob}_{1}$.

###### Proof.

Note first that the statement of the claim makes sense since any two linking
loops at smooth points of ${\rm Wob}_{1}$ are conjugate in $\pi_{1}(X_{1}-{\rm
Wob}_{1})$ and hence are homologous. This follows immediately since ${\rm
Wob}_{1}$ is irreducible and hence the smooth locus of ${\rm Wob}_{1}$ is
connected. Using the tubular neighborhood theorem we can view the linking
loops as two different fibers of the circle bundle in the normal bundle
$N_{{\rm Wob}_{1}^{\operatorname{smooth}}/X_{1}}$ and hence they are homotopy
equivalent up to a conjugation via a path connecting the two points in ${\rm
Wob}_{1}^{\operatorname{smooth}}$ over which these circle fibers sit.

Next recall that if $M$ is a connected, oriented, not necessarily compact,
$C^{\infty}$-manifold and if $Z\subset M$ is a connected, oriented
$C^{\infty}$ submanifold of dimension $d$, which is closed in the Eucledian
topology, then $Z$ defines a $d$-dimensional cohomology class $[Z]\in
H^{d}(M;\mathbb{Z})$. Indeed, let $Z\subset\mathfrak{T}\subset M$ be a tubular
neighborhood of $Z$. Then for every $k$ we have canonical identifications
$H^{k}(M,M-Z;{\mathbb{Z}})=H^{k}(\mathfrak{T},\partial\mathfrak{T};{\mathbb{Z}})$
(by excision) and also
$H^{k}(\mathfrak{T},\partial\mathfrak{T};{\mathbb{Z}})=H^{k-d}(Z;{\mathbb{Z}})$
(by the Thom isomorphism theorem). In particular we get that

$H^{d}((M,M-Z;{\mathbb{Z}})=H^{d}(\mathfrak{T},\partial\mathfrak{T};{\mathbb{Z}})=H^{0}(Z;{\mathbb{Z}})={\mathbb{Z}},$

where the last equality holds since $Z$ is connected. But from the long exact
cohomology sequence of the pair $(M,M-Z)$ we have a canonical map

$H^{d}(M,M-Z;{\mathbb{Z}})\to H^{d}(M;{\mathbb{Z}})$ (57)

and we define $[Z]\in H^{d}(M;{\mathbb{Z}})$ to be the image of
$1\in{\mathbb{Z}}=H^{d}(M,M-Z;{\mathbb{Z}})$ under the map (57). The cap
product

$[Z]\cap(-):H_{k}(M;{\mathbb{Z}})\to H_{k-d}(Z;{\mathbb{Z}})\to
H_{k-d}(M;{\mathbb{Z}})$

fits in the classical _tube exact sequence_ in homology

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{k-d+1}(Z;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{tube}_{Z/M}}$$\textstyle{H_{k}(M-Z;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{k}(M;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[Z]\cap(-)}$$\textstyle{H_{k-d}(Z;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots,}$

where the middle map is induced from the inclusion $M-Z\subset M$ and the tube
map $\mathsf{tube}_{Z/M}$ sends a $(k-d+1)$-cycle $A$ in $Z$ to the $k$-cycle
in $M-Z$ which is the total space of the $S^{d-1}$-bundle
$\partial\mathfrak{T}_{|A}\subset\partial\mathfrak{T}\subset M-Z$.

Taking this into account, consider $Z\subset M$ to be the $C^{\infty}$
manifolds underlying the smooth complex varietties $({\rm
Wob}_{1}-\operatorname{Sing}({\rm
Wob}_{1}))\subset(X_{1}-\operatorname{Sing}({\rm Wob}_{1}))$. Thus $Z$ and $M$
are oriented, $Z$ is of real codimension $2$, and since $X_{1}$ and ${\rm
Wob}_{1}$ are irreducible, both $Z$ and $M$ are connected. Furthermore
$M-Z=X_{1}-{\rm Wob}_{1}$ as topoloical spaces and so the piece of the tube
sequence corresponding to $k=1$ reads

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(Z;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{tube}_{Z/M}}$$\textstyle{H_{1}(X_{1}-{\rm
Wob}_{1};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(M;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[Z]\cap(-)}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

Since $\operatorname{Sing}({\rm Wob}_{1})$ is a compact subvariety of complex
codimension $2$ in $X_{1}$ we have that
$H_{1}(M;{\mathbb{Z}})=H_{1}(X_{1}-{\rm
Wob}_{1};{\mathbb{Z}})=H_{1}(X_{1};{\mathbb{Z}})=0$. Therefoe

$\mathsf{tube}_{Z/M}:H_{0}(Z:{\mathbb{Z}})\to H_{1}(X_{1}-{\rm
Wob}_{1};{\mathbb{Z}})$

is surjective. But the $H_{0}(Z;{\mathbb{Z}})={\mathbb{Z}}$ is generated by
the class of of a point in $Z$, and by the definition of $\mathsf{tube}_{Z/M}$
for any point $pt\in Z$, we have that $\mathsf{tube}_{Z/M}(pt)$ is the class
of the circle in $X_{1}-{\rm Wob}_{1}$ linking to $Z={\rm
Wob}_{1}-\operatorname{Sing}({\rm Wob}_{1})$ this point. This proves the
claim. ∎

Finally, observe that a general line $\ell$ will intersect ${\rm Wob}_{1}$
transversally at a set of $8$ smooth points, and so any simple loop in $\ell$
that goes once around a point $x\in\ell\cap{\rm Wob}_{1}$ will be a a loop in
$X_{1}$ that links to ${\rm Wob}_{1}$ at $x$. Thus the image of
$H_{1}(\ell^{\circ},{\mathbb{Z}})$ in $H_{1}(X_{1}-{\rm
Wob}_{1},{\mathbb{Z}})$ contains a linking loop and so the map (56) is
surjective. This implies that $\chi$ is trivial, and hence $V\cong V^{\prime}$
which completes the proof of the proposition. ∎

Apply this now to the Hecke transform, a rank $16$ local system on $X_{1}-{\rm
Wob}_{1}$.

######  Lemma 8.18.

The rank $16$ Hecke transform local system on $X_{1}-{\rm Wob}_{1}$ decomposes
as a direct sum of two rank $8$ local systems which are isomorphic, when
restricted to a general line, to the local system constructed in Section 4.

###### Proof.

The arguments of this section show that the restriction of the rank $16$ local
system to a general line decomposes as a direct sum of two copies of the rank
$8$ local system constructed in Section 4. Furthermore, such a decomposition
can be obtained by considering the two pieces ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})$ and ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})$. We
obtain a family of decompositions over the lines, with the property that when
two lines intersect the decompositions correspond. Now, a complete
intersection of $X_{1}$ with two general hyperplanes is an elliptic curve that
can degenerate into a cycle of $4$ lines
$Z=\ell_{1}\cup\ell_{2}\cup\ell_{3}\cup\ell_{4}$. We get the decomposition on
each of these lines and the decompositions coincide on the intersection
points. This gives a decomposition of the rank $16$ local system over the
singular curve $Z$, into two pieces of rank $8$. The map $\pi_{1}(Z-Z\cap{\rm
Wob}_{1})\rightarrow\pi_{1}(X_{1}-{\rm Wob}_{1})$ is surjective, so we get a
decomposition of local systems on $X_{1}-{\rm Wob}_{1}$. ∎

######  Corollary 8.19.

The rank $16$ Hecke transform local system on $X_{1}-{\rm Wob}_{1}$ decomposes
as a direct sum of two rank $8$ local systems which are isomorphic to the
local system constructed in Section 4.

