# Period-index problem for hyperelliptic curves

J. N. Iyer  Institute of Mathematical Sciences
C.I.T. Campus
Taramani, Chennai 600113. India<EMAIL_ADDRESS>and  R. Parimala
with an appendix by S. Ramanan  Department of Mathematics & Computer Science
Emory University
400 Dowman Drive NE
Atlanta, GA 30322, USA<EMAIL_ADDRESS>Chennai Mathematical Institute
H1, SIPCOT IT Park,
Siruseri Kelambakkam 603103 India<EMAIL_ADDRESS>
###### Abstract.

Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a
rational point. We prove that the index and exponent coincide for elements in
the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli
space of rank 2 stable vector bundles with odd determinant on a smooth
projective hyperelliptic curve $C$ of genus $g$ with a rational point over any
field of characteristic not two with the Grassmannian of $(g-1)$-dimensional
linear subspaces in the base locus of a certain pencil of quadrics is
established, making a result of ([10]) rational. We establish a twisted
version of this isomorphism and we derive as a consequence a weak Hasse
principle for the smooth intersection $X$ of two quadrics in
${\mathbb{P}}^{5}$ over a number field: if $X$ contains a line locally, then
$X$ has a $k$-rational point.

Let $k$ be a field and $\operatorname{\mathrm{Br}}(k)$ the Brauer group of
$k$. There are two numerical invariants attached to a Brauer class
$\alpha\in\operatorname{\mathrm{Br}}(k)$; $index(\alpha)=\sqrt{[D:k]}$ if
$\alpha$ is represented by a central division algebra $D$ over $k$ and
$period(\alpha)=$ order of $\alpha$ in $\operatorname{\mathrm{Br}}(k)$. There
has been extensive study of uniform bounds for index of algebras in terms of
their periods over fields which are of arithmetic or geometric interest ([9],
[17], [32], [15]). Let $p$ be a prime and $K$ a field of characteristic not
equal to $p$. The Brauer $p$-dimension of $K$ denoted by
$\operatorname{\mathrm{Br}}\\!_{p}$dim$(K)$ is the least integer $d$ such that
for every finite extension $L/K$ and every
$\alpha\in~{}_{p}\operatorname{\mathrm{Br}}(L)$, index$(\alpha)$ divides
$p^{d}$. Here ${}_{p}\mathrm{Br}(L)$ denotes the $p$-torsion subgroup of
$\operatorname{\mathrm{Br}}(L)$.

Bounding Brauer dimension has deep consequences in the study of homogeneous
spaces under connected linear algebraic groups. A theorem of de-Jong/Lieblich
that period = index for function fields of surfaces over algebraically closed
fields is critical to the solution of Conjecture II of Serre for exceptional
groups of type $D_{4}$, $E_{6}$, $E_{7}$ due to Gille ([13, IV.2]). Bounding
the Brauer $2$-dimension of function fields of all curves over totally
imaginary number fields would lead to finiteness of the $u$-invariant of such
fields ([20]). Finiteness of the $u$-invariant of $k(t)$, $k$ totally
imaginary number field, is an open question.

Let $k$ be a totally imaginary field with the ring of integers $\mathscr{O}$.
Let $C/k$ be a smooth projective geometrically integral curve over $k$ with
$\mathscr{C}/\mathscr{O}$ a regular proper model. In ([20]), finiteness of
$\operatorname{\mathrm{Br}}_{p}$dim$(k(C))$ for all $C/k$ is reduced to
bounding indices of $p$-torsion elements in
$\operatorname{\mathrm{Br}}(\mathscr{C})$, i.e. ‘unramified elements’ in
$\operatorname{\mathrm{Br}}(k(C))$, for all $C/k$. By a theorem of
Grothendieck, for a smooth projective curve $C$ over $k$, the image of
$\operatorname{\mathrm{Br}}(\mathscr{C})$ is zero in
$\operatorname{\mathrm{Br}}(k_{\nu}(C))$ for every finite place $\nu$ of $k$.
Let
$\Sha\operatorname{\mathrm{Br}}(k(C))=Ker(H^{2}(k(C),{\mathbb{G}}_{m})\to\prod_{\nu\in\Omega_{K}}H^{2}(k_{\nu}(C),{\mathbb{G}}_{m}))$.
Note that
$\Sha\operatorname{\mathrm{Br}}(k(C))\subseteq\operatorname{\mathrm{Br}}(\mathscr{C})\subseteq\operatorname{\mathrm{Br}}(C)$
([29]) and
$\Sha\operatorname{\mathrm{Br}}(k(C))=\operatorname{\mathrm{Br}}(\mathscr{C})$
if $k$ is a totally imaginary number field. We state the following conjecture
concerning the period/index bounds for elements in
$\Sha\operatorname{\mathrm{Br}}(k(C))$:

######  Conjecture.

For elements in $\Sha\operatorname{\mathrm{Br}}(k(C))$, the index and period
coincide.

