# Simplicity of Tangent bundles on the moduli spaces
of symplectic and orthogonal bundles over a curve

Insong Choe∗ Department of Mathematics, Konkuk University, 1 Hwayang-dong,
Gwangjin-Gu, Seoul 143-701, Republic of Korea<EMAIL_ADDRESS>,  George
H. Hitching  Oslo Metropolitan University, Postboks 4, St. Olavs plass, 0130
Oslo, Norway<EMAIL_ADDRESS>and  Jaehyun Hong  Center for Complex
Geometry, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

The variety of minimal rational tangents associated to Hecke curves was used
by J.-M. Hwang [8] to prove the simplicity of the tangent bundle on the moduli
of vector bundles over a curve. In this paper, we use the tangent maps of the
symplectic and orthogonal Hecke curves to prove an analogous result for
symplectic and orthogonal bundles. In particular, we show the nondegeneracy of
the associated variety of minimal rational tangents, which implies the
simplicity of the tangent bundle on the moduli spaces of symplectic and
orthogonal bundles over a curve. We also show that for large enough genus, the
tangent map is an embedding for a general symplectic or orthogonal bundle.

Résumé. La variété des tangentes des courbes minimales rationnelles associés
aux courbes de Hecke, a été utilisée par J.-M. Hwang [8] pour prouver la
simplicité du fibré tangent á l’espace de modules des fibrés vectoriels sur
une courbe. Nous utilisons les applications tangentes des courbes de Hecke
symplectiques et orthogonales pour demontrer un resultat analogue pour les
fibrés symplectiques et orthogonaux. En particulier, nous prouvons que la
variété des tangentes des courbes rationnelles minimales associée est
nondegénérée ; ce qui implique la simplicité des fibrés tangents des espaces
de modules des fibrés symplectiques et orthogonaux sur une courbe. Nous
montrons d’ailleurs, pour genre suffisamment grand, que l’application tangente
est un plongement pour un fibré symplectique ou orthogonal générique.

###### Key words and phrases:

symplectic bundle, orthogonal bundle, minimal rational tangents

###### 2020 Mathematics Subject Classification:

14D20, 53C10

##  1\. Introduction

Let $C$ be a smooth projective curve of genus $g\geq 2$ over the complex
numbers. Let $\mathcal{M}:={\mathcal{S}U}_{C}(n,d)$ be the moduli space of
semistable vector bundles over $C$ of rank $n$ with fixed determinant of
degree $d$. Note that $\mathcal{M}$ is a Fano variety of Picard number 1 and
moreover smooth if $n$ and $d$ are coprime.

It was shown in [8, Corollary 1] that for $g\geq 4$, the tangent bundle of the
smooth part $\mathcal{M}^{\circ}\subset\mathcal{M}$ is simple. The strategy
was to exploit certain minimal rational curves called Hecke curves and the
associated variety of minimal rational tangents. More precisely, it is shown
that the variety of minimal rational tangents $\mathcal{C}_{W}$ at a generic
point $W\in\mathcal{M}$ is non-degenerate in $\mathbb{P}(T_{W}\mathcal{M})$
and this implies the simplicity of the tangent bundle (cf. Proposition 2.6).

The goal of this paper is to prove the analogous result for the moduli spaces
$\mathcal{M}S_{C}(n,L)$ of symplectic bundles and $\mathcal{M}O_{C}(n,L)$ of
orthogonal bundles. The symplectic and orthogonal versions of Hecke curves
were constructed in [3]. In the same paper, these curves were shown to be the
minimal rational curves in the ambient varieties. Based on this, we establish
the following results in this paper:

  * •

The smoothness of the symplectic and orthogonal Hecke curves (§3)

  * •

The nondegeneracy of the tangent map on the variety of minimal rational
tangents of these Hecke curves (§4)

  * •

The very-ampleness of the associated complete linear system (§5)

In particular, as a corollary of the nondegeneracy in §4, we show that the
tangent bundles of $\mathcal{M}S_{C}(n,L)$ and $\mathcal{M}O_{C}(n,L)$ are
simple, under a certain genus bound.

We would like to add a word of warning for the arguments that will follow.
Inside the moduli space ${\mathcal{S}U}_{C}(n,d)$ of vector bundles, the locus
of symplectic/orthogonal bundles form a closed subvariety, and a
symplectic/orthogonal Hecke curve can be thought of as either a special kind
of Hecke curve on ${\mathcal{S}U}_{C}(n,d)$ or its variation. So one might
expect that the results for Hecke curves in [8], [9], or [13] directly imply
the same results for the symplectic or orthogonal setting.

But in most discussions of a Hecke curve on ${\mathcal{S}U}_{C}(n,d)$, one
assumes that it passes through a generic point, such as a $(1,1)$-stable
bundle [12, Definition 5.1]. And it is unclear if a generic point of
$\mathcal{M}S_{C}(n,L)$ and/or $\mathcal{M}O_{C}(n,L)$ is $(1,1)$-stable as a
vector bundle. By simple dimension comparison, it is still possible that the
subvarieties $\mathcal{M}S_{C}(n,L)$ and/or $\mathcal{M}O_{C}(n,L)$ are
entirely contained in the non-$(1,1)$-stable locus. For this reason, we cannot
tell from the outset if the symplectic and orthogonal Hecke curves share the
same properties as the Hecke curves passing through $(1,1)$-stable locus. This
is why we later devise arguments based on $\delta$-stability (cf. [3, §4.1])
on $\mathcal{M}S_{C}(n,L)$ and $\mathcal{M}O_{C}(n,L)$.

##  2\. Preliminary results

In this section, we gather the notations and preliminary results relevant to
our discussion. Let $C$ be a smooth projective curve of genus $g\geq 2$.

######  Notation 2.1.

Given a subspace $\Lambda\subset V^{*}$ of a dual vector space,
$\Lambda^{\perp}\subset V$ denotes the kernel:

$\Lambda^{\perp}=\\{v\in V\>:\>\lambda(v)=0\ \text{for all
}\lambda\in\Lambda\\}.$

Also for a subspace $U\subset V$, $U^{\perp}\subset V^{*}$ denotes the
annihilator:

$U^{\perp}=\\{\phi\in V^{*}\>:\>\phi(u)=0\ \text{for all }u\in U\\}.$

When $V$ is equipped with a bilinear form $\omega\colon V\otimes
V\to\mathbb{C}$, we define

$\ker(\omega):=\\{v_{0}\in V\>:\>\omega(v_{0},v)=0\ \text{for all }v\in V\\}$

###  2.1. Hecke modification

Let $W$ be a vector bundle over $C$. Choose a subspace $\Lambda\subset
W_{x}^{*}$ for some $x\in C$. The Hecke modification $W^{\Lambda}$ of $W$
along $\Lambda$ is given by the kernel of the composition map $W\rightarrow
W_{x}\rightarrow W_{x}/\Lambda^{\perp}$. There is an exact sequence of
sheaves:

$0\rightarrow
W^{\Lambda}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}W\rightarrow(W_{x}/\Lambda^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0$

whose restriction to the fiber at $x$ is given by

$0\rightarrow\operatorname{Ker}(\phi_{x}\colon W_{x}^{\Lambda}\rightarrow
W_{x})\rightarrow W^{\Lambda}_{x}\rightarrow W_{x}\rightarrow
W_{x}/\Lambda^{\perp}\rightarrow 0.$

Then the locally free sheaf $W^{\Lambda}$ corresponds to a vector bundle with
$\det(W^{\Lambda})=\det(W)\otimes\mathcal{O}_{C}(-kx)$, where $k$ is the
dimension of $\Lambda$ in $W_{x}^{*}$.

###  2.2. Hecke curves on ${\mathcal{S}U}_{C}(n,d)$

The main reference for this subsection is [8].

Let $W$ be a vector bundle over $C$. For a subspace $\theta\subset W_{x}^{*}$
of dimension one, by abuse of notation, let $W^{\theta}$ denote the Hecke
modification of $W$ along $\theta$. Then we have

$0\rightarrow W^{\theta}\rightarrow
W\rightarrow(W_{x}/\theta^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

For a subspace $\ell\subset W^{\theta}_{x}$ of dimension one, the Hecke
modification $V^{\ell}$ of $V:=(W^{\theta})^{*}$ along $\ell$ fits into the
exact sequence

(1) $0\rightarrow
V^{\ell}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\ell^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

In particular for $\ell_{0}:=\operatorname{Ker}(W_{x}^{\theta}\rightarrow
W_{x})$, we have $(V^{\ell_{0}})^{*}\simeq W$.

For any two-dimensional subspace $U$ with $\ell_{0}\subset U\subset
W^{\theta}_{x}$, the subspace $U^{\perp}$ has codimension two in $V_{x}$, and
is contained in $\ell_{0}^{\perp}$. Hence the family

$\\{(V^{\ell})^{*}\>:\>{[\ell]\in\mathbb{P}(U)}\\}$

parameterized by
$\mathbb{P}(U)=\mathbb{P}(V_{x}/U^{\perp})\cong\mathbb{P}^{1}$, is a
deformation of $W$. If we choose a generic $W\in{\mathcal{S}U}_{C}(n,d)$, then
this family gives a smooth rational curve through $W$ called a Hecke curve. It
was shown in [13] that the Hecke curves have minimal degree among the rational
curves passing through a generic $W\in{\mathcal{S}U}_{C}(n,d)$.

Note that the parameter space of Hecke curves passing through
$W\in{\mathcal{S}U}_{C}(n,d)$ is given by a double fibration

${\mathbb{P}}(T_{\pi}^{*})\to{\mathbb{P}}(W^{*})\stackrel{{\scriptstyle\pi}}{{\to}}C,$

where $T_{\pi}$ is the vertical tangent bundle of
$\pi\colon{\mathbb{P}}(W^{*})\to C$. In the previous notation, this
corresponds to the composition map

$(U/\ell_{0}\subset W_{x}^{\theta}/\ell_{0})\mapsto(\theta\subset
W_{x}^{*})\mapsto x.$

In particular for $n=2$, this double fibration boils down to the ruled surface
${\mathbb{P}}(W^{*})$.