###### Proof.

The lemma gives the decomposition, and the pieces restrict to a general line
to the given rank $8$ local system. By Lemma 8.15 and Proposition 8.16, the
pieces of the decomposition are globally isomorphic to the rank $8$ local
system constructed in Section 4. ∎

##  9 Hecke transformation from $X_{1}$ to $X_{0}$

The standard diagram for the direction of the Hecke transformation
$(X_{1}\rightarrow X_{0})$ shows the big Hecke correspondence fitting into

$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X_{1}}$$\textstyle{X_{0}\times{\overline{C}}.}$

Fix a point $a=(A,t)\in{\overline{C}}$ where $A^{\otimes
2}={\mathcal{O}}_{C}(t-\boldsymbol{\mathsf{p}})$ and consider the Hecke
correspondence ${\overline{\mathcal{H}}}(a)$ with its diagram

$\textstyle{{\overline{\mathcal{H}}}(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X_{1}}$$\textstyle{X_{0}.}$

The objective in this section is to pull-back the constructed Higgs bundle
from $X_{1}$ to ${\overline{\mathcal{H}}}(a)$ and take the higher direct image
along $q$ to $X_{0}$.

The big abelianized Hecke fits into a diagram of the form

$\textstyle{{\widehat{\mathcal{H}}}^{\rm
ab}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\rm
ab}}$$\scriptstyle{q^{\rm
ab}}$$\textstyle{Y_{1}}$$\textstyle{Y_{0}\times{\widehat{C}}.}$

The two points $\widehat{a}$ and $\tau\widehat{a}$ over $a\in{\overline{C}}$
give two abelianized Hecke varieties ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})$ and ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})$.
These are the same as the varieties with the same notation in the previous
section, and recall that their disjoint union is denoted by

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a}):={\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})\sqcup{\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a}).$

The map ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})\rightarrow Y_{1}$ is
the blow-up along $16$ lines contained in the wobbly locus, while the map
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})\rightarrow Y_{0}$ is the blow-
up along a curve isomorphic to ${\widehat{C}}$ (and that will be called by the
same name) inside $Y_{0}$.

###  9.1 Restriction to a line

As before, fix a general line $\ell\subset X_{0}$. Let
${\overline{\mathcal{H}}}(a)_{\ell}$ be the inverse image of $\ell$ in the
Hecke variety ${\overline{\mathcal{H}}}(a)$. As $\ell$ is general, it does not
meet the image in $X_{0}$ of the curve ${\widehat{C}}\subset Y_{0}$, so if
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ denotes the inverse
image of $\ell$ in ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$, the
projection induces an isomorphism

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}Y_{0,\ell}.$

The same holds for the other piece ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$.

Recall from Corollary 7.10 that for $\ell$ general, the images of
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ and
${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})_{\ell}$ are disjoint in the
spectral variety

$\Sigma:=Y_{1}\times_{X_{1}}{\overline{\mathcal{H}}}(a)_{\ell}\rightarrow{\overline{\mathcal{H}}}(a)_{\ell}.$

Thus, we may treat each piece separately. We will look mainly at
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ with the understanding
that the arguments for ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$ are the same.

######  Lemma 9.1.

The Hecke correspondence ${\overline{\mathcal{H}}}(a)_{\ell}$ over $\ell$ maps
by a closed immersion onto a hyperplane section that we will denote by
$H_{\ell}\subset X_{1}$. The image of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ is a curve of genus $25$. The map

$\phi:{\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}\rightarrow\ell$

has degree $8$. It has: $16$ ramification points over the intersection points
of $\ell$ with the trope planes, and $4$ ramification points over each of the
$4$ intersection points in $\ell\cap\mathsf{Kum}$. It has $32$ other ’movable’
ramification points not mapping to points in the wobbly ${\rm Wob}_{0}$, so
there are altogether $64$ branch points.

###### Proof.

Recall that one of the quadrics in the pencil is identified with the
Grassmanian of lines in ${\mathbb{P}}^{3}$, embedded in ${\mathbb{P}}^{5}$ by
the Plücker coordinates. The line $\ell$ itself corresponds to a vector in
$v(\ell)\in\bigwedge^{2}{\mathbb{C}}^{4}$. The condition for another line
$\mathsfit{m}\subset{\mathbb{P}}^{3}$ to meet $\ell$ is that its vector
$v(\mathsfit{m})\bigwedge^{2}{\mathbb{C}}^{4}$ satisfies
$v(\mathsfit{m})\wedge v(\ell)=0$ in
$\bigwedge^{4}{\mathbb{C}}^{4}\cong{\mathbb{C}}$. This is a linear condition
on $v(\mathsfit{m})$ so it corresponds to a hyperplane in ${\mathbb{P}}^{6}$.
The image of ${\overline{\mathcal{H}}}(a)_{\ell}$ in $X_{1}$ corresponds to
the subset of points in $X_{1}$ whose corresponding line meets $\ell$, in
other words it is this hyperplane intersected with $X_{1}$. This yields the
hyperplane section $H_{\ell}$.

Since $\ell$ is general, its inverse image in ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})$ is the same as its inverse image in $Y_{0}$ because the
abelianized Hecke is the blow-up of $Y_{0}$ on a subset that maps to a curve
in $X_{0}$ that will be missed by a general $\ell$. Now, we may proceed to
calculate the normal bundle of this curve in $Y_{0}$. It is the pullback of
the normal bundle of $\ell$ in ${\mathbb{P}}^{3}$ which is to say
${\mathcal{O}}(1)\oplus{\mathcal{O}}(1)$, so that has degree $2$ on $\ell$.
Its pullback has degree $16$. Now, $Y_{0}$ is the blow-up of an abelian
variety at $16$ points; the exceptional divisors map to the trope planes so
our general $\ell$ meets each of the exceptional divisors once. The canonical
bundle of $Y_{0}$ is twice the exceptional divisor, so it has degree $32$. The
canonical bundle of our curve is the canonical of $Y_{0}$ restricted to the
curve, plus a divisor of degree $16$, so it has degree $32+16=48$. Therefore,
the image of ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ in
$H_{\ell}$ has genus $25$.

The ramification of this curve over $\ell$ is just the restriction to $\ell$
of the ramification of $Y_{0}/X_{0}$. We know that has fixed pieces including
a simple ramification over each trope plane, plus four simple ramifications
over general points of $\mathsf{Kum}$. The remaining ramification points are
really movable, as was noted in Lemma 5.13 (see also Corollary 4.23 for the
corresponding statement in the other direction). ∎

The ramification points of the horizontal divisor lie over points of
$\left(\bigcup_{\kappa}\mathsf{Trope}_{\kappa}\right)\cap\ell$. They come from
$16$ lines inside ${\rm Wob}_{1}$, and correspond to the first $16$
ramification points mentioned in Lemma 9.1. The locations of these lines in
${\rm Wob}_{1}$ depend on the choice of point $a=(A,t)$ used to make the Hecke
correspondence.

We note that the modular spectral covering $\Sigma$ has two ramification
points over each general point of the wobbly locus, so these $16$ points
correspond to $32$ points in the full abelianized Hecke. They are distrubuted
as $16$ in each of the two pieces ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ and ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$.