If $C$ is an elliptic curve, $\Sha\operatorname{\mathrm{Br}}(k(C))$ is
isomorphic to the Tate-Shafarevich group of $C$ and the conjecture is true
([4], [25]). We prove this conjecture for the $2$-torsion subgroup for genus 2
curves.

######  Theorem 0.1.

(cf. 5.4) Let $k$ be a number field and $C$ a smooth projective geometrically
integral curve over $k$ of genus 2 with a $k$-rational point. Then for every
element $\tilde{\alpha}$ in ${}_{2}\Sha\operatorname{\mathrm{Br}}(C)$, index
of $\tilde{\alpha}$ divides 2.

As a consequence of our approach to the period-index question for
hyperelliptic curves, we also obtain following result of independent interest.

######  Theorem 0.2.

(cf. 10.1) Let $k$ be a global field of characteristic not $2$. Let $X$ be a
smooth intersection of two quadrics $Q_{1}$ and $Q_{2}$ in $\mathbb{P}^{5}$.
Suppose $disc(Q_{1})=1$. If $X(k_{\nu})$ contains a line for all places $\nu$
of $k$, then, $X(k)$ is nonempty.

The Hasse principle for smooth intersection of quadrics in
$\mathbb{P}_{k}^{5}$ is an open question. Wittenberg ([38]) proves that
assuming the finiteness of Tate-Shafarevich groups and Schinzel’s hypothesis,
the Hasse principle holds for smooth intersections of two quadrics in
$P^{5}_{k}$.

We follow the approach of Lieblich to study twisted moduli spaces associated
to a Brauer class in order to bound its index. Using this technique, Lieblich
proves period index bounds for unramified Brauer classes of function fields of
surfaces over finite fields (([17])). Here is a sketch of the contents of the
paper.

In §1, we recall some properties of twisted sheaves and moduli spaces/stacks.
In §2, we recall the Hecke correspondence, introduced by Narasimhan and
Ramanan ([22]). We describe the correspondence between the moduli spaces of
vector bundles of rank two with odd and even degree determinants for curves of
genus 2. In §3 we define a twist of this correspondence with respect to a
Galois cocycle with values in ${}_{2}\mathrm{Pic}C$. In §4, we record a Hasse
principle result, due to Colliot-Thélène, for certain conic fibrations over
${\mathbb{P}}^{3}$, which is used in §5. In §5, if $g=2$, using the twisted
Hecke correspondence, we obtain period=index statements for
${}_{2}\Sha\operatorname{\mathrm{Br}}(k(C))$ (cf. 5.4) stated at the beginning
of the introduction. In particular, if $k$ is a totally imaginary number
field, then period and index coincide for elements in
$\operatorname{\mathrm{Br}}(\mathscr{C})$.

In §6, over a general field of characteristic not $2$, we recall several
results on a nonsingular pencil $Q=tQ_{1}-Q_{2}$ of quadrics on a vector space
of dimension $2g+2$ and the asssociated hyperelliptic curve $C$ defined by
$y^{2}=f(t)$ where $f(t)=disc(tQ_{1}-Q_{2})$. We describe the group of
automorphisms $\operatorname{\mathbf{Aut}}^{+}_{Q}$ of the pencil and an
isomorphism
$\theta_{Q}:{}_{2}\mathrm{Pic}C\simeq\operatorname{\mathbf{Aut}}^{+}_{Q}$. In
§7, given a Galois cocycle $\gamma$ with values in
$\operatorname{\mathbf{Aut}}^{+}_{Q}$, we describe the twisted pencil
$Q^{\gamma}$ which is defined over a central simple algebra of degree $2g+2$.
If the algebra is split, we recover the usual pencil of quadrics. Let $C$ be a
smooth hyperelliptic curve which admits a $k$\- rational point $P$ and
$L={\mathscr{O}}((2g+1)P)$. Let $M_{C}(2,L)$ denote the moduli space of rank 2
stable locally free sheaves on $C$ with determinant $L$. There is a
description due to Desale-Ramanan ([10]) of $M_{C}(2,L)$ over $\overline{k}$
in terms of the space $I_{g-1}(Q_{0})$, the $(g-1)$-dimensional Grassmannian
of linear subspaces contained in the base locus of a pencil of quadrics
$Q_{0}=tQ_{1}-Q_{2}$ in $\mathbb{P}^{2g+1}$; the hyperelliptic curve $C$ is
defined by $y^{2}=(-1)^{g+1}det(Q_{0})$. In §8, we recall a construction of
Ramanan (cf. A) of a nonsingular pencil $Q_{\eta}$ whose associated
hyperelliptic curve is $C$, as well as the isomorphism over $k$:

$\psi:M_{C}(2,L)\to I_{g-1}(Q_{\eta}).$

We describe a twist of this isomorphism by a Galois cocycle $\beta$ with
values in ${}_{2}\mathrm{Pic}C$:

$\psi^{\beta}:M^{\beta}_{C}(2,L)\to I_{g-1}(Q^{\beta}_{\eta})$

Here $Q_{\eta}^{\beta}$ is a pencil of twisted quadrics.

Let $k$ be a number field and $C/k$ a smooth projective hyperelliptic curve of
genus $g$ with $C(k)\neq\emptyset$. Let
$\tilde{\alpha}\in~{}_{2}\Sha\operatorname{\mathrm{Br}}(k(C))$. Then
$\tilde{\alpha}$ is in the image of
$\operatorname{\mathrm{Br}}(C)\to\operatorname{\mathrm{Br}}(k(C))$ ([29]). We
choose a lift $\alpha\in H^{2}(C,\mu_{2})$ of $\tilde{\alpha}$ with
$\alpha_{\overline{k}}=0$, where $\overline{k}$ is the algebraic closure of
$k$. This is possible since $C(k)\neq\emptyset$. Let $\beta$ be a one cocycle
with values in ${}_{2}\mathrm{Pic}C$ associated to $\alpha$ (cf. §1,
$(\star\star)$). In §9, we prove (cf. 9.4) that the twisted pencil
$Q_{\eta}^{\beta}$ is indeed a pencil of quadrics. This uses a result of
Shankar-Wang ([34]). Thus we have an identification of the twisted moduli
space $M_{C}^{\beta}(2,L)$ with the space $I_{g-1}(Q_{\eta}^{\beta})$ with
$Q_{\eta}^{\beta}$ a genuine pencil of quadrics.

In §10, we use the identification of the twisted moduli space
$M^{\beta}_{C}(2,L)$ with the base locus $X$ of the pencil
$(Q^{\beta}_{\eta})$ in the genus 2 case and the results of §5 to derive a
variant of a Hasse principle for a smooth intersection of two quadrics in
${\mathbb{P}}^{5}$ stated in the beginning.

Acknowledgments:
The first author was partially supported by the IMSc-DAE project Complex
Analysis and Algebraic Geometry.
The second author was partially supported by the NSF grant DMS-1801951.
The authors thank J.-L. Colliot-Thélène, Max Lieblich, V. Suresh and J.-P.
Tignol for their generous contributions to this paper.

##  1\. Twisted sheaves and moduli spaces/stacks

We recall some properties of moduli stacks of twisted sheaves. We refer to the
papers [17],[18], [19] of Lieblich for details.

###  1.1. $\mu_{n}$-gerbes

Let $n$ be coprime to char$(k)$ and $C/k$ a smooth projective geometrically
integral curve.

######  Definition 1.1.

A $\mu_{n}$-gerbe on $C$ is a gerbe $\mathscr{C}\to C$ together with an
isomorphism $(\mu_{n})_{*}\simeq\mathscr{I}(\mathscr{C})$, where
$\mathscr{I}(\mathscr{C})$ denotes the inertial stack of $\mathscr{C}$.

We record the following results on $\mu_{n}$-gerbs on $C$, due to Giraud
([14], Section IV.3).

######  Proposition 1.2.

There is a natural bijection between the set of isomorphism classes of
$\mu_{n}$-gerbes on $C$ and $H^{2}(C,\mu_{n})$.

We simply call $\alpha\in H^{2}(C,\mu_{n})$ a $\mu_{n}$-gerbe under the above
correspondence.