###  2.3. Kodaira–Spencer map

The main references of this subsection are [11] and [12]. Consider the above
family $\\{V^{\ell}:[\ell]\in\mathbb{P}(U)\\}$ as a deformation of
$V^{\ell_{0}}\cong W^{*}$. Since the map $\beta_{x}\colon V_{x}^{\ell_{0}}\to
V_{x}$ surjects onto $\ell_{0}^{\perp}$, we have the induced pull-back map

$\widehat{\beta_{x}}\colon\operatorname{Hom}(\ell_{0}^{\perp}/U^{\perp},V^{\prime})\to\operatorname{Hom}(V_{x}^{\ell_{0}},V^{\prime})$

for any vector space $V^{\prime}$.

######  Proposition 2.2.

The Kodaira–Spencer map of the family $\\{V^{\ell}:[\ell]\in\mathbb{P}(U)\\}$
is given by

$\displaystyle\operatorname{Hom}(\ell_{0}^{\perp}/U^{\perp},V_{x}/\ell_{0}^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\longrightarrow}}\operatorname{Hom}(V_{x}^{\ell_{0}},V_{x}/\ell_{0}^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\ell_{0}})),$

where $\delta$ is induced from the sequence (1).

Later we will also need to consider a slight generalization of the above
family, where the Hecke modification is taken for subspaces of dimension two.
For a subspace $\Theta\subset W_{x}^{*}$ of dimension 2, the Hecke
modification $W^{\Theta}$ of $W$ along $\Theta$ can be put into the following
exact sequence:

$0\rightarrow W^{\Theta}\rightarrow
W\rightarrow(W_{x}/\Theta^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

Let $\wp_{0}$ denote the kernel of $W_{x}^{\Theta}\rightarrow W_{x}$. The
following is a straightforward generalization of Proposition 2.2.

######  Proposition 2.3.

Let $U$ be a subspace of dimension 4 with $\wp_{0}\subset U\subset
W_{x}^{\Theta}$; that is,
$U^{\perp}\subset\wp_{0}^{\perp}\subset(W_{x}^{\Theta})^{*}=:V_{x}$. Then the
family $\\{V^{\wp}:[\wp]\in Gr(2,V_{x}/U^{\perp})\\}$ is a deformation of
$W^{*}=V^{\wp_{0}}$, and its Kodaira–Spencer map

$T_{\wp_{0}}(Gr(2,V_{x}/U^{\perp}))=\operatorname{Hom}(\wp_{0}^{\perp}/U^{\perp},V_{x}/\wp_{0}^{\perp})\rightarrow
H^{1}(C,\operatorname{End}(V^{\wp_{0}}))$

is given by the composition

$\displaystyle\operatorname{Hom}(\wp_{0}^{\perp}/U^{\perp},V_{x}/\wp_{0}^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\rightarrow}}\operatorname{Hom}(V_{x}^{\wp_{0}},V_{x}/\wp_{0}^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\wp_{0}})),$

where $\widehat{\beta_{x}}$ and $\delta$ are induced from

$0\rightarrow
V^{\wp_{0}}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\wp_{0}^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

###  2.4. Symplectic Hecke curves on $\mathcal{M}S_{C}(n,L)$

In this subsection, we recall the construction of symplectic Hecke curves,
following [3], to which we refer the reader for the details.

For a line bundle $L$ on $C$, an $L$-valued symplectic bundle of rank $n$ is a
vector bundle $W$ of (even) rank $n$ equipped with an $L$-valued symplectic
form $\omega\colon W\otimes W\rightarrow L.$ Let $\mathcal{M}S_{C}(n,L)$ be
the moduli space of $L$-valued symplectic bundles of rank $n$. By the morphism
forgetting the symplectic forms, the moduli space $\mathcal{M}S_{C}(n,L)$ can
be thought of as a subvariety of ${\mathcal{S}U}_{C}(n,\frac{1}{2}n\ell)$,
where $\ell=\deg(L)$. To avoid the coincidence
$\mathcal{M}S_{C}(2,L)={\mathcal{S}U}_{C}(2,\ell)$, we assume $n\geq 4$
throughout this paper.

The construction of symplectic Hecke curves on $\mathcal{M}S_{C}(n,L)$ closely
follows the previous construction of Hecke curves on
${\mathcal{S}U}_{C}(n,d)$, keeping track of the deformation of symplectic
forms. For a subspace $\theta\subset W_{x}^{*}$ of dimension one, let
$W^{\theta}$ be the Hecke modification of $W$ along $\theta$, fitting into the
sequence

$0\rightarrow W^{\theta}\rightarrow
W\rightarrow(W_{x}/\theta^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

Noting that every 1-dimensional subspace $\theta$ is isotropic, we get an
induced $L^{*}(x)$-valued skew-symmetric form on $V:=(W^{\theta})^{*}$:

$\omega^{\theta}\colon(W^{\theta})^{*}\rightarrow W^{\theta}\otimes L^{*}(x).$

Then $\ker\omega^{\theta}_{x}$ has codimension two in
$V_{x}=(W^{\theta})^{*}_{x}$.

For a subspace $\ell\subset W^{\theta}_{x}$ of dimension one, we have the
sequence

$0\rightarrow V^{\ell}\rightarrow
V\rightarrow(V_{x}/\ell^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

Then the bundle $V^{\ell}$ has a skew-symmetric form induced from
$\omega^{\theta}$, and it is an $L^{*}$-valued nondegenerate (symplectic) form
if and only if $\ker\omega^{\theta}_{x}\subset\ell^{\perp}$. Now the family

$\\{V^{\ell}\>:\>\ker\omega^{\theta}_{x}\subset\ell^{\perp}\subset V_{x}\\}$

of $L$-valued symplectic bundles are parameterized by
$\mathbb{P}(V_{x}/\ker\omega^{\theta}_{x})\simeq\mathbb{P}^{1}$. In
particular, if $\ell_{0}$ is in the kernel of $W_{x}^{\theta}\rightarrow
W_{x}$, then $V^{\ell_{0}}\cong W^{*}$.

Under the assumption that $g\geq 3$ and $W\in\mathcal{M}S_{C}(n,L)$ is a
generic point, by [3, Lemma 4.5] the dual family

$\\{(V^{\ell})^{*}\>:\>\ell\in\mathbb{P}(V_{x}/\ker\omega^{\theta}_{x})\\}$

gives a rational curve on $\mathcal{M}S_{C}(n,L)$ passing through $W$, called
a symplectic Hecke curve. Also it was shown in [3, Theorem 5.2] that these
curves have minimal degree among the rational curves passing through a generic
point $W\in\mathcal{M}S_{C}(n,L)$.

Later we need the following fact.

######  Proposition 2.4.

Assume $g\geq 3$. For a generic point $W\in\mathcal{M}S_{C}(n,L)$, every
symplectic Hecke curve passing through $W$ is contained in the smooth locus of
$\mathcal{M}S_{C}(n,L)$.

###### Proof.

By [3, Lemma 4.5], any symplectic Hecke curve passing through a generic point
$W$ stays inside the locus of stable symplectic bundles. To see that it is
contained in the smooth locus, it suffices to show that it does not touch the
locus of non-regularly stable symplectic bundles which are of the form
$W_{1}\perp W_{2}$ for some stable symplectic subbundles $W_{1}$ and $W_{2}$.

This can be checked by dimension count: The locus of non-regularly stable
bundles is contained in a finite union of the images of
$\mathcal{M}S_{C}(n_{1},L)\times\mathcal{M}S_{C}(n_{2},L)$, where
$n_{1}+n_{2}=n$. Since symplectic Hecke curves passing through a point $W_{0}$
in this subvariety are parameterized by $\mathbb{P}(W_{0}^{*})$, it suffices
to check the inequality:

$\operatorname{dim}\mathbb{P}(W_{0}^{*})+\operatorname{dim}\mathcal{M}S_{C}(n_{1},L)+\operatorname{dim}\mathcal{M}S_{C}(n_{2},L)\
<\ \operatorname{dim}\mathcal{M}S_{C}(n,L)$

for any even integers $n_{1},n_{2}$ with $n_{1}+n_{2}=n$. This boils down to
$n<n_{1}n_{2}(g-1)$, which holds for $g\geq 3$. ∎

###  2.5. Orthogonal Hecke curves on $\mathcal{M}O_{C}(n,L)$

Again, the main reference in this subsection for a construction of orthogonal
Hecke curves will be [3].

For a line bundle $L$ on $C$, an $L$-valued orthogonal bundle of rank $n$ is a
vector bundle $W$ of rank $n$ equipped with an $L$-valued orthogonal form
$b\colon W\otimes W\rightarrow L.$ Let $\mathcal{M}O_{C}(n,L)$ be the moduli
space of $L$-valued orthogonal bundles of rank $n$. The moduli space
$\mathcal{M}O_{C}(2r,L)$ has several irreducible components, due to the
invariants $\det(W)$ and the 2nd Stiefel–Whitney class $w_{2}(W)$ (see [2,
§2]). By the morphism forgetting the orthogonal forms, each irreducible
component of the moduli space $\mathcal{M}O_{C}(n,L)$ is sent to a subvariety
of ${\mathcal{S}U}_{C}(n,\frac{1}{2}n\ell)$, where $\ell=\deg(L)$.

As in [3], we assume $n\geq 5$ throughout this paper. The reason behind this
convention is that the moduli space $\mathcal{M}O_{C}(n,L)$ has Picard number
one for $n\geq 5$, while $\mathcal{M}O_{C}(4,L)$ has Picard number two.
Accordingly, the minimality of the orthogonal Hecke curves was discussed in
[3] for $n\geq 5$. We remark that there is a standard construction of
orthogonal bundles of low rank from vector bundles, described in [5].

The construction of orthogonal Hecke curves on $\mathcal{M}S_{C}(n,L)$ is a
little bit different from that of Hecke curves on ${\mathcal{S}U}_{C}(n,d)$:
the dimension of the involved subspaces are doubled.