######  Remark 9.2.

The horizontal divisor in $H_{\ell}$ is ${\rm Wob}_{1}\cap H_{\ell}$. It has
nodes and cusps coming from the nodes and cusps of ${\rm Wob}_{1}$. However,
for a general $\ell$ these do not meet the image of
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$. In particular, they
are not going to contribute singularities to the higher direct image – this is
another version of the “apparent singularities” encountered in the previous
chapter.

The proof of this statement is that the nodal and cuspidal loci of the wobbly
locus ${\rm Wob}_{1}$ are curves; they pull back to curves in $Y_{1}$. The
centers of the blow-up ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$ are the
$16$ lines in ${\rm Wob}_{1}$ that get blown up to form
$\boldsymbol{\mathit{Exc}}_{0,\kappa}$. These are transverse to both the nodal
and cuspidal loci. Therefore, the pullbacks of the nodal and cuspidal loci in
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$ are $1$-dimensional. Their
$1$-dimensional images in $X_{0}$ do not meet a general line $\ell$ so
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ does not meet the nodal
and cuspidal loci of ${\overline{W}}_{1}$ and we get the statement of the
remark.

Among the hypotheses of Subsection 3.11 is the following statement.

######  Lemma 9.3.

For a general line $\ell$, the intersection of the plane $H_{\ell}$ with any
of the $16$ lines in ${\rm Wob}_{1}$ that provide ramification of the Hecke
correspondence, is a general point on that line. At such a point, the value of
the spectral $1$-form $\alpha$ on the vertical direction in the tangent space
of $Y_{1}$, at either of the two ramification points of $Y_{1}/X_{1}$ over
this point, is nonzero.

###### Proof.

For the first part, we note that one of these $16$ lines in ${\rm Wob}_{1}$
gets blown up to an $\mathbb{F}_{1}$-surface, that then maps to the
corresponding trope plane in $X_{0}={\mathbb{P}}^{3}$. The exceptional divisor
blows down to a point, that we will call the origin, in the trope plane
(corresponding to the intersection in $Y_{0}$ of the plane with the
${\widehat{C}}$ curve). The location of the intersection of $H_{\ell}$ with
the line corresponds to the slope of the line from this origin to the
intersection point of $\ell$ with the trope plane. For $\ell$ general, this
direction is a general point of the line.

Inside ${\mathcal{P}}_{2}$, the curve ${\widehat{C}}$ passes through $16$
points that are blown up to get $Y_{0}$. The identification
${\mathcal{P}}_{2}\cong{\mathcal{P}}_{3}$ depends on $a$. Instead of blowing
up the points, we blow up the curve ${\widehat{C}}\subset{\mathcal{P}}_{3}$ to
get $Y_{1}$, and this generates $16$ lines inside
$\boldsymbol{\mathit{E}}_{1}\subset Y_{1}$. Let $\mathfrak{v}$ denote one of
these lines. The normal bundle of $\mathfrak{v}$ in $Y_{1}$ is
${\mathcal{O}}\oplus{\mathcal{O}}(-1)$, with the ${\mathcal{O}}$ direction
being the normal bundle of $\mathfrak{v}$ in $\boldsymbol{\mathit{E}}_{1}$.

Let us look at how this maps to the normal bundle of the image line
$\mathsfit{v}\subset{\rm Wob}_{1}$. The normal bundle of a line in
${\mathbb{P}}^{5}$ is ${\mathcal{O}}(1)^{\oplus 4}$, and to get the normal
bundle of the line in the intersection of two quadrics we take the kernel of a
map ${\mathcal{O}}(1)^{\oplus 4}\rightarrow{\mathcal{O}}(2)^{\oplus 2}$. For a
typical line, this kernel will be ${\mathcal{O}}^{\oplus 2}$.

However, for a line in the wobbly locus, that we recall counts twice in the
set of four lines through each point of $X_{1}$, we claim that the kernel is
${\mathcal{O}}(-1)\oplus{\mathcal{O}}(1)$. This may be seen by recalling that
the lines in the wobbly locus are the tangent lines to the copy of
${\overline{C}}$ that forms the cuspical locus of ${\rm Wob}_{1}$, and when we
move to first order along this ${\overline{C}}$ the tangent line undergoes a
deformation that vanishes at that point in the normal direction. Thus, these
first order deformations are sections of the normal bundle that vanish. But,
the bundle ${\mathcal{O}}^{\oplus 2}$ does not have any nonzero sections that
vanish somewhere. The only other possibility is the bundle
${\mathcal{O}}(-1)\oplus{\mathcal{O}}(1)$ since it has to be a subbundle of
${\mathcal{O}}(1)^{\oplus 4}$. This proves the claim.

As we move in the curve ${\overline{C}}$, the line moves in ${\rm Wob}_{1}$ so
the normal direction has sections, and it has to be the ${\mathcal{O}}(1)$
subbundle. The map ${\mathcal{O}}\rightarrow{\mathcal{O}}(1)$ has a zero at
the points where the line $\mathsfit{v}$ crosses the cuspidal locus of ${\rm
Wob}_{1}$. However, by looking at the local picture of the covering
$Y_{1}\rightarrow X_{1}$ near such a cuspidal locus, we can see that the map
from the full tangent space of $Y_{1}$ into the tangent space of $X_{1}$ has
image of dimension $2$, so the map from the full normal bundle of
$\mathfrak{v}\subset Y_{1}$ to the ${\mathcal{O}}(1)$ piece in the normal
bundle of $\mathsfit{v}\subset X_{1}$ is surjective. We conclude that the
bundle of vertical directions, which is the kernel of this map

${\mathcal{O}}\oplus{\mathcal{O}}(-1)\rightarrow{\mathcal{O}}(1),$

is ${\mathcal{O}}(-2)$ sitting in ${\mathcal{O}}\oplus{\mathcal{O}}(-1)$ as a
saturated subbundle. In particular it is not contained in any subbundle of the
form ${\mathcal{O}}(-1)$. The map from this bundle of vertical normal
directions, into the space of normal directions at the original point of
${\mathcal{P}}_{3}$, therefore does not have image in a plane. Thus, the
tautological $1$-form, that is a nonzero linear form on the tangent space of
${\mathcal{P}}_{3}$, does not vanish on the vertical normal direction at a
general point of $\mathfrak{v}$. This proves the second part of the lemma, in
view of the generality statement of the first part. ∎

Recall that the fiber of $q:{\overline{\mathcal{H}}}(a)_{\ell}\rightarrow\ell$
over a general point $\ell$ is a smooth conic, but over points of the Kummer
it degenerates to a union of two lines.

######  Lemma 9.4.

The branch points of ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$
lying over points of the Kummer, all map to points of the Kummer K3 surface
$H_{\mathsf{Kum}}\subset{\overline{\mathcal{H}}}(\widehat{a})$, that is to say
in ${\overline{\mathcal{H}}}(a)_{\ell}$ they map to points where the two lines
in the fiber meet.

###### Proof.

The reasoning for this is as follows: the horizontal divisor in
${\overline{\mathcal{H}}}(a)_{\ell}$ is the intersection of
${\overline{\mathcal{H}}}(a)_{\ell}$ (considered as a hyperplane $H_{\ell}$ in
$X_{1}$) with the wobbly ${\rm Wob}_{1}$. But, the lines in the fibers over
points of the Kummer, represent general lines in $X_{1}$ since $\ell$
intersects the Kummer in general points. A general line in $X_{1}$ will
intersect ${\rm Wob}_{1}$ transversally.