######  Definition 1.3.

A $\mu_{n}$-gerbe $\alpha$ is essentially trivial if its image in
$H^{2}(C,\mathbf{G}_{m})$ is zero.

######  Definition 1.4.

Let $L$ be an invertible sheaf on $C$. The gerbe $[L]^{\frac{1}{n}}$ is the
stack over $C$ whose objects over a $C$\- scheme $T$ are pairs
$(\mathscr{L},\phi)$ where $\mathscr{L}$ is an invertible sheaf on $T$ and
$\phi:\mathscr{L}^{n}\to L_{T}$ is an isomorphism. The gerbe
$[L]^{\frac{1}{n}}$ is a $\mu_{n}$-gerbe.

We recall the following facts on essentially trivial $\mu_{n}$-gerbs from
([18, §2.3.4]). The Kummer sequence

$1\to\mu_{n}\to\mathbf{G}_{m}\buildrel n\over{\to}\mathbf{G}_{m}\to 1$

yields an exact sequence

$\operatorname{\mathrm{Pic}}C\buildrel
n\over{\to}\operatorname{\mathrm{Pic}}C\to H^{2}(C,\mu_{n})\to
H^{2}(C,\mathbf{G}_{m}).$

Every essentially trivial $\mu_{n}$-gerbe $\alpha$ is isomorphic to
$[L]^{\frac{1}{n}}$ where $L$ is an invertible line bundle on $C$ with
$\partial(L)=\alpha$. There is an invertible $\alpha$-twisted sheaf
$\mathscr{L}_{\alpha}$ such that $\mathscr{L}_{\alpha}^{n}\simeq L$, uniquely
determined up to tensoring with a line bundle in
$\operatorname{\mathrm{{}_{2}Pic}}C$. Given an $\alpha$-twisted sheaf
$\mathscr{F}$, $\mathscr{F}\otimes\mathscr{L}_{\alpha}^{-1}$ is isomorphic to
the pull-back to the gerbe $\alpha$ of a sheaf $\mathscr{F}$ on $C$. This
association can be used to define stability of $\alpha$-twisted sheaves.

###  1.2.  Twisted sheaves and Stability

We recall the definition of stability and semistability for locally free
sheaves on $C$. For a vector bundle $E$ on $C$, the slope $\mu(E)$ is the
rational number $\frac{deg(E)}{rank(E)}$.. The vector bundle $E$ is semistable
if $\mu(F)\leq\mu(E)$ for every locally free subbundle $F$ of $E$ and stable
if strict inequality holds. We shall now define the corresponding notions for
twisted sheaves.

Let $\alpha\in H^{2}(C,\mu_{n})$ and $\mathscr{F}$ a locally free
$\alpha$-twisted sheaf of rank $n$. Then determinant of $\mathscr{F}$, denoted
by det$(\mathscr{F})$, has no twist and is a pullback of a line bundle $L$ on
$C$. We write det$(\mathscr{F})=L$.

For an invertible $\alpha$-twisted sheaf $M$, $M^{\otimes n}=L$ where $L$ is a
line bundle on $C$. We define deg$(M)=\frac{1}{n}deg(L)$. We define the degree
of a locally free $\alpha$-twisted sheaf $\mathscr{F}$ to be the degree of
det$(\mathscr{F})$.

######  Definition 1.5.

The slope $\mu(\mathscr{F})$ of a locally free $\alpha$-twisted sheaf
$\mathscr{F}$ is $\frac{deg(\mathscr{F})}{rank(\mathscr{F})}$.

An $\alpha$-twisted locally free sheaf $\mathscr{F}$ is semistable (resp.
stable) if for every $\alpha$-twisted subbundle $\mathscr{G}$ of
$\mathscr{F}$,

$\mu(\mathscr{G})\leq\mu(\mathscr{F})\,\,\,({resp.}\mu(\mathscr{G})<\mu(\mathscr{F})).$

Let $\alpha\in H^{2}(C,\mu_{2})$. Since $H^{2}(C_{k^{s}},\mathbf{G}_{m})=0$,
$\alpha_{k^{s}}$ is essentially trivial and there is a line bundle $L$ on
$C_{k^{s}}$ such that the image of
$[L]\in(\operatorname{\mathrm{Pic}}C_{k^{s}}/2)\to H^{2}(C_{k^{s}},\mu_{2})$
is $\alpha$. Thus there is an $\alpha$-twisted invertible sheaf
$\mathscr{L}_{\alpha}$ over $k^{s}$. The stability /semistability of an
$\alpha$-twisted locally free sheaf $\mathscr{F}$ is equivalent to the
stability/semistability of the locally free sheaf
$\mathscr{F}\otimes\mathscr{L}_{\alpha}^{-1}$ over $C_{k^{s}}$.