For an isotropic subspace $\Theta\subset W_{x}^{*}$ of dimension two, let
$W^{\Theta}$ be the Hecke modification:

$0\rightarrow W^{\Theta}\rightarrow
W\rightarrow(W_{x}/\Theta^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

Then there is an $L^{*}(x)$-valued symmetric form

$b^{\Theta}\colon(W^{\Theta})^{*}\rightarrow W^{\Theta}\otimes L^{*}(x)$

on $V:=(W^{\Theta})^{*}$ such that $\ker b^{\Theta}_{x}$ has codimension four
in $V_{x}=(W^{\Theta})^{*}_{x}$.

For an isotropic subspace $\wp\subset W^{\Theta}_{x}$ of dimension two with
$\wp^{\perp}\subset V_{x}$, we have the Hecke modification

$0\rightarrow V^{\wp}\rightarrow
V\rightarrow(V_{x}/\wp^{\perp})\otimes\mathcal{O}_{x}\rightarrow 0.$

Then $V^{\wp}$ is equipped with a symmetric form induced from $b^{\Theta}$,
and it is an $L^{*}$-valued nondegenerate (orthogonal) form if and only if
$\ker b^{\Theta}_{x}\subset\wp^{\perp}$. In particular when $\wp_{0}$ is in
the kernel of $W_{x}^{\Theta}\rightarrow W_{x}$, we have $V^{\wp_{0}}\cong
W^{*}$.

Note that the space of two-dimensional isotropic subspaces $\wp\subset
W_{x}^{\Theta}$ such that

$\ker b^{\Theta}_{x}\ \subset\ \wp^{\perp}\ \subset\ V_{x}\ =\
(W^{\theta})^{*}_{x}$

is the isotropic Grassmannian of 2-dimensional subspaces of $V_{x}/\ker
b_{x}^{\Theta}\cong\mathbb{C}^{4}$, which is a disjoint union of two
projective lines. Let $\mathrm{IG}(2,V_{x}/\ker b_{x}^{\Theta})$ be the line
containing the point $\wp_{0}^{\perp}/\ker b_{x}^{\Theta}$. Then the family

$\\{V^{\wp}\>:\>\wp\in\mathrm{IG}(2,V_{x}/\ker b_{x}^{\Theta})\\}$

of $L^{*}$-valued orthogonal bundles gives a deformation of $W^{*}$.

Under the assumption that $g\geq 5,n\geq 5$ and that
$W\in\mathcal{M}O_{C}(n,L)$ is a generic point, by [3, Lemma 4.7] the dual
family $\\{(V^{\wp})^{*}\\}$ gives a rational curve on $\mathcal{M}O_{C}(n,L)$
passing through $W$, called an orthogonal Hecke curve. Also it was shown in
[3, theorem 5.3] that these curves have minimal degree among the rational
curves passing through a generic point $W\in\mathcal{M}O_{C}(n,L)$.

Again, we show the following.

######  Proposition 2.5.

Assume $g\geq 5$ and $n\geq 5$. For a generic point
$W\in\mathcal{M}O_{C}(n,L)$, every orthogonal Hecke curve passing through $W$
is contained in the smooth locus of $\mathcal{M}O_{C}(n,L)$.

###### Proof.

By [3, Lemma 4.7], any orthogonal Hecke curve passing through a generic point
$W$ stays inside the locus of stable orthogonal bundles. To see that it is
contained in the smooth locus, it suffices to show that it does not touch the
locus of non-regularly stable orthogonal bundles which are of the form
$W_{1}\perp W_{2}$ for some stable orthogonal subbundles $W_{1}$ and $W_{2}$.

This can be checked by dimension count: The locus of non-regularly stable
bundles is contained in a finite union of the images of
$\mathcal{M}O_{C}(n_{1},L)\times\mathcal{M}O_{C}(n_{2},L)$, where
$n_{1}+n_{2}=n$. Since orthogonal Hecke curves passing through a point
$[W_{0}]$ in this subvariety are parameterized by $\mathrm{IG}(2,W_{0}^{*})$,
it suffices to check the inequality:

$\operatorname{dim}\mathrm{IG}(2,W_{0}^{*})+\operatorname{dim}\mathcal{M}O_{C}(n_{1},L)+\operatorname{dim}\mathcal{M}O_{C}(n_{2},L)\
<\ \operatorname{dim}\mathcal{M}O_{C}(n,L)$

for any integers $n_{1},n_{2}$ with $n_{1}+n_{2}=n$. Since
$\operatorname{dim}IG(2,W_{0}^{*})=2n-6$, this boils down to
$2n-6<n_{1}n_{2}(g-1)$, which holds for $g,n\geq 5$. ∎

###  2.6. Minimal rational curves

Let $M$ be a projective variety. Let $\mathcal{K}$ be an irreducible component
of the Hilbert scheme of complete curves on $M$ such that generic members of
$\mathcal{K}$ cover an open subset of the smooth locus $M^{\circ}$ of $M$. For
a generic point $x\in M$, denote by $\mathcal{K}_{x}$ the subscheme of
$\mathcal{K}$ consisting of members of $\mathcal{K}$ passing through $x$.
Assume that for a generic point $x\in M$, every member of $\mathcal{K}_{x}$ is
an irreducible smooth rational curve contained in $M^{\circ}$ and
$\mathcal{K}_{x}$ is an irreducible complete variety. In this case, we call
$\mathcal{K}$ a minimal rational component of $M$.

A covering family of rational curves having minimal degree gives a minimal
rational component. More precisely, an irreducible component $\mathcal{K}$ is
a minimal rational component of $M$ if it satisfies the following conditions:

  1. (i)

For a generic $x\in M$, every member of $\mathcal{K}_{x}$ is irreducible
smooth rational curve contained in $M^{\circ}$.

  2. (ii)

The locus swept out by the curves in $\mathcal{K}$ is dense in $M$.

  3. (iii)

For a fixed ample line bundle $\xi$ on $M$, the degree of members of
$\mathcal{K}$ with respect to $\xi$ is minimal among the curves in an
irreducible family satisfying (1) and (2).

Let $x\in M$ be a generic point. Define the tangent map
$\tau_{x}\colon\mathcal{K}_{x}\dashrightarrow\mathbb{P}(T_{x}(M))$ by

$\tau_{x}([R])=[T_{x}R]\in\mathbb{P}(T_{x}(M)),$

where $R$ is a smooth rational curve in $M^{\circ}$ passing through $x$. The
closure $\mathcal{C}_{x}$ of the image
${\tau_{x}(\mathcal{K}_{x})}\subset\mathbb{P}(T_{x}(M))$ is called the variety
of minimal rational tangents (VMRT for short) at $x$ associated with
$\mathcal{K}$.

The following is [8, Theorem 2], which connects the theory of VMRT and the
simplicity of the tangent bundle.

######  Proposition 2.6.

Let $M$ be a Fano variety which has a minimal rational component
$\mathcal{K}$. If the VMRT $\mathcal{C}_{x}$ at a generic point $x\in M$ is
non-degenerate in $\mathbb{P}T_{x}M$, then the tangent bundle $T(M^{\circ})$
is simple.

For the moduli space of vector bundles $\mathcal{M}={\mathcal{S}U}_{C}(n,d)$,
it is proven in [8] that for $g\geq 4$, the irreducible component
$\mathcal{K}$ of the Hilbert scheme of $\mathcal{M}$ containing Hecke curves
is a minimal rational component of $\mathcal{M}$. In this case, given a
generic point $W\in\mathcal{M}$, $\mathcal{K}_{W}$ is given by

$\mathcal{K}_{W}=\bigcup_{[\theta]\in\mathbb{P}(W^{*})}\mathbb{P}(W_{x}^{\theta}/\ell_{0})\
\simeq\ \mathbb{P}(T_{\pi}^{*})$

where $T_{\pi}$ is the vertical tangent bundle of
$\pi\colon\mathbb{P}(W^{*})\rightarrow C$. In particular for $n=2$, we have
$\mathcal{K}_{W}=\mathbb{P}(W^{*})$.

Moreover it is shown in [9, Theorem 3.1, Theorem 3.7] that the tangent map at
a generic point $W\in\mathcal{M}$ is biregular to the image for $g\geq 5$ and
birational for $g=4$.

Finally we discuss the case of $\mathcal{M}S_{C}(n,L)$ and
$\mathcal{M}O_{C}(n,L)$. By the result [3, Theorem 5.2] on the minimality of
degree, we can see that there is a minimal rational component $\mathcal{K}$ of
$\mathcal{M}S_{C}(n,L)$ containing symplectic Hecke curves such that
$\mathcal{K}_{W}$ for a generic element $W$ is given by

$\displaystyle\mathcal{K}_{W}$ $\displaystyle=$
$\displaystyle\mathbb{P}(W^{*}).$

Similarly by [3, Theorem 5.3], there is a minimal rational component
$\mathcal{K}$ of $\mathcal{M}O_{C}(n,L)$ containing orthogonal Hecke curves
such that $\mathcal{K}_{[W]}$ for a generic element $[W]$ is given by

$\mathcal{K}_{W}=IG(2,W^{*}).$

##  3\. Smoothness of Hecke curves

In this section, we show the smoothness of the symplectic and orthogonal Hecke
curves.

######  Proposition 3.1.

Assume $g\geq 4$ and $n\geq 4$. Then any symplectic Hecke curve passing
through a generic point $W\in\mathcal{M}S_{C}(n,L)$ is smooth.