We note that it is necessary to have four Jordan blocks of size two in the
monodromy of the Hecke-transformed local system (the rank $8$ piece
corresponding to our chosen branch of $H^{\rm ab}$ out of two, that we are
hoping is our chosen flat bundle of rank $8$ on $X_{0}$) at each point of the
Kummer. Otherwise, the Hecke transform back in the other direction from
$X_{0}$ to $X_{1}$ will not have the right rank.

So, these all have to come from simple ramification points of
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ over the Kummer points.
However, if those ramification points were to occur on smooth points of the
lines, that wouldn’t contribute anything to the monodromy in the direct image
(i.e. the residue of the Higgs field) since the map is not singular at those
locations. This heuristic argument suggests that ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ should have four points over each point where a fiber
of $\phi$ breaks into two lines.

This can be shown in terms of bundles. Suppose $L\in{\mathcal{P}}_{2}$ is a
line bundle on ${\widetilde{C}}$ whose direct image $V=\pi_{*}L$ is stable
with trivial determinant, so it is a point in $X_{0}$, and suppose that $V$
contains a line subbundle $U$ of degree $0$. Then we claim that $V$ also
contains $U^{-1}$.

To see this, we note that the group ${\mathbb{Z}}/2\times{\mathbb{Z}}/2$ acts
on ${\widetilde{C}}$, as may be seen for example by expressing

${\widetilde{C}}=C\times_{{\mathbb{P}}^{1}}{\mathbb{P}}^{1}$

as a fiber product with the double cover
${\mathbb{P}}^{1}\rightarrow{\mathbb{P}}^{1}$ ramified at two points (so the
ramification of ${\widetilde{C}}/C$ is the preimage in $C$ of those two points
of ${\mathbb{P}}^{1}$). Explicitly the non-trivial elements of
${\mathbb{Z}}/2\times{\mathbb{Z}}/2$ are given by the covering involution
$\tau:{\widetilde{C}}\to{\widetilde{C}}$ for thr map $\pi:{\widetilde{C}}\to
C$, the covering involution $\sigma:{\widetilde{C}}\to{\widetilde{C}}$ for the
hyperelliptic map
$\mathsf{h}_{{\widetilde{C}}}:{\widetilde{C}}\to{\mathbb{P}}^{1}$, and their
composition $\rho=\tau\circ\sigma$.

Now, note that by construction the group element
$\rho\in{\mathbb{Z}}/2\times{\mathbb{Z}}/2$ acts trivially on the Prym of
${\widetilde{C}}/C$ but acts by $-1$ on $\operatorname{Jac}(C)$. Applying
$\rho$ to the map $\pi^{*}(U)\rightarrow L$ we obtain a map
$\pi^{*}(U^{-1})\rightarrow L$ and hence by adjunction we get an injective map
$U^{-1}\rightarrow V$. This will give by degree considerations $V=U\oplus
U^{-1}$. In the Hecke correspondence this will correspond to a point at the
intersection of the two ${\mathbb{P}}^{1}$ components of the fiber of $q$ over
$V$ (we recall that the affine parts of the two lines themselves were bundles
that were semistable but not polystable).

We can also note that
$L\cong\pi^{*}(U)\otimes{\mathcal{O}}_{{\widetilde{C}}}(a+b)$ for an effective
degree $2$ divisor $a+b$ on ${\widetilde{C}}$, amd that the image divisor
$\pi(a)+\pi(b)$ is fixed by the determinant condition for $V$. Thus, there are
four choices of $(a,b)$ lifting this divisor to ${\widetilde{C}}$. We get the
four claimed branches of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ going through a crossing point in
${\overline{\mathcal{H}}}(a)_{\ell}$. ∎

######  Proposition 9.5.

The map ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\rightarrow\Sigma$ identifies the
abelianized Hecke as the upper critical locus (see Proposition 12.1)

${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}\cong\widetilde{{\rm
Crit}}\left({\overline{\mathcal{H}}}(a)_{\ell}/\ell,\left(\mathscr{F}_{0,{\overline{\mathcal{H}}}(a)_{\ell}},\Phi_{0}\right)\right).$

These are smooth curves in $\Sigma$ that decompose into a disjoint union of
two pieces corresponding to $\widehat{a}$ and $\tau\widehat{a}$.

###### Proof.

This is similar to the proof of Proposition 8.1, using Lemma 9.3 for
smoothness of the upper critical locus at ramification points of the
horizontal divisor.

For other points, we note that the abelianized Hecke is identified with
$Y_{0}$ outside of $\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$, but
$\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$ has image equal to a curve in
$X_{0}$ that does not intersect a general $\ell$ (as will be pointed out again
in Lemma 9.6 below). Thus, the inverse image of a general line is smooth in
this open subset of the abelianized Hecke. This shows that
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$ is
smooth away from the ramification points of the horizontal divisor.

The pointwise identification between the abelianized Hecke and the upper
critical locus, whose proof is the same as for Proposition 8.1, therefore
gives an identification of subschemes (again, neither of them has embedded
points). We have seen that one or the other is smooth at every point, so the
are both smooth. Disjointness of the two pieces comes from Corollary 7.10 . ∎

We next turn to the calculation of the direct image. Suppose given a spectral
line bundle $\mathscr{L}_{1}$ over $Y_{1}$ whose direct image to $X_{1}$
corresponds to our parabolic Higgs bundle. Recall that this means, more
precisely, that $f_{1*}(\mathscr{L}_{1})$ is the parabolic level $0$ piece of
the Higgs bundle, with parabolic level $1/2$ piece equal to
$f_{1*}(\mathscr{L}_{1}(\boldsymbol{\mathit{E}}_{1}))$.

The direct image formula for the holomorphic Dolbeault complex leads to a line
bundle over ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$. At points
where ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ goes through the
crossing points of the fibers over Kummer points, the contribution is just
$\mathscr{L}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}\otimes\omega_{{{\overline{\mathcal{H}}}(a)}_{\ell}/\ell}$,
since the horizontal divisor does not intervene. Recall that
$\omega_{{{\overline{\mathcal{H}}}(a)}_{\ell}/\ell}$ may also be viewed as the
quotient of the forms with logarithmic singularities along the fiber, modulo
logarithmic forms from the base.

In the notations of Subsection 3.11, the present situation is covered by the
situation there, and there are no points of type 3.11.1(e). Indeed we are in
the “parabolic” case where there is a parabolic structure with weights $0,1/2$
on the source space, and such points are not allowed as the horizontal divisor
does not have nodes.

The direct image calculations were summarized in Theorem 3.17 and proven in
Theorem 12.10 based on Proposition 12.7. Near a point $\mathbf{q}$ where the
horizontal divisor (which is ${\rm Wob}_{1}\cap H_{\ell}$) has a ramification
over a point of a trope plane, the curve ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$ also passes through that point, and the contribution
for the direct image is

$\mathscr{L}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}\otimes\omega_{H_{\ell}/\ell}(\mathbf{q}).$

This formula would also hold near a point $\mathbf{q}$ where the upper
critical locus intersects the ramification divisor $\mathsf{Ram}\subset\Sigma$
over points where the horizontal divisor is étale over the base. Indeed, the
proof of Theorem 12.10 included a discussion of that possibility. The
ramification divisor is the same as the pullback of
$\boldsymbol{\mathit{E}}_{1}$ to $\Sigma$. However, in fact, these kinds of
points do not occur:

######  Lemma 9.6.

For general $\ell$, the curve ${\widehat{\mathcal{H}}}^{\rm
ab}(\tilde{a})_{\ell}$ viewed inside $Y_{1}$, only meets the divisor
$\boldsymbol{\mathit{E}}_{1}$ at points $\mathbf{q}$ where the horizontal
divisor has a ramification over a point of a trope plane intersected with
$\ell$.

###### Proof.

The pullback of $\boldsymbol{\mathit{E}}_{1}$ to ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})$ is the sum
$\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})\cup\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$
of the strict transform $\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$ and the
exceptional divisors over the $16$ lines that form
$\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$. However,
$\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$ is contracted and maps to a curve
inside $Y_{0}$, hence also in $X_{0}$. A general line does not meet this
curve. Thus, taking the inverse image of a general line $\ell$ inside
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})$ and projecting back to
$Y_{1}$, gives a curve that only meets $\boldsymbol{\mathit{E}}_{1}$ at points
of the $16$ lines that generate ramification over a trope plane. ∎

Either using this lemma, or in any case by the remark of the preceding
paragraph, the contribution to the higher direct image coming from points near
$\boldsymbol{\mathit{E}}_{1}$ is the restriction to the curve $G$ of

$\mathscr{L}_{1}(\boldsymbol{\mathit{E}}_{1})|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}\otimes\omega_{H_{\ell}/\ell}.$

This globalizes to other points of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}$. For the points lying over the Kummer, notice that
the relative dualizing sheaf is the same as the relative sheaf of logarithmic
differentials that enters into the higher direct image calculations, so the
spectral line bundle is obtained by taking $\mathscr{L}_{1}$ and tensoring
with $\omega_{H_{\ell}/\ell}$ near these points. Since these points are not on
$\boldsymbol{\mathit{E}}_{1}$ (as follows from Lemma 9.4), the contributions
for those points are given by the same expression. Putting these all together,
we obtain the computation of the spectral line bundle:

######  Proposition 9.7.

Taking the Hecke transform and restricting to the line $\ell$, the spectral
line bundle on ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$ is the
line bundle

$\mathscr{L}_{1}(\boldsymbol{\mathit{E}}_{1})|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}\otimes\omega_{H_{\ell}/\ell}.$

The same holds for the spectral line bundle on ${\widehat{\mathcal{H}}}^{\rm
ab}(\tau\widehat{a})_{\ell}$, and these two are disjoint in $\Sigma$, so this
expression gives the spectral line bundle over the disjoint union
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a},\tau\widehat{a})_{\ell}$.

###  9.2 Calculation of the pushforward

Calculate in the same way as at the end of the previous chapter. Recall that
if we write $\Xi$ for the theta divisor on the Prym, or its pullbacks to
$Y_{0}$, $Y_{1}$ or ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$, we have
$\boldsymbol{\mathit{F}}_{0}=2\Xi-\boldsymbol{\mathit{E}}_{0}$ and
$\boldsymbol{\mathit{F}}_{1}=4\Xi-\boldsymbol{\mathit{E}}_{1}$. Also

$\boldsymbol{\mathit{E}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}=\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$

whereas

$\boldsymbol{\mathit{E}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}=\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})+\boldsymbol{\mathit{Exc}}_{1}(\widehat{a}),$

and

$\boldsymbol{\mathit{F}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}=2\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})$
$\boldsymbol{\mathit{F}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}=4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a}).$

######  Lemma 9.8.

For the relative canonical class over the degree $0$ moduli space, we have

$\omega_{{\overline{\mathcal{H}}}(a)/X_{0}}={\mathcal{O}}_{X_{0}}(2)|_{{\overline{\mathcal{H}}}(a)}\otimes{\mathcal{O}}_{X_{1}}(-1)|_{{\overline{\mathcal{H}}}(a)}$

and this pulls back to ${\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}(\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))$.

###### Proof.

As in Lemma 8.13,

$\omega_{{\overline{\mathcal{H}}}(a)}={\mathcal{O}}_{X_{0}}(-2)|_{{\overline{\mathcal{H}}}(a)}\otimes{\mathcal{O}}_{X_{1}}(-1)|_{{\overline{\mathcal{H}}}(a)}.$

On the other hand, $\omega_{X_{0}}={\mathcal{O}}_{X_{0}}(-4)$, giving the
first formula. Then

$\omega_{{\overline{\mathcal{H}}}(a)/X_{0}}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}={\mathcal{O}}(2\boldsymbol{\mathit{F}}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}-\boldsymbol{\mathit{F}}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})})$
$={\mathcal{O}}(2(2\Theta-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))-(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})))={\mathcal{O}}(\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})).$

∎

Recall that the spectral line bundle on $Y_{0}$ is
$\mathscr{L}_{0}={\mathcal{O}}_{Y_{0}}(2\boldsymbol{\mathit{F}}_{0}+\boldsymbol{\mathit{E}}_{0})$
and the spectral line bundle on $Y_{1}$ is
$\mathscr{L}_{1}={\mathcal{O}}_{Y_{1}}(\boldsymbol{\mathit{F}}_{1})$. As
before, we get

$\mathscr{L}_{0}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}={\mathcal{O}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))$

and

$\mathscr{L}_{1}|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})}={\mathcal{O}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})).$

From Proposition 9.7, the spectral line bundle of the Hecke transform
restricted over $\ell$ is the bundle

$\displaystyle\mathscr{L}_{1}\otimes$
$\displaystyle{\mathcal{O}}_{Y_{1}}(\boldsymbol{\mathit{E}}_{1})|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}\otimes\omega_{H_{\ell}/\ell}$
$\displaystyle={\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{1}(\widehat{a}))\otimes{\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})+\boldsymbol{\mathit{Exc}}_{1}(\widehat{a}))$
$\displaystyle\hskip 108.405pt\otimes{\mathcal{O}}_{H^{\rm
ab}}(\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))|_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}$
$\displaystyle={\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(4\Xi+\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a})).$

This compares with the spectral line bundle coming from $Y_{0}$ which is
${\mathcal{O}}_{{\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a})_{\ell}}(4\Xi-\boldsymbol{\mathit{Exc}}_{0}(\widehat{a}))$.

As before, the divisor $\boldsymbol{\mathit{E}}_{1}$ is the exceptional
divisor of blowing-up $Y_{1}$ along a copy of the curve ${\widehat{C}}$. The
image of the curve in ${\mathbb{P}}^{3}$ does not intersect a general line
$\ell$ so $\boldsymbol{\mathit{Exc}}_{1}(\widehat{a})$ does not intersect
${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$. Thus, the spectral
line bundle of the Hecke transform coincides with the spectral line bundle
coming from $Y_{0}$ on ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$.

######  Proposition 9.9.

Define the line bundle ${\mathcal{U}}$ over ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$ as the pullback of
$\mathscr{L}_{1}(\boldsymbol{\mathit{E}}_{1})$ on $Y_{1}$, tensored with
$\omega_{H_{\ell}/\ell}$. Then ${\mathcal{U}}$ is the spectral line bundle for
the $L^{2}$ Dolbeault higher direct image Higgs bundle on $\ell$. Over each of
the two pieces of ${\widehat{\mathcal{H}}}^{\rm
ab}(\widehat{a},\tau\widehat{a})_{\ell}$, ${\mathcal{U}}$ is isomorphic to the
restriction of the spectral line bundle that we construct for $X_{0}$, from
$Y_{0}$ to ${\widehat{\mathcal{H}}}^{\rm ab}(\widehat{a})_{\ell}$
(respectively ${\widehat{\mathcal{H}}}^{\rm ab}(\tau\widehat{a})_{\ell}$).