###### Proof.

From the construction, the family

(2) $\\{V^{\ell}\>:\>\ker\omega^{\theta}_{x}\subset\ell^{\perp}\subset
V_{x}\\}$

gives a deformation of $V^{\ell}$ along a subspace
$U:=(\ker\omega_{x}^{\theta})^{\perp}\cong\mathbb{C}^{2}$. By Proposition 2.3,
the Kodaira–Spencer map

(3)
$T_{\ell}(\mathbb{P}(V_{x}/U^{\perp}))=\operatorname{Hom}(\ell^{\perp}/U^{\perp},V_{x}/\ell^{\perp})\rightarrow
H^{1}(C,\operatorname{End}(V^{\ell}))$

associated to this family is given by the composition

(4)
$\displaystyle\operatorname{Hom}(\ell^{\perp}/U^{\perp},V_{x}/\ell^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\rightarrow}}\operatorname{Hom}(V_{x}^{\ell},V_{x}/\ell^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\ell})),$

where $\widehat{\beta_{x}}$ and $\delta$ are induced from

(5) $0\rightarrow
V^{\ell}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\ell^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

Note that this can be geometrically understood as the composition map

$T_{\ell}(\mathbb{P}(V_{x}/U^{\perp}))\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\longrightarrow}}T_{[\beta]}\mathrm{Quot}(V)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H^{1}(C,\operatorname{ad}(V^{\ell}))=T_{[V^{\ell}]}\mathcal{M}S_{C}(n,L^{*}),$

where the tangent space $T_{[\beta]}\mathrm{Quot}(V)$ of the Quot scheme of
$V$ is given by

$T_{[\beta]}\mathrm{Quot}(V)=H^{0}(C,\mathrm{Hom}(V^{\ell},(V_{x}/\ell^{\perp})\otimes\mathcal{O}_{x}))\cong\mathrm{Hom}(V_{x}^{\ell},V_{x}/\ell^{\perp}).$

Hence to show that the map

$\phi_{U}\colon\mathbb{P}(U)\cong\mathbb{P}^{1}\to\mathcal{M}S_{C}(n,L)$

which gives the symplectic Hecke curve (2) is an immersion, we need to show
that the map (3) is injective.

Since $\widehat{\beta_{x}}$ is injective by definition of Quot schemes, we
need to check that $\delta$ is injective. The map $\delta$ fits into the long
exact sequence associated to (5) tensored by $(V^{\ell})^{*}$:

$0\to H^{0}(C,\mathrm{End}(V^{\ell}))\to H^{0}(C,(V^{\ell})^{*}\otimes
V)\to\mathrm{Hom}(V^{\ell},(V_{x}/\ell^{\perp}))\stackrel{{\scriptstyle\delta}}{{\to}}H^{1}(C,\mathrm{End}(V^{\ell}))$

Hence $\delta$ is injective everywhere if we know:

  * •

$\operatorname{dim}H^{0}(C,\mathrm{End}(V^{\ell}))=1$ for all $\ell$ and

  * •

$\operatorname{dim}H^{0}(C,(V^{\ell})^{*}\otimes V)=1$ for all $\ell$.

The first condition holds if $V^{\ell}$ is regularly stable. By [3, Lemma
4.2], the second condition holds if every point $V$ is a generic point and
$g\geq 3$.

Now it remains to show that the map $\phi_{U}$ is injective. It was shown in
[3, Lemma 4.5] that $\phi_{U}$ is generically injective if $g\geq 3$ and
$n\geq 4$. Its proof can be slightly modified to show the injectiveness (under
a stronger bound on $g$ and $n$). The point of the proof was to choose
$W=(V^{\ell_{0}})^{*}$ as a “1-stable” symplectic bundle (see [3, §4.1]). By
the same argument, if we choose $W$ to be 2-stable, then every bundle
$V^{\ell}$ can be shown to be 1-stable, and hence $V^{\ell_{1}}\cong
V^{\ell_{2}}$ implies $\ell_{1}=\ell_{2}$. By [3, Lemma 4.1] a generic point
of $\mathcal{M}S_{C}(n,L)$ is 2-stable for $g\geq 4,n\geq 4$ and we are done.
∎

######  Proposition 3.2.

Assume $g\geq 5$ and $n\geq 5$. Then any orthogonal Hecke curve passing
through a generic point $W\in\mathcal{M}O_{C}(n,L)$ is smooth.

###### Proof.

From the construction of orthogonal Hecke curves, the family

(6)
$\\{V^{\wp}\>:\>\operatorname{Ker}\omega^{\Theta}_{x}\subset\wp^{\perp}\subset
V_{x}\\}$

gives a deformation of $V^{\wp}$ along a subspace
$U:=(\operatorname{Ker}\omega_{x}^{\Theta})^{\perp}\cong\mathbb{C}^{4}$. By
Proposition 2.3, the Kodaira–Spencer map

(7)
$T_{\wp}Gr(2,V_{x}/U^{\perp})=\operatorname{Hom}(\wp^{\perp}/U^{\perp},V_{x}/\wp^{\perp})\rightarrow
H^{1}(C,\operatorname{End}(V^{\wp}))$

associated to this family is given by the composition

(8)
$\displaystyle\operatorname{Hom}(\wp^{\perp}/U^{\perp},V_{x}/\wp^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\rightarrow}}\operatorname{Hom}(V_{x}^{\wp},V_{x}/\wp^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\wp})),$

where $\widehat{\beta_{x}}$ and $\delta$ are induced from

(9) $0\rightarrow
V^{\wp}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\wp^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

Note that this can be geometrically understood as the composition map

$T_{\ell}Gr(2,V_{x}/U^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\longrightarrow}}T_{[\beta]}\mathrm{Quot}(V)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H^{1}(C,\operatorname{ad}(V^{\wp}))=T_{[V^{\wp}]}\mathcal{M}O_{C}(n,L^{*}),$

where the tangent space $T_{[\beta]}\mathrm{Quot}(V)$ of the Quot scheme of
$V$ is given by

$T_{[\beta]}\mathrm{Quot}(V)=H^{0}(C,\mathrm{Hom}(V^{\wp},(V_{x}/\wp^{\perp})\otimes\mathcal{O}_{x}))\cong\mathrm{Hom}(V_{x}^{\wp},V_{x}/\wp^{\perp}).$

Hence to show the immersedness of the orthogonal Hecke curve (6), it suffices
to show that the map (7) is injective.

Since $\widehat{\beta_{x}}$ is injective by definition of Quot schemes, we
need to check that $\delta$ is injective. The map $\delta$ fits into the long
exact sequence associated to (9) tensored by $(V^{\wp})^{*}$:

$0\to H^{0}(C,\mathrm{End}(V^{\wp}))\to H^{0}(C,(V^{\wp})^{*}\otimes
V)\to\mathrm{Hom}(V^{\wp},(V_{x}/\wp^{\perp}))\stackrel{{\scriptstyle\delta}}{{\to}}H^{1}(C,\mathrm{End}(V^{\wp}))$

Hence $\delta$ is injective everywhere if we know:

  * •

$\operatorname{dim}H^{0}(C,\mathrm{End}(V^{\wp}))=1$ for all $\wp$ and

  * •

$\operatorname{dim}H^{0}(C,(V^{\wp})^{*}\otimes V)=1$ for all $\wp$.

The first condition holds if $V^{\wp}$ is regularly stable. The second
condition holds if $V$ is general and $g\geq 3$ by [3, Lemma 4.2].

Now to show that the injectiveness. as in the symplectic case, it suffices to
choose $W$ to be 3-stable in order that every bundle $V^{\wp}$ is 2-stable. By
[3, Lemma 4.1] a generic point of $\mathcal{M}O_{C}(n,L)$ is 3-stable for
$g\geq 5,n\geq 5$ and we are done. ∎

##  4\. Nondegeneracy of the tangent map

In this section, we discuss the tangent map of the variety of minimal rational
tangents for the moduli spaces $\mathcal{M}S_{C}(2r,L)$ and
$\mathcal{M}O_{C}(n,L)$. In particular, we study the complete linear system
which defines the tangent map. This confirms that the image of the tangent map
is nondegenerate, and as a consequence we get the simpleness of the tangent
bundle of the moduli space. Basically we follow the computations in [7] of the
Kodaira–Spencer map of the Hecke curves on the moduli space
$\mathcal{{\mathcal{S}U}}_{C}(2,d)$.

###  4.1. Symplectic bundles

The tangent space of ${\mathcal{S}U}_{C}(n,d)$ at a stable bundle $W$ is given
by $H^{1}(C,\mathrm{End}_{0}(W))$, where $\mathrm{End}_{0}(W)$ is the vector
bundle of traceless endomorphisms of $W$. By a similar argument as [1, Lemma
2.2], the tangent space of $\mathcal{M}S_{C}(n,L)$ at a regularly stable
symplectic bundle $W$ is given by $H^{1}(C,L\otimes\mathrm{Sym}^{2}W^{*})$. In
this context, we put $\operatorname{ad}(W)=L\otimes\mathrm{Sym}^{2}W^{*}$.

######  Proposition 4.1.

Assume $g\geq 4$ and $n\geq 4$ as in Proposition 3.1. Let $\mathcal{K}$ be the
minimal rational component consisting of symplectic Hecke curves on
$\mathcal{M}S_{C}(2r,L)$. Then for a generic $W\in\mathcal{M}S_{C}(2r,L)$, the
tangent map

$\tau_{W}\colon\mathcal{K}_{W}=\mathbb{P}W^{*}\to\mathbb{P}T_{W}\mathcal{M}S_{C}(n,L)=\mathbb{P}H^{1}(C,\mathrm{Sym}^{2}W^{*}\otimes
L)$

is the composition $\Phi_{W}\circ\iota$, where $\Phi_{W}$ is given by the
complete linear system $|\mathcal{O}(1)\otimes\pi^{*}K_{C}|$. Also,
$\iota(\theta)=\theta\otimes(\omega(\theta,\,\cdot\,))$ for $\theta\in
W_{x}^{*}$, and the image of $\iota(\theta)$ in
$\mathbb{P}(H^{1}(C,\mathrm{Sym}^{2}W^{*}\otimes L))$ is given by the linear
functional $H^{0}(C,\mathrm{Sym}^{2}W\otimes L^{*}\otimes K_{C})\rightarrow
K_{C,x}$ taking the trace of endomorphisms of $W_{x}$.

######  Remark 4.2.

The upper arrow $\Psi_{W}$ is the natural mapping of the ruled variety
${\mathbb{P}}(\mathrm{End}_{0}(W))$, which is not necessarily defined
everywhere at this stage, but it will turn out in §5 to be a morphism and
furthermore an embedding under certain genus assumption. On the other hand,
the lower arrow $\Phi_{W}$ is a morphism by Proposition 3.1.