###### Proof.

The above calculations show this, it is also described in Section 7. Notice
that the proof in this direction is significantly less complicated than in the
previous section since we do not need to deal with the points of type
3.11.1(e). ∎

##  10 The big Hecke correspondences

In this section we will consider the “big” Hecke correspondences fitting into
a diagram of the form, for the $(X_{1}\rightarrow X_{0})$ direction:

$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X_{1}}$$\textstyle{X_{0}\times{\overline{C}}}$

or similarly in the opposite direction $(X_{0}\rightarrow X_{1})$ pictured
below. We would like to show that the local system obtained by pulling back
our local system from $(X_{1},{\rm Wob}_{1})$ then taking $R^{1}q_{*}$, is an
exterior tensor product of our local system on $(X_{0},{\rm Wob}_{0})$ by the
initially given rank two local system on $C$, pulled back to ${\overline{C}}$.

The main part of the proof on the spectral data was given in Subsection 7.4.
The objective of this section is to prove some complementary statements
designed to deal with possible apparent singularities of the higher direct
image operation. For this, we introduce the notion of _effective discriminant
divisor_ , this is the part of the discriminant divisor on which the higher
direct image local system really does have singularities.

###  10.1 From $X_{1}$ to $X_{0}$

For the moment we will work in the direction from $X_{1}$ to $X_{0}$ as
pictured in the previous diagram. Let $\Delta\subset
X_{0}\times{\overline{C}}$ be the discriminant divisor of the map $q$ with
respect to the pair $({\overline{\mathcal{H}}},p^{-1}{\rm Wob}_{1})$. This
includes points in $X_{0}\times{\overline{C}}$ over which the map $q$ is not
smooth, and points over which the horizontal divisor $p^{-1}{\rm Wob}_{1}$ is
not étale. The singularities of the $R^{1}q_{*}$ local system are _a priori_
contained in $\Delta$. Let $\Delta_{\rm eff}$ be the _effective_ singular
divisor, namely the divisor over which the local system has singularities.
Thus $\Delta_{\rm eff}\subset\Delta$.

From the previous section we know the following statement:

######  Proposition 10.1.

If $a$ is a general point of ${\overline{C}}$ then along the fiber
$X_{0}\times\\{a\\}$, the singular divisor $\Delta_{\rm eff}$ consists of just
${\rm Wob}_{0}\subset X_{0}$.

This proposition readily implies the following:

######  Corollary 10.2.

The effective singular divisor consists of ${\rm
Wob}_{0}\times{\overline{C}}$, possibly union with a finite number of fibers
of the form $X_{0}\times\\{a_{i}\\}$ for points $a_{i}\in{\overline{C}}$.

###### Proof.

In general a divisor such as $\Delta_{\rm eff}$ in a product
$X_{0}\times{\overline{C}}$ decomposes as

$\Delta_{\rm eff}=\Delta_{\rm eff}^{\rm vert}+\Delta_{\rm eff}^{\rm
horiz}+\Delta_{\rm eff}^{\rm mov}$

where $\Delta_{\rm eff}^{\rm vert}$ is a sum of vertical components
$X_{0}\times\\{a_{i}\\}$, $\Delta_{\rm eff}^{\rm horiz}$ is a sum of divisors
of the form $D_{i}\times{\overline{C}}$, and $\Delta_{\rm eff}^{\rm mov}$ is
given by a moving family of divisors parametrized by ${\overline{C}}$. Over a
general point of ${\overline{C}}$, the intersection of these divisors with the
fiber will be: for $\Delta_{\rm eff}^{\rm vert}$, empty; for $\Delta_{\rm
eff}^{\rm horiz}$, the union of the $D_{i}$; and for $\Delta_{\rm eff}^{\rm
mov}$, a divisor that moves as a function of the point. The proposition says
that these all consist of just ${\rm Wob}_{0}\subset X_{0}$, not moving as a
function of the point of ${\overline{C}}$. It follows that $\Delta_{\rm
eff}^{\rm mov}=\varnothing$ and $\Delta_{\rm eff}^{\rm horiz}={\rm
Wob}_{0}\times{\overline{C}}$. There remains the possibility of a nonempty
$\Delta_{\rm eff}^{\rm vert}$. ∎

In order to rule out the possibility of having a vertical piece in
$\Delta_{\rm eff}$ we’ll just rule that out for the full discriminant divisor.

######  Proposition 10.3.

The discriminant $\Delta$ of the map $q$ from
$({\overline{\mathcal{H}}},p^{-1}{\rm Wob}_{1})$ to
$X_{0}\times{\overline{C}}$ does not contain any vertical pieces of the form
$X_{0}\times\\{a\\}$.

###### Proof.

We need to show that for any point $a\in{\overline{C}}$, the full fiber
$X_{0}\times\\{a\\}$ is not contained in the discriminant.

The point $a\in{\overline{C}}$ corresponds to a pair $a=(A,t)$ where $A$ is a
line bundle of degree $0$ and $t\in C$ such that $A^{\otimes
2}(\boldsymbol{\mathsf{p}})={\mathcal{O}}_{C}(t)$ (recall that
$\boldsymbol{\mathsf{p}}$ is our fixed Weierstrass point). We need to show
that for a general point ${\mathcal{F}}\in X_{0}$, the Hecke curve
corresponding to $({\mathcal{F}},(A,t))$ is smooth and intersects ${\rm
Wob}_{1}$ transversally.

A general ${\mathcal{F}}$ is stable. A bundle obtained by a Hecke
transformation is the kernel in

$0\rightarrow{\mathcal{E}}\rightarrow{\mathcal{F}}\otimes
A(\boldsymbol{\mathsf{p}})\rightarrow{\mathbb{C}}_{t}\rightarrow 0$

where the quotient corresponds to a rank $1$ quotient of the fiber
$({\mathcal{F}}\otimes A(\boldsymbol{\mathsf{p}}))_{t}$ over the point $t$. If
${\mathcal{F}}$ is stable of degree $0$ then its maximal degree line
subbundles have degree $-1$, so any line subbundle of ${\mathcal{E}}$ must
have degree $\leq 0$. Thus, ${\mathcal{E}}$ is stable. The Hecke curve is
therefore isomorphic to the space of such rank $1$ quotients, so it is
${\mathbb{P}}^{1}$ and is hence smooth.

A little more precisely, let
${\overline{\mathcal{H}}}^{s}\subset{\overline{\mathcal{H}}}$ be the moduli
space of line bundles $B\in{\overline{C}}$ paired with inclusions
${\mathcal{E}}\subset{\mathcal{F}}\otimes A(\boldsymbol{\mathsf{p}})$ of
colength $1$ such that ${\mathcal{E}}$ and ${\mathcal{F}}$ are stable. Let
$X_{0}^{s}\subset X_{0}$ denote the open subset of stable bundles. By the
previous paragraph, ${\overline{\mathcal{H}}}^{s}$ is the inverse image in
${\overline{\mathcal{H}}}$ of $X_{0}^{s}\times{\overline{C}}$. The projection
from here to $X_{0}\times{\overline{C}}$ is a ${\mathbb{P}}^{1}$-bundle over
the open subset $X_{0}^{s}\times{\overline{C}}$. This was also verified
synthetically in section 2.4.

Thus, ${\overline{\mathcal{H}}}$ is smooth over
$X_{0}^{s}\times{\overline{C}}$. This shows that for any $a=(A,t)$ and for a
general ${\mathcal{F}}$ the Hecke curve over $({\mathcal{F}},a)$ is smooth.
Next, we would like to understand its intersection with $p^{-1}{\rm Wob}_{1}$.
Recall that ${\mathcal{E}}\in{\rm Wob}_{1}$ if and only if there exists a line
subbundle $B\subset{\mathcal{E}}$ of degree $0$ such that $B^{\otimes
2}(\boldsymbol{\mathsf{p}})$ is effective. If ${\mathcal{E}}$ is a Hecke
transformation of $({\mathcal{F}},(A,t))$ and $B$ is such a line subbundle
then we get an injection $B\hookrightarrow{\mathcal{F}}\otimes
A(\boldsymbol{\mathsf{p}})$ or equivalently

$L:=B\otimes A^{\vee}(-\boldsymbol{\mathsf{p}})\hookrightarrow{\mathcal{F}}.$

Here $L$ is a line bundle of degree $-1$. If ${\mathcal{F}}$ is stable of
degree $0$, that is the maximal degree of a locally free subsheaf, in
particular $L\subset{\mathcal{F}}$ is a saturated locally free subsheaf, i.e.
a strict subbundle.

In the other direction, given a degree $-1$ subbundle $L\subset{\mathcal{F}}$
there is a unique rank $1$ quotient over the point $t$ such that $B=L\otimes
A(\boldsymbol{\mathsf{p}})$ maps into the kernel of ${\mathcal{F}}\otimes
A(\boldsymbol{\mathsf{p}})\rightarrow{\mathbb{C}}_{t}$. Note that

$B^{\otimes 2}(\boldsymbol{\mathsf{p}})=L^{\otimes 2}\otimes A^{\otimes
2}(3\boldsymbol{\mathsf{p}})=L^{\otimes 2}(2\boldsymbol{\mathsf{p}}+t).$

This makes an isomorphism between the set of intersection points of the Hecke
curve with ${\rm Wob}_{1}$, and the set of solutions $L$ of the pair of
conditions

  1. (1)

$h^{0}(L^{\vee}\otimes{\mathcal{F}})>0$

  2. (2)

$h^{0}(L^{\otimes 2}(2\boldsymbol{\mathsf{p}}+t))>0$.

We will look at the solutions as a subset of $\operatorname{Jac}^{-1}(C)$ the
Jacobian of line bundles of degree $-1$ on $C$. Solutions of Condition (1)
form a divisor $D_{{\mathcal{F}}}$ in the linear system $|2\Theta|$. Indeed,
$D_{{\mathcal{F}}}$ is the Narasimhan-Ramanan point corresponding to
${\mathcal{F}}$ [NR69]. We have

$\textstyle{\operatorname{Jac}^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{3}}$$\textstyle{={\mathbb{P}}\left(H^{0}(\operatorname{Jac}^{-1}(C),{\mathcal{O}}(2\Theta)^{\vee}\right)}$$\textstyle{\cup}$$\textstyle{\cup}$$\textstyle{D_{{\mathcal{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{{\mathcal{F}}}}$

where the
${\mathbb{P}}^{3}={\mathbb{P}}\left(H^{0}(\operatorname{Jac}^{-1}(C),{\mathcal{O}}(2\Theta)^{\vee}\right)$
here is the dual of
$X_{0}={\mathbb{P}}\left(H^{0}(\operatorname{Jac}^{-1}(C),{\mathcal{O}}(2\Theta)\right)$
and $H_{{\mathcal{F}}}$ is the hyperplane corresponding to the point
${\mathcal{F}}\in X_{0}$. The top map is the mapping given by $|2\Theta|$ that
is basepoint-free [Mum07a, BL04].

The second Condition (2) corresponds to the pullback of
$C\hookrightarrow\operatorname{Jac}^{1}$ by the composed map

$\textstyle{\operatorname{Jac}^{-1}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{\otimes
2}}$$\textstyle{\operatorname{Jac}^{-2}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\otimes{\mathcal{O}}(2\boldsymbol{\mathsf{p}}+t)}$$\textstyle{\operatorname{Jac}^{1}(C).}$

Let ${\overline{C}}^{t}\subset\operatorname{Jac}^{-1}$ denote this pullback
curve.

Solutions of both conditions together correspond to the intersection of these
two subspaces. It corresponds to mapping
${\overline{C}}^{t}\rightarrow{\mathbb{P}}^{3}$ by the composed map

${\overline{C}}^{t}\rightarrow\operatorname{Jac}^{-1}\rightarrow{\mathbb{P}}^{3}$

and then pulling back a hyperplane section $H_{{\mathcal{F}}}$. If
${\mathcal{F}}$ is a general point of $X_{0}$ then this is a general
hyperplane section.

The map ${\overline{C}}^{t}\rightarrow{\mathbb{P}}^{3}$ is nonconstant, so the
pullback of a general hyperplane section is reduced, consisting of a
collection of distinct points. The number of points is the degree of the map,
which we claim is $16$. To prove that, note that our arguments below will show
that it can not be $>16$ otherwise the scheme-theoretical intersection of the
Hecke curve, a conic, with ${\rm Wob}_{1}$ would be too big. The degree is the
intersection number of the pushforward of $2\Theta$ by the squaring map. Let
us push forward the original $\Theta$ divisor, which is just
$C\subset\operatorname{Jac}^{1}$. The square is the set of divisors of the
form $2x$ for $x\in C$, translated back to $\operatorname{Jac}^{1}$ as the set
of divisors of the form $2x-t$. We want to know when this is effective, i.e.
how many pairs $(x,y)$ solve $2x-t=y$. This may be written as the equation
$2x+y^{\prime}=t+2\boldsymbol{\mathsf{p}}$, so it is the set of ramification
points of the trigonal curve associated to the linear system
$|{\mathcal{O}}(t+2\boldsymbol{\mathsf{p}})|$. We know that this is $8$, that
is the intersection of the pushforward of $\Theta$ with $C$ has $8$ points, so
degree $\geq 8$. Thus, the intersection of the pushforward of $2\Theta$ with
$C$ has degree $\geq 16$. This completes the proof that the degree of the map
${\overline{C}}^{y}\rightarrow{\mathbb{P}}^{3}$ is $16$.

Thus, there are $16$ distinct solutions $L$ of the two conditions. Each point
$L$ yields a line bundle $B=L\otimes A(\boldsymbol{\mathsf{p}})$, with
$(B,s)\in{\overline{C}}$ for some $s\in C$, and for distinct
$L\in\operatorname{Jac}^{-1}$ the line bundles $B$ are distinct. They
correspond therefore to distinct points in ${\overline{C}}$, hence to distinct
points of the normalization of ${\rm Wob}_{1}$ that fibers over
${\overline{C}}$. We next note that since our $16$ points were obtained by
intersecting the image ${\overline{C}}\rightarrow{\mathbb{P}}^{3}$ with a
general hyperplane section, any pair of two points are general with respect to
each other. Pairs of points can be glued together under the normalization map
for ${\rm Wob}_{1}$, but this generality condition implies that our $16$
solutions $L$, leading to $16$ line bundles $A$, can not contain pairs of
points that are glued together. This shows that they give $16$ distinct points
in ${\rm Wob}_{1}$.

Now, the scheme theoretic intersection of a Hecke curve with ${\rm Wob}_{1}$
has length $16$, so if there are $16$ distinct points then they have to be
reduced points in the scheme theoretic intersection. This implies that
$p^{-1}{\rm Wob}_{1}$ is unramified, hence etale over a general points of
$X_{0}^{s}\times{\overline{C}}$. This completes the proof of the proposition.
∎

######  Corollary 10.4.

The effective singular locus $\Delta_{\rm eff}$ of the $R^{1}q_{*}$ local
system on $X_{0}\times{\overline{C}}$ is ${\rm Wob}_{0}\times{\overline{C}}$.