###### Proof.

Recall that $\ker\omega^{\theta}$ is a subspace of $V_{x}$ of codimension two,
and after we put $U^{\perp}=\ker\omega^{\theta}_{x}\subset V_{x}$, the family
$\\{V^{\ell}:[\ell]\in\mathbb{P}(V_{x}/U^{\perp})\\}$ is a deformation of
$W^{*}=V^{\ell_{0}}$. Applying Proposition 2.2 to the kernel of
$\omega^{\theta}$, we get that the Kodaira-Spencer map

$T_{\ell_{0}}(\mathbb{P}(V_{x}/U^{\perp}))=\operatorname{Hom}(\ell_{0}^{\perp}/U^{\perp},V_{x}/\ell_{0}^{\perp})\rightarrow
H^{1}(C,\operatorname{End}(V^{\ell_{0}}))$

for the family $\\{V^{\ell}:[\ell]\in\mathbb{P}(V_{x}/U^{\perp})\\}$, is given
by the composition

$\displaystyle\operatorname{Hom}(\ell_{0}^{\perp}/U^{\perp},V_{x}/\ell_{0}^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\rightarrow}}\operatorname{Hom}(V_{x}^{\ell_{0}},V_{x}/\ell_{0}^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\ell_{0}})),$

where $\widehat{\beta_{x}}$ and $\delta$ are induced from

$0\rightarrow
V^{\ell_{0}}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\ell_{0}^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

Furthermore, as in the case of the moduli space of vector bundles discussed in
[7], the element $\delta(\widehat{\beta}_{x}(v))$ in
$H^{1}(C,\mathrm{End}(V^{\ell_{0}}))$ for
$v\in\operatorname{Hom}(\ell_{0}^{\perp}/U^{\perp},V_{x}/\ell_{0}^{\perp})$,
is represented by the cocycle $\left\\{\dfrac{e_{2}^{*}\otimes e_{1}}{z}\text{
on }\mathcal{U}_{0}\cap\mathcal{U}_{j}\right\\}$. Here,
$\\{\mathcal{U}_{0},\mathcal{U}_{1},\dots,\mathcal{U}_{N}\\}$ is a coordinate
covering of $C$ such that

  * •

all the involved vector bundles are trivial on $\mathcal{U}_{0}$ and
$\mathcal{U}_{j}$ for $1\leq j\leq N$,

  * •

$x\in\mathcal{U}_{0}$ and $x\not\in\mathcal{U}_{j}$ for $1\leq j\leq N$ so
that on each $\mathcal{U}_{j}$, we identify $V^{\ell_{0}}$ and $V$ via
$V^{\ell_{0}}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V$,

  * •

$z$ is a coordinate on $\mathcal{U}_{0}$ centered at $x$,

  * •

$\\{e_{1},e_{2},\dots,e_{n}\\}$ and $\\{f_{1},f_{2},\dots,f_{n}\\}$ are frames
of $V^{\ell_{0}}|_{\mathcal{U}_{0}}$ and $V|_{\mathcal{U}_{0}}$, respectively
such that $e_{1,x}\in\theta\subset V_{x}^{\ell_{0}}=W^{*}_{x}$,
$f_{2,x}\in\ell_{0}^{\perp}/U^{\perp}\subset
V_{x}/U^{\perp}=(W_{x}^{\theta})^{*}/U^{\perp}$ and

  * •

$\beta$ sends $e_{1},e_{2},\dots,e_{n}$ to $zf_{1},f_{2},\dots,f_{n}$.

Thus $\delta(\widehat{\beta}_{x}(v))$ corresponds to the image of
$\iota(\theta)=\theta\otimes\omega(\theta,\,\cdot\,)$ via the duality
$H^{1}(C,\mathrm{End}_{0}(W))\simeq
H^{0}(C,K_{C}\otimes\mathrm{End}_{0}(W))^{*}$ induced by the residue pairing.
∎

###  4.2. Orthogonal bundles

By a similar argument as [1, Lemma 2.2], the tangent space of
$\mathcal{M}O_{C}(n,L)$ at a regularly stable orthogonal bundle $W$ is given
by $H^{1}(C,L\otimes\wedge^{2}W^{*})$. In this context, we put
$\operatorname{ad}(W)=L\otimes\wedge^{2}W^{*}$.

######  Proposition 4.3.

Assume $g\geq 5$ and $n\geq 5$ as in Proposition 3.2. Let $\mathcal{K}$ be the
minimal rational component consisting of orthogonal Hecke curves on
$\mathcal{M}O_{C}(n,L)$. Then for a generic $W\in\mathcal{M}O_{C}(n,L)$, the
tangent map $\tau_{W}$

$\tau_{W}\colon\mathcal{K}_{W}=IG(2,W^{*})\to\mathbb{P}T_{W}\mathcal{M}O_{C}(n,L)=\mathbb{P}H^{1}(C,\wedge^{2}W^{*}\otimes
L)$

is the composition $\Phi_{W}\circ\iota$, where $\Phi_{W}$ is given by the
complete linear system $|\mathcal{O}(1)\otimes\pi^{*}K_{C}|$. Also,
$\iota(v_{1}\wedge v_{2})=b(v_{1},\,\cdot\,)\otimes
v_{2}-b(v_{2},\,\cdot\,)\otimes v_{1}$ for $[v_{1}\wedge v_{2}]\in
IG(2,W^{*}_{x})$ and the image of $\iota(v_{1}\wedge v_{2})$ in
$\mathbb{P}(H^{1}(C,\wedge^{2}W^{*}\otimes L))$ is given by the linear
functional $H^{0}(C,\wedge^{2}W^{*}\otimes L^{*}\otimes K_{C})\rightarrow
K_{C,x}$ taking the trace of endomorphisms of $W_{x}$.

######  Remark 4.4.

As before, the upper arrow $\Psi_{W}$ is not necessarily defined everywhere at
this stage, but it will turn out in §5 to be a morphism and furthermore an
embedding under certain genus assumption. On the other hand, the lower arrow
$\Phi_{W}$ is a morphism by Proposition 3.2.

###### Proof.

Recall that the kernel of $b^{\Theta}$ is a subspace of codimension 4, and if
we put $U^{\perp}=\operatorname{Ker}b^{\Theta}_{x}\subset
V_{x}=(W_{x}^{\Theta})^{*}$, then the family $\\{V^{\wp}:[\wp]\in
IG(2,V_{x}/U^{\perp})\\}$ is a deformation of $W^{*}=V^{\wp_{0}}$. As in the
proof of Proposition 3.2, we apply Proposition 2.3 to the kernel of
$b^{\Theta}$ and we get that the Kodaira-Spencer map

$T_{\wp_{0}}(Gr(2,V_{x}/U^{\perp}))=\operatorname{Hom}(\wp_{0}^{\perp}/U^{\perp},V_{x}/\wp_{0}^{\perp})\rightarrow
H^{1}(C,\operatorname{End}(V^{\wp_{0}}))$

for the family $\\{V^{\wp}:[\wp]\in Gr(2,V_{x}/U^{\perp})\\}$ is given by the
composition

$\displaystyle\operatorname{Hom}(\wp_{0}^{\perp}/U^{\perp},V_{x}/\wp_{0}^{\perp})\stackrel{{\scriptstyle\widehat{\beta_{x}}}}{{\rightarrow}}\operatorname{Hom}(V_{x}^{\wp_{0}},V_{x}/\wp_{0}^{\perp})\stackrel{{\scriptstyle\delta}}{{\rightarrow}}H^{1}(C,\mathrm{End}(V^{\wp_{0}})),$

where $\widehat{\beta_{x}}$ and $\delta$ are induced from

$0\rightarrow
V^{\wp_{0}}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V\rightarrow(V_{x}/\wp_{0}^{\perp})\otimes\mathcal{O}_{x}\rightarrow
0.$

Choose a coordinate covering
$\\{\mathcal{U}_{0},\mathcal{U}_{1},\dots,\mathcal{U}_{N}\\}$ of $C$ such that

  * •

all the involved vector bundles are trivial on $\mathcal{U}_{0}$ and
$\mathcal{U}_{j}$ for $1\leq j\leq N$,

  * •

$x\in\mathcal{U}_{0}$ and $x\not\in\mathcal{U}_{j}$ for $1\leq j\leq N$ so
that on each $\mathcal{U}_{j}$, we identify $V^{\wp_{0}}=W^{*}$ and
$V=(W^{\Theta})^{*}$ by
$V^{\wp_{0}}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}V$,

  * •

$z$ is a coordinate on $\mathcal{U}_{0}$ centered at $x$,

  * •

$\\{e_{1},e_{2},\dots,e_{n}\\}$ and $\\{f_{1},f_{2},\dots,f_{n}\\}$ are frames
of $V^{\wp_{0}}|_{\mathcal{U}_{0}}$ and $V|_{\mathcal{U}_{0}}$ respectively
such that $e_{1,x},e_{2,x}\in\Theta\subset V_{x}^{\wp_{0}}=W^{*}_{x}$,
$f_{3,x},f_{4,x}\in\wp_{0}^{\perp}/U^{\perp}\subset
V_{x}/U^{\perp}=(W_{x}^{\Theta})^{*}/U^{\perp}$ and

  * •

$\beta$ sends $e_{1},e_{2},e_{3},e_{4}$ to $zf_{1},zf_{2},f_{3},f_{4}$.

Then the tangent space $T_{\wp_{0}}(IG(2,V_{x}/U^{\perp}))$ is generated by

$v=f_{3,x}^{*}\otimes f_{1,x}-f_{4,x}^{*}\otimes
f_{2,x}\in\operatorname{Hom}(\wp_{0}^{\perp}/U^{\perp},V_{x}/\wp_{0}^{\perp}).$

Furthermore, $v\circ\beta$ maps $e_{1,x}$, $e_{2,x}$, $e_{3,x}$, $e_{4,x}$ to
$0$, $0$, $f_{1,x}$, $f_{2,x}$ up to a constant multiple. Thus
$\widehat{\beta_{x}}(v)$ can be extended to $\tilde{v}$ with
$\tilde{v}(e_{1})=0$ and $\tilde{v}(e_{2})=0$ and $\tilde{v}(e_{3})=f_{1}$ and
$\tilde{v}(e_{4})=f_{2}$. Then $\delta(\widehat{\beta_{x}}(v))$ is defined by
the cocycle

$\widehat{v}_{j}=(\beta|_{\mathcal{U}_{0}\cap\mathcal{U}_{j}})^{-1}\circ\widetilde{v}\
\in H^{0}(\mathcal{U}_{0}\cap\mathcal{U}_{j},\operatorname{End}V^{\ell_{0}}),$

where $\beta|_{\mathcal{U}_{0}\cap\mathcal{U}_{j}}\colon
V^{\ell_{0}}|_{\mathcal{U}_{0}\cap\mathcal{U}_{j}}\rightarrow
V|_{\mathcal{U}_{0}\cap\mathcal{U}_{j}}$is the isomorphism. Therefore,
$\delta(\widehat{\beta}_{x}(v))$ is represented by the cocycle
$\left\\{\dfrac{e_{3}^{*}\otimes e_{1}-e_{4}^{*}\otimes e_{2}}{z}\text{ on
}\mathcal{U}_{0}\cap\mathcal{U}_{j}\right\\}$, where
$\\{e_{1}^{*},e_{2}^{*},\dots,e_{n}^{*}\\}$ is the dual frame.

Thus $\delta(\widehat{\beta}_{x}(v))$ corresponds to the image of
$\iota(v_{1}\wedge v_{2})=b(v_{1},\,\cdot\,)\otimes
v_{2}-b(v_{2},\,\cdot\,)\otimes v_{1}$ via the duality
$H^{1}(C,\mathrm{End}_{0}(W))\simeq
H^{0}(C,K_{C}\otimes\mathrm{End}_{0}(W))^{*}$ induced by the residue pairing.

∎

###  4.3. Simplicity of the tangent bundles

Now we can prove the simplicity of the tangent bundles. Let us denote by
$\mathcal{M}^{\circ}$ the smooth locus of the moduli space
$\mathcal{M}S_{C}(2r,L)$ ($\mathcal{M}O_{C}(n,L)$, respectively) of $L$-valued
symplectic (orthogonal, respectively) bundles on $C$ of genus $g$.

We exclude the case of symplectic bundles of rank two, which are nothing but
vector bundles of rank two. In this case, the simplicity of the tangent bundle
of $\mathcal{S}U_{C}(2,d)$ has been proven in [8]. Moreover when $d$ is odd,
the stability of the tangent bundle has been proven in [7]. Also recall that
the rank of orthogonal bundles are assumed to be at least 5 to guarantee the
minimality of the orthogonal Hecke curves (see §2.5).

######  Theorem 4.5.

Assume that $g\geq 4,n\geq 4$ for symplectic case, and $g\geq 5,n\geq 5$ for
orthogonal case. Then the tangent bundle $T(\mathcal{M}^{\circ})$ is simple.

###### Proof.

By Proposition 4.1 and Proposition 4.3, the tangent maps of symplectic and
orthogonal Hecke curves are given by the corresponding complete linear
systems. Hence for each case, the VMRT at a generic point is non-degenerate,
and the wanted result follows from Proposition 2.6. ∎

##  5\. Biregularity of the tangent map

Let $W$ be an $L$-valued symplectic or orthogonal bundle over $C$. Throughout
this section, we will assume that an orthogonal bundle of even rank, say $2r$,
admits an isotropic subbundle of rank $r$. As shown in [2, Lemma 2.5], this is
equivalent to that $\det(W)=L^{r}$. Accordingly, $\mathcal{M}O_{C}(2r,L)$
denotes one of the moduli components which parameterizes $L$-valued orthogonal
bundles of rank $2r$ with determinant $L^{r}$. On the other hand, every
orthogonal bundle of odd rank, say $2r+1$, admits an isotropic subbundle of
rank $r$ by [2, Lemma 2.7]. Hence $\mathcal{M}O_{C}(2r+1,L)$ denotes any
moduli component which parameterizes $L$-valued orthogonal bundles of rank
$2r+1$.

As discussed in the previous section, we write

$\operatorname{ad}W\ =\ \begin{cases}\mathrm{Sym}^{2}W\otimes L^{*}\hbox{ if
$W$ is symplectic;}\\\ \wedge^{2}W\otimes L^{*}\hbox{ if $W$ is
orthogonal.}\end{cases}$

In either case, $\operatorname{ad}W$ is a self-dual subbundle of
$\mathrm{End}_{0}W$.

Let $\pi\colon{\mathbb{P}}(\operatorname{ad}W)\to C$ be the associated
projective bundle of $\operatorname{ad}W$. We consider the natural map

$\Phi_{W}\colon{\mathbb{P}}(\operatorname{ad}W)\ {\dashrightarrow}\
{\mathbb{P}}H^{0}(C,K_{C}\otimes\operatorname{ad}W)^{*},$

given by the complete linear system
$|{\mathcal{O}}_{{\mathbb{P}}(\operatorname{ad}W)}(1)\otimes\pi^{*}K_{C}|$.
Our goal will be to prove that $\Phi_{W}$ is an embedding. In fact, we show a
stronger statement: the map

$\Psi_{W}\colon{\mathbb{P}}(\mathrm{End}_{0}W)\ \dashrightarrow\
{\mathbb{P}}H^{0}(C,K_{C}\otimes\mathrm{End}_{0}W)^{*}$

is an embedding. More precisely:

######  Theorem 5.1.

Let $\ell:=\deg(L)\in\\{0,1\\}$. The map $\Psi_{W}$ is an embedding for:
(1) a generic $W\in\mathcal{M}S_{C}(2r,L)$ if $r\geq 2$ and $g\geq 5+2\ell$,
(2) a generic $W\in\mathcal{M}O_{C}(2r,L)$ if either ($r=3,\>g\geq 8$) or
($r\geq 4,\>g\geq 7$),
(3) a generic $W\in\mathcal{M}O_{C}(2r+1,L)$ if either ($r=2,\ g\geq 14$) or
($r\geq 3,\ g\geq 9$).

######  Corollary 5.2.

Let $\mathcal{M}$ be one of the moduli spaces in the above theorem. Let

$\tau_{W}\colon\mathcal{K}_{x}\to{\mathbb{P}}(T_{W}\mathcal{M})$

be the tangent map of the minimal rational component $\mathcal{K}$ associated
to the symplectic or orthogonal Hecke curves. Then $\tau_{W}$ is an embedding
of the corresponding VMRT under the same assumption as in Theorem 5.1.

###### Proof.

This follows from the fact that $\Psi_{W}$, hence $\Phi_{W}$, is an embedding,
together with the pictures of Proposition 4.1 and Proposition 4.3. ∎

The remaining parts of this section are devoted to proving Theorem 5.1.
Following [9, Proof of Theorem 3.1], we shall use the fact that $\Psi_{W}$ is
an embedding if and only if

(12) $h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}W)=0\hbox{ for all
}D\in C^{(2)},$

where $C^{(2)}$ parameterizes the effective divisors of degree two.

We first show the following:

######  Lemma 5.3.

Let ${\mathcal{E}}\to B\times C$ be a family of vector bundles over $C$. Then
the subset

$\left\\{b\in
B:h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes{\mathcal{E}}_{b})=0\hbox{ for all
}D\in C^{(2)}\right\\}$

is open in $B$, where ${\mathcal{E}}_{b}={\mathcal{E}}|_{\\{b\\}\times C}$.

###### Proof.

The complement of the locus in question is

$\left\\{b\in B:h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes{\mathcal{E}}_{b})\geq
1\hbox{ for some }D\in C^{(2)}\right\\}.$

This is the image in $B$ of the closed set

$\left\\{(b,D)\in B\times
C^{(2)}:h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes{\mathcal{E}}_{b})\geq
1\right\\}$

by the projection from $B\times C^{(2)}$. As $C^{(2)}$ is projective and in
particular complete, this projection is closed. The statement follows. ∎

Applying Lemma 5.3 to a suitable étale cover of each of the moduli spaces in
question, we see that it suffices to exhibit a single bundle $V$ with property
(12). We now assemble some vanishing results which we shall use to this end.

######  Lemma 5.4.

Let $E_{1}$ and $E_{2}$ be vector bundles such that for all $D\in C^{(2)}$ we
have

  1. (i)

$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}E_{1})=0$

  2. (ii)

$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}E_{2})=0$

  3. (iii)

$h^{0}(\operatorname{Hom}(E_{1},E_{2}(D)))=0$

  4. (iv)

$h^{0}(\operatorname{Hom}(E_{2},E_{1}(D)))=0$

Let $\Pi\subseteq H^{1}(\operatorname{Hom}(E_{2},E_{1}))$ be a subspace of
dimension at least $2\cdot\mathrm{rk}\,(E_{1})\cdot\mathrm{rk}\,(E_{2})+3$.
Then if $0\to E_{1}\to W\to E_{2}\to 0$ is an extension whose class $\delta$
is a general element of $\Pi$, we have
$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}W)=0$ for all $D\in
C^{(2)}$.

###### Proof.

Let $W$ be an extension whose class is a general element of $\Pi$. Let $D$ be
an element of $C^{(2)}$, and suppose that $\alpha\colon W\to W(D)$ is a
nonzero map. We shall show that under the hypotheses above,
$\alpha=\mathrm{Id}_{W}\otimes\mathbf{s}$ for some $\mathbf{s}\in
H^{0}({{{\mathcal{O}}}_{C}}(D))$. This will suffice to prove the statement.

By (iii), the restriction $\alpha|_{E_{1}}$ factorizes via $E_{1}(D)$.
Therefore, we have a diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\alpha}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$$\textstyle{E_{2}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By (i), then, $\alpha|_{E_{1}}=\mathrm{Id}_{E_{1}}\otimes\mathbf{s}$ for some
$\mathbf{s}\in H^{0}({{{\mathcal{O}}}_{C}}(D))$. (If $C$ is nonhyperelliptic
then $(\mathbf{s})=D$.) Therefore, $\alpha-\mathrm{Id}_{W}\otimes\mathbf{s}$
is a map $W\to W(D)$ vanishing on $E_{1}$. Hence

(13) $\alpha-\mathrm{Id}_{W}\otimes\mathbf{s}\ =\ \beta\circ q$

for some $\beta\in H^{0}(\operatorname{Hom}(E_{2},W(D)))$.