######  Corollary 10.5.

The $R^{1}q_{*}$ local system of rank $16$ on $X_{0}\times{\overline{C}}$
decomposes as an exterior tensor product of the rank $8$ local system we have
constructed on $X_{0}$ with a rank $2$ local system on ${\overline{C}}$ whose
spectral curve is ${\widehat{C}}\rightarrow{\overline{C}}$.

###### Proof.

We have seen that on $X_{0}\times\\{a\\}$ the Higgs bundle is a direct sum of
two copies of the rank $8$ Higgs bundle we construct over $X_{0}$.
Furthermore, the direct sum consists of two copies that are preserved by the
Higgs field in the ${\overline{C}}$ direction, indeed the Higgs field comes
from the section of the sheaf of total differentials on the upper relative
critical locus which is a disjoint union. The two components vary in a
covering given by ${\widehat{C}}\rightarrow{\overline{C}}$. Also, the spectral
$1$-form is the canonical $1$-form over ${\widehat{C}}$. This may be seen by
restricting to horizontal copies of ${\widehat{C}}$ inside
$Y_{0}\times{\overline{C}}$ which map to translates of standard copies of
${\widehat{C}}$ in $Y_{1}$, on which the spectral $1$-form is the canonical
one for ${\widehat{C}}$.

It follows that the Higgs bundle is not a direct sum of two copies of a rank
$8$ Higgs bundle on $X_{0}\times{\widehat{C}}$. In view of the theorem on
irreducible representations of product groups, the only other possibility is
that it is an exterior tensor product. The spectral curve of the rank $2$
local system on ${\overline{C}}$ is ${\widehat{C}}$ with embedding given by
the canonical $1$-form on ${\widehat{C}}$, so this identifies the spectral
curve as a curve in $T^{\vee}{\overline{C}}$. ∎

###  10.2 From $X_{0}$ to $X_{1}$

Consider next the diagram

$\textstyle{{\overline{\mathcal{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{b}$$\textstyle{X_{0}}$$\textstyle{X_{1}\times{\overline{C}}.}$

In this case, $b$ is a fibration with fibers ${\mathbb{P}}^{1}$, so there is
no discriminant for the map $b$. Let $\Delta\subset X_{1}\times{\overline{C}}$
now denote the discriminant for the horizontal divisor $d^{-1}{\rm Wob}_{0}$
over $X_{1}\times{\overline{C}}$. Here again, let $\Delta_{\rm
eff}\subset\Delta$ denote the effective singularities of the local system
$R^{1}b_{*}$ of the pullback by $b$ of the local system we construct over
$(X_{0},{\rm Wob}_{0})$.

As before, the effective discriminant decomposes into potentially nonempty
pieces as

$\Delta_{\rm eff}=\Delta_{\rm eff}^{\rm vert}+\Delta_{\rm eff}^{\rm
horiz}+\Delta_{\rm eff}^{\rm mov}.$

######  Lemma 10.6.

We have $\Delta_{\rm eff}^{\rm mov}=\emptyset$, and $\Delta_{\rm eff}^{\rm
horiz}={\rm Wob}_{1}\times{\overline{C}}$.

###### Proof.

We have seen in chapter 8 that the effective discriminant in the fiber over a
general point $a\in{\overline{C}}$ is just ${\rm Wob}_{1}\times\\{a\\}\subset
X_{1}\times\\{a\\}$. ∎

We would like to rule out the possibility of a component $X_{1}\times\\{a\\}$
in $\Delta_{\rm eff}^{\rm vert}$. As before, for this we show that the full
discriminant $\Delta$ itself does not contain any vertical components. We
would like to show that for any point $a\in{\overline{C}}$, and for a general
${\mathcal{F}}\in X_{0}$, the Hecke line associated to $({\mathcal{F}},a)$
intersects ${\rm Wob}_{0}$ transversally.

Since it is a line in ${\mathbb{P}}^{3}$, it will be transverse to the trope
planes unless it is contained in one of them. Therefore, we would like to show
that the general Hecke line is not contained in a trope plane, and that it
intersects the Kummer in $4$ distinct points.

The bundle ${\mathcal{F}}$ is a stable bundle of determinant
${\mathcal{O}}_{C}(\boldsymbol{\mathsf{p}})$. If $a=(A,t)$ with $A^{\otimes
2}(\boldsymbol{\mathsf{p}})={\mathcal{O}}_{C}(t)$, then the bundles in the
Hecke line are kernels of the form

$0\rightarrow{\mathcal{E}}\rightarrow{\mathcal{F}}\otimes
A\rightarrow{\mathbb{C}}_{t}\rightarrow 0$

noting that the determinant of ${\mathcal{E}}$ is $A^{\otimes
2}(\boldsymbol{\mathsf{p}}-t)={\mathcal{O}}_{C}$.

Notice that ${\mathcal{E}}$ is semistable, indeed if it had a subbundle of
degree $1$ that would give a degree $1$ subbundle of ${\mathcal{F}}$
contradicting stability of ${\mathcal{F}}$.

We have that ${\mathcal{E}}$ is in a trope plane if there is a line subbundle
$B\subset{\mathcal{E}}$ of degree $-1$ such that $B^{\otimes
2}(2\boldsymbol{\mathsf{p}})={\mathcal{O}}_{C}$. The $16$ trope planes
correspond to the $16$ solutions of this equation. If this is the case, it
gives

$L:=B\otimes A^{\vee}\rightarrow{\mathcal{F}}.$

The set of possible $L$’s is finite.

If $h^{0}(L^{\vee}\otimes{\mathcal{F}})\geq 2$ then there is a morphism
$L^{\oplus 2}\hookrightarrow{\mathcal{F}}$, with cokernel a nontrivial
skyscraper sheaf. If a point $y\in C$ is in the support of this skyscraper
shead, then we will get a map $L(y)\rightarrow{\mathcal{F}}$, showing that
${\mathcal{F}}\in\mathsf{Kum}$. Thus, for general ${\mathcal{F}}$ there is at
most one map up to scalars from $L$ to ${\mathcal{F}}$. For similar reasons,
the image is a saturated subsheaf of ${\mathcal{F}}$. Thus, for a given $L$
there is at most one rank $1$ quotient of $({\mathcal{F}}\otimes A)_{t}$ whose
kernel contains $B=L\otimes A$. As there are finitely many $L$, the Hecke
transformed bundles ${\mathcal{E}}$ can not all be in the wobbly locus. This
shows that, for a general ${\mathcal{F}}$, the Hecke line is not contained
entirely in a trope plane.

Let’s now look at the intersection of the trope line with the Kummer. The
point of $X_{0}$ corresponding to the bundle ${\mathcal{E}}$ is a point of the
Kummer if ${\mathcal{E}}$ has a subbundle of degree $0$. So we need to
consider the possibility that there is a subbundle

$B\hookrightarrow{\mathcal{E}}$

with $A$ of degree $0$. This gives

$L:=B\otimes A^{\vee}\hookrightarrow{\mathcal{F}}$

with $L$ of degree $0$. In that case, note that the inclusion has to be
strict, since ${\mathcal{F}}$ can’t contain a subbundle of degree $1$. Then,
there is a unique rank $1$ quotient of ${\mathcal{F}}_{t}$ such that this