If $\beta\neq 0$ then by (iv) we see that $q^{\prime}\circ\beta$ is a nonzero
map $E_{2}\to E_{2}(D)$. By (ii), we have

$q^{\prime}\circ\beta\ =\ \mathrm{Id}_{E_{2}}\otimes\mathbf{s}_{D^{\prime}}$

for some $D^{\prime}\in|D|$ and $\mathbf{s}_{D^{\prime}}\in
H^{0}({{{\mathcal{O}}}_{C}}(D))$ satisfying
$(\mathbf{s}_{D^{\prime}})=D^{\prime}$. (If $C$ is nonhyperelliptic then
$D^{\prime}=D$.) In particular, the map
$\mathrm{Id}_{E_{2}}\otimes\mathbf{s}_{D^{\prime}}\colon E_{2}\to
E_{2}(D^{\prime})$ lifts to $W$. This means that

$\delta(W)\ \in\
\ker\left(\mathrm{Id}_{E_{2}}\otimes\mathbf{s}_{D^{\prime}}\colon
H^{1}(\operatorname{Hom}(E_{2}(D),E_{1}(D)))\to
H^{1}(\operatorname{Hom}(E_{2}(D-D^{\prime}),E_{1}(D)))\right).$

In view of the exact sequence

$0\ \to\ E_{2}^{*}\otimes E_{1}\ \to\ E_{2}^{*}\otimes E_{1}(D^{\prime})\ \to\
E_{2}^{*}\otimes E_{1}(D^{\prime})|_{D^{\prime}}\ \to\ 0,$

this kernel has dimension at most
$2\cdot\mathrm{rk}\,(E_{2})\cdot\mathrm{rk}\,(E_{1})$. The union

$\bigcup_{D^{\prime}\in
C^{(2)}}\ker\left(\left(\mathrm{Id}_{E_{2}}\otimes\mathbf{s}_{D^{\prime}}\right)^{*}\right)$

is therefore of dimension at most
$2\cdot\mathrm{rk}\,(E_{2})\cdot\mathrm{rk}\,(E_{1})+2<\operatorname{dim}\Pi$.
But since $\delta(W)$ was assumed to be general in $\Pi$, we may assume that
the map $\mathrm{Id}_{E_{2}}\otimes\mathbf{s}_{D^{\prime}}$ does not lift to
$W$. Therefore, we must have $\beta=0$.

By (13), we obtain $\alpha=\mathrm{Id}_{W}\otimes\mathbf{s}$, as desired. ∎

######  Lemma 5.5.

Suppose $t$ and $s$ are integers with $s\geq 0$ and $t+2s<g$. Let $N$ be a
generic line bundle in $\mathrm{Pic}^{t}(C)$. Then $h^{0}(N(D))=0$ for all
$D\in C^{(s)}$, where $C^{(s)}$ parameterizes the effective divisors of degree
$s$.

###### Proof.

If $t+s<0$ then this is clear. Otherwise: If $h^{0}(C,N(D))\neq 0$ for some
$D\in C^{(s)}$, then $N$ is of the form ${{{\mathcal{O}}}_{C}}(D_{1}-D)$ for
some $D_{1}\in C^{(t+s)}$ and $D\in C^{(s)}$. Hence the locus of such $N$ is
of dimension at most $t+2s$ in $\mathrm{Pic}^{t}(C)$. (We take
$C^{(0)}=\\{{{{\mathcal{O}}}_{C}}\\}$.) By hypothesis, this locus is not dense
in $\mathrm{Pic}^{t}(C)$. The statement follows. ∎

######  Lemma 5.6.

Let $r,s,t$ be integers with $r\geq 2$, $s\geq 0$ and $t+rs<(r-1)(g-1)$. Let
$G$ be a generic stable bundle of rank $r$ and degree $t$. Then
$h^{0}(G(D))=0$ for all $D\in C^{(s)}$.

###### Proof.

A nonzero section of $G(D)$ gives a sheaf injection
${{{\mathcal{O}}}_{C}}(-D)\to G$. This implies that the first Segre invariant
of $G$ is bounded by

$s_{1}(F)\ \leq\ \deg(G)-\mathrm{rk}\,(G)\cdot\deg({{{\mathcal{O}}}_{C}}(-D))\
=\ t+rs.$

But as $G$ is generic, by [10, Satz 2.2] we have

$s_{1}(G)\ \geq\ (r-1)(g-1).$

Hence we have an inequality $(r-1)(g-1)\leq t+rs$, which contradicts the
assumption. Thus $h^{0}(C,G(D))=0$ for all $D$. ∎

######  Lemma 5.7.

Let $E$ be a generic stable bundle over $C$ of rank $r\geq 1$ and degree $e$.
(1) If $e\geq 0,s\geq 1$ and $2e+2s<g$, then $h^{0}(C,E\otimes E(D))=0$ for
all $D\in C^{(s)}$.
(2) If $e<0,s\geq 1$ and $2s<g$, then $h^{0}(C,E\otimes E(D))=0$ for all $D\in
C^{(s)}$.

###### Proof.

By Lemma 5.3, the locus of points $E$ which satisfy the desired vanishing
property is open. Thus it suffices to exhibit a single $E$ with the desired
property. (Note that even if such an $E$ is unstable, we can continuously
deform it to get a generic stable one with the same vanishing property.)

We first assume $e\geq 0$ and proceed by induction on $r$. For $r=1$, this
holds by Lemma 5.5. Suppose now that $r\geq 2$. By induction hypothesis, there
is a stable vector bundle $F$ of rank $r-1$ and degree $e$ with the desired
vanishing property, which can be assumed to be generic in the moduli. Fix a
generic line bundle $L_{0}\in\mathrm{Pic}^{0}(C)$ and put $E=L_{0}\oplus F$.
Then $E\otimes E(D)$ has four direct summands, and it suffices to show all of
them have no sections.

By induction we may assume that $h^{0}(C,F\otimes F(D))=0$ for all $D\in
C^{(2)}$. Also we note that $h^{0}(C,(L_{0})^{2}(D))=0$ for all $D\in C^{(2)}$
by Lemma 5.5. Finally we show that $h^{0}(C,F\otimes L_{0}(D))=0$ for all $D$.
If $F$ is a line bundle, the condition for the vanishing is $e+2s<g$ by Lemma
5.5, and this holds by assumption. If $\mathrm{rk}\,(F)=r-1\geq 2$, then the
condition for the vanishing is $e+(r-1)s<(r-2)(g-1)$ by Lemma 5.6. Also this
holds by assumption, noting that $s\geq 1$ and $r=\mathrm{rk}\,(F)+1\geq 3$.

It follows that for a generic extension $E$ given above, we have
$h^{0}(C,E\otimes E(D))=0$ for all $D\in C^{(s)}$.

To finish, we need to find a bundle of degree $e<0$ with the vanishing
property. By the part (1), we may choose a stable bundle $E$ of rank $r$ and
degree 0 satisfying $h^{0}(E\otimes E(D))=0$ for all $D\in C^{(s)}$. Let
$\widetilde{E}$ be obtained by an elementary transformation
$0\to\widetilde{E}\to E\to\tau\to 0$, where $\deg(\tau)=e$. Then
$\widetilde{E}\otimes\widetilde{E}(D)$ is a subsheaf of $E\otimes E(D)$, and
by the vanishing result for $E$, we obtain
$h^{0}(C,\widetilde{E}\otimes\widetilde{E}(D))=0$. This completes the proof. ∎

Also we record the following cohomology vanishing result [9, Proposition 3.2],
which will be used later:

######  Proposition 5.8.

For a general stable vector bundle $F$ of arbitrary rank and degree,
$H^{0}(C,(\operatorname{End}_{0}F)(D))=0$ for any effective divisor $D$ of
degree $d$ whenever $g\geq\frac{3}{2}d+2$. ∎

Now we apply these results to show the desired vanishing property for a
generic symplectic and orthogonal bundle of even rank. Consider an extension

$(\ast)\ \ \ \ \ 0\to E\to W\to E^{*}\otimes L\to 0.$

for $E\in{\mathcal{S}U}(r,e)$ which is a generic stable bundle. Recall that,
by [6, Criterion 2.1], we get a symplectic bundle $W$ if we choose $(\ast)$ in
$H^{1}(\mathrm{Sym}^{2}E\otimes L^{*})$, and an orthogonal bundle if we choose
$(\ast)$ in $H^{1}(\wedge^{2}E\otimes L^{*})$. Also in both cases, $E\to W$ is
an isotropic subbundle. (From now on whenever we discuss orthogonal bundles of
even rank, we consider those bundles with an isotropic subbundle of the half
rank only.)

Once we find a symplectic/orthogonal bundle $W$ in this extension which has
the desired vanishing property (12), by deformation this will show that the
vanishing property holds for a general stable symplectic/orthogonal bundle
with the same topological invariants. The only topological invariants of a
symplectic bundle are rank and degree, while we need to additionally consider
the 2nd Stiefel–Whitney class for an orthogonal bundles. Note that we may
assume $L={\mathcal{O}}_{C}$ when $\deg(L)$ is even and
$L={\mathcal{O}}_{C}(x)$ for some $x\in C$ when $\deg(L)$ is odd.

If $L={\mathcal{O}}_{C}$ so that $\deg(W)=0$, the moduli space
$\mathcal{M}O_{C}(2r,{\mathcal{O}}_{C})$ has two components classified by the
2nd Stiefel–Whitney class $w_{2}(W)$ such that the degree of any rank $r$
isotropic subbundle of $W$ has the same parity as $w_{2}(W)$. On the other
hand, if $L={\mathcal{O}}_{C}(x)$ so that $\deg(W)=r$, then by [2, §2] every
orthogonal bundle $W\in\mathcal{M}O_{C}(2r,{\mathcal{O}}_{C}(x))$ has rank $r$
isotropic subbundles both of even degree and of odd degree.

Hence it suffices to show that a bundle $W$ obtained by a generic extension
$(\ast)$ has the vanishing property (12) in each of the following cases:

  * •

For symplectic bundles: $e=\deg E=0$ and $L={\mathcal{O}}_{C}$ or
${\mathcal{O}}_{C}(x)$.

  * •

For orthogonal bundles: ($L={\mathcal{O}}_{C}$ and $e=-1,0$) or
($L={\mathcal{O}}_{C}(x)$ and $e=0$).

######  Proposition 5.9.

Let $\ell=\deg(L)\in\\{0,1\\}$.

  1. (1)

For $r\geq 2$, suppose $g\geq 5+2\ell$. If $(\ast)$ is a symplectic extension
defined by a generic class in $H^{1}(\mathrm{Sym}^{2}E\otimes L^{*})$ where
$e:=\deg E=0$, then
$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}W)=0$ for all $D\in
C^{(2)}$.

  2. (2)

Suppose either ($r=3,\>g\geq 8$) or ($r\geq 4,\>g\geq 7$). If $(\ast)$ is an
orthogonal extension defined by a generic class in $H^{1}(\wedge^{2}E\otimes
L^{*})$ where $e:=\deg E\in\\{-1,0\\}$, then
$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}W)=0$ for all $D\in
C^{(2)}$.

###### Proof.

For the vanishing, let us apply Lemma 5.4 with $E_{1}=E^{*}$ and
$E_{2}=E\otimes L$. As $E$ is general in moduli and $g\geq 5$, conditions (i)
and (ii) follow from Proposition 5.8.

Conditions (iii) and (iv) read $h^{0}(E\otimes E\otimes L(D))=0$ and
$h^{0}(E^{*}\otimes E^{*}\otimes L^{*}(D))=0$ for any $D\in C^{(2)}$,
respectively. These vanishing conditions are checked by Lemma 5.7 under the
assumption $g\geq 5+2\ell$ in each case for $L={\mathcal{O}}_{C}$ or
$L={\mathcal{O}}_{C}(x)$. To get a conclusion from Lemma 5.4, it will suffice
to check that $h^{1}(\mathrm{Sym}^{2}E\otimes L^{*})$ and
$h^{1}(\wedge^{2}E\otimes L^{*})$ are bigger than $2r^{2}+2$, respectively.

In the symplectic case, by Riemann–Roch the desired inequality would follow
from

$2r^{2}+3\ \leq\ \frac{1}{2}r(r+1)(g-1+\deg(L)).$

This reads

$\frac{g-1+\ell}{2}\geq\frac{2r^{2}+3}{r(r+1)},$

which holds for $r\geq 2$ and $g\geq 5-\ell$.

In the orthogonal case, the desired inequality would follow from

$2r^{2}+3\ \leq\ -(r-1)e+\frac{1}{2}r(r-1)(g-1+\deg(L)).$

This reads

$\frac{g-1+\ell}{2}\geq\frac{2r^{2}+3+(r-1)e}{r(r-1)},$

which holds if either ($r=3,\>g\geq 8$) or ($r\geq 4,\>g\geq 7$). ∎

Now we discuss the case of orthogonal bundles of odd rank. In this case, we
require some more vanishing results. To get a better genus bound, we assume
here $e=\deg E\in\\{-2,-1\\}$ instead of $\\{-1,0\\}$.

######  Lemma 5.10.

Suppose $g\geq 8$. Suppose $r\geq 2$, and let $E$ be a generic stable bundle
of rank $r$ and degree $e$, where $e\in\\{-2,-1\\}$. Let $0\to E\to
F\to{{{\mathcal{O}}}_{C}}\to 0$ be a generic extension. Then
$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}F)=0$ for all $D\in
C^{(2)}$.

###### Proof.

Let us apply Lemma 5.4 with $E_{1}=E$ and $E_{2}={{{\mathcal{O}}}_{C}}$.
Condition (i) follows as above from [9, Proposition 3.2] since $E$ is general
and $g\geq 5$. Condition (ii) is trivial. Conditions (iii) and (iv) follow
from Lemma 5.6 under the assumption $g\geq 8$.

It remains to check that
$\operatorname{dim}H^{1}(\operatorname{Hom}({{{\mathcal{O}}}_{C}},E))>2r+3$.
As $E$ is stable of negative degree, $h^{0}(E)=0$. By Riemann–Roch,
$h^{1}(E)=-e+r(g-1)\geq r(g-1)+1$. Using the inequalities $r\geq 2$ and $g\geq
5$, one checks that this exceeds $2r+3$. The statement now follows from Lemma
5.4. ∎

######  Lemma 5.11.

Let $E$ and $F$ be as in Lemma 5.10, and suppose $g\geq 8$. Then for all $D\in
C^{(2)}$ we have $h^{0}(E^{*}\otimes F^{*}(D))=0$ and $h^{0}(F\otimes
E(D))=0$.

###### Proof.

By construction of $F$, for each $D\in C^{(2)}$ we have exact sequences

$H^{0}(E\otimes E(D))\ \to\ H^{0}(F\otimes E(D))\ \to\ H^{0}(E(D))\ \to\
\cdots$

and

$H^{0}(E^{*}(D))\ \to\ H^{0}(E^{*}\otimes F^{*}(D))\ \to\ H^{0}(E^{*}\otimes
E^{*}(D))\ \to\ \cdots$

By Lemma 5.7, we have $h^{0}(E\otimes E(D))=0=h^{0}(E^{*}\otimes E^{*}(D))$
for all $D\in C^{(2)}$. The vanishing $h^{0}(E^{*}(D))=0=h^{0}(E(D))$ is a
consequence of Lemma 5.6. The statement follows. ∎

By [2, Lemma 2.4], for any $L$-valued orthogonal bundle $W$ of odd rank, there
is a line bundle $N$ such that $W\otimes N$ is an ${\mathcal{O}}_{C}$-valued
orthogonal bundle of trivial determinant. So we may work only for the moduli
component of ${\mathcal{O}}_{C}$-valued orthogonal bundles of trivial
determinant. As in the even rank case, the moduli space
$\mathcal{M}O_{C}(2r+1,{\mathcal{O}}_{C})$ has two components classified by
the 2nd Stiefel–Whitney class $w_{2}(W)$. By [4, Theorem 3.1], the degree of
any rank $n$ isotropic subbundle of $W$ has the same parity as $w_{2}(W)$. To
construct such orthogonal bundles of rank $2r+1$ as extensions, we use some
results from [4, § 3]. Let $0\to E\xrightarrow{j}F\to{{{\mathcal{O}}}_{C}}\to
0$ be an extension as above, and let $\Pi_{j}$ be the subspace of
$H^{1}(F\otimes E)$ as defined in [4, § 3], which contains
$H^{1}(C,\wedge^{2}E)$ as a subspace of codimension 1. By [4, Lemma 3.2], an
extension $0\to E\to W\to F^{*}\to 0$ defined by a class contained in
$\Pi_{j}\setminus H^{1}(C,\wedge^{2}E)$ is an orthogonal bundle.

######  Proposition 5.12.

Suppose either ($r=2,\ g\geq 14$) or ($r\geq 3,\ g\geq 9$). Let $0\to E\to
W\to F^{*}\to 0$ be a stable orthogonal bundle of rank $2r+1$ as above, whose
extension whose class is general in $\Pi_{j}$. Then
$h^{0}({{{\mathcal{O}}}_{C}}(D)\otimes\mathrm{End}_{0}W)=0$ for all $D\in
C^{(2)}$.

###### Proof.

Again, we use Lemma 5.4; this time with $E_{1}=E$ of rank $r$, degree $e$ and
$E_{2}=F^{*}$ of rank $r+1$, degree $-e$ where $e\in\\{-2,-1\\}$. Condition
(i) follows from Proposition 5.8 as before. Condition (ii) follows from Lemma
5.10. Conditions (iii) and (iv) follow from Lemma 5.11.

Lastly, we must show that $\operatorname{dim}(\Pi_{j})\geq
2\cdot\mathrm{rk}\,(E)\cdot\mathrm{rk}\,(F^{*})+3=2r(r+1)+3$. By Riemann–Roch,

$\operatorname{dim}(\Pi_{j})\ =\ h^{1}(\wedge^{2}E)+1\ \geq\
-(r-1)e+\frac{r(r-1)}{2}(g-1)+1.$

Hence it suffices to have

$-(r-1)e+\frac{r(r-1)}{2}(g-1)+1\ \geq\ 2r(r+1)+3.$

Simplifying this by using $e\leq-1$, we get the wanted inequality if

(14) $\frac{r(r-1)}{2}(g-1)\geq 2r^{2}+r+3.$

This holds for ($r=2,g\geq 14$) or ($r\geq 3,g\geq 9$). ∎

###### Proof of Theorem 5.1.

By Lemma 5.3, it suffices to exhibit a single element $W_{0}$ of each of the
moduli spaces satisfying the vanishing property (12). This follows from
Proposition 5.9 and 5.12. ∎

######  Remark 5.13.

(1) The genus bound in Theorem 5.1 is not optimal, and can be improved simply
by computing inequalities above more accurately. For instance, the above
inequality (14) reads $g\geq\frac{2(2r^{2}+r+3)}{r(r-1)}+1$, which becomes
$g\geq 6$ in the limit $r\to\infty$.
(2) We did not consider orthogonal bundles of rank $\leq 4$ according to the
convention in §2.5. But all the arguments in §5 are valid for arbitrary rank,
hence we can apply the same argument to get a very ampleness result for
orthogonal bundles of rank $\leq 4$. For example, we could state Proposition
5.9 (2) for $r=2$, in which case the genus assumption would be $g\geq 12$.
Hence the map $\Psi_{W}$ is an embedding for a generic
$W\in\mathcal{M}O_{C}(4,L)$ if $g\geq 12$.
(3) Theorem 5.1 shows that the VMRT of $\mathcal{M}S_{C}(n,L)$ and
$\mathcal{M}O_{C}(n,L)$ at a generic point $W$ is biregular to
${\mathbb{P}}(W^{*})$ and $IG(2,W)$ respectively, under the assumption on the
genus. As remarked in [3, Remark 6.2], this improves the involved genus bound
in [3, §6].

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