###  1.3. Moduli stacks

Let $\alpha\in H^{2}(C,\mu_{2})$ and $L$ a line bundle on $C$. The moduli
stack $\mathscr{M}_{C}^{\alpha}(n,L)$ of rank $n$ $\alpha$-twisted stable
locally free sheaves determinant $L$ on $C$ is defined in [17]. It was shown
to be an Artin stack locally of finite presentation over $k$ (see [17]). In
fact, the stack $\mathscr{M}_{C}^{\alpha}(n,L)$ is a $\mu_{n}$-gerbe over the
smooth quasi-projective variety $M_{C}^{\alpha}(n,L)$ which is the coarse
moduli space of the stack $\mathscr{M}_{C}^{\alpha}(n,L)$. This variety admits
a locally factorial compactification $M^{\alpha,ss}_{C}(n,L)$ with a closed
complement of codimension at least one. In fact $M^{\alpha,ss}_{C}(n,L)$ is
the coarse moduli space of stack of rank $n$ locally free semistable
$\alpha$-twisted sheaves of determinant $L$.

We record the following result from ([17, 3.1.2.1]).

######  Theorem 1.6.

Let $\alpha\in H^{2}(C,\mu_{n})$ with image $\tilde{\alpha}\in
H^{2}(C,\mathbf{G}_{m})$. Then index$(\tilde{\alpha})$ divides $n$ if and only
if there is a stable locally free $\alpha$-twisted sheaf of rank $n$ and
trivial determinant.

There is another way of describing the stack of $\alpha$-twisted sheaves and
the associated moduli space $M_{C}^{\alpha}(n,L)$ using Galois twists which we
shall explain next. Let $f:C\to k$ be a smooth projective geometrically
integral curve with $C(k)\neq\emptyset$. The Leray spectral sequence yields
decomposition

$H^{2}(C,\mathbf{G}_{m})=H^{2}(k,\mathbf{G}_{m})\oplus
H^{1}(k,\operatorname{\mathrm{Pic}}C)$ $(\star)$

$H^{2}(C,\mu_{n})=H^{2}(k,\mu_{n})\oplus
H^{1}(k,\operatorname{\mathrm{{}_{n}Pic}}C)\oplus R^{2}f_{*}\mu_{n}$
$(\star\star)$

The sheaf $R^{2}f_{*}\mu_{n}$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$. The
splitting of the natural map $H^{2}(C,\mu_{n})\to\mathbb{Z}/n\mathbb{Z}$ is
given by $[1]$ mapping to the gerbe $[\mathscr{O}(P)]^{\frac{1}{n}}$.

Let $\alpha\in H^{2}(C,\mu_{n})$ with image $\tilde{\beta}\in
H^{1}(k,\operatorname{\mathrm{{}_{n}Pic}}C)$ under projection in
$(\star\star)$. Let $\beta:\Gamma_{k}\to\operatorname{\mathrm{{}_{n}Pic}}C$ be
a cocycle representing $\tilde{\beta}$. Let $\mathscr{M}_{C}(n,L)$ (resp.
$\mathscr{M}_{C}^{ss}(n,L)$) denote the stack of rank $n$ stable (resp.
semistable) locally free sheaves of $C$ with determinant $L$. Tensoring gives
an action of $\operatorname{\mathrm{{}_{n}Pic}}C$ on these moduli stacks and
we have the twists $\mathscr{M}_{C}^{\beta}(n,L)$ (resp.
$\mathscr{M}_{C}^{ss,\beta}(n,L)$) of these moduli stacks by the cocycle
$\beta$. We also have the twisted $M_{C}^{\beta}(n,L)$ and
$M_{C}^{ss,\beta}(n,L)$. In view of ([17, 3.2.2.6]), the moduli space
$M_{C}^{\alpha}(n,L)$ of $\alpha$-twisted stable sheaves of rank $n$ and
determinant $L$ is precisely the twist $M_{C}^{\beta}(n,L)$ of the moduli
space $M_{C}(n,L)$ by the cocycle $\beta$. We shall identify the space
$M_{C}^{\alpha}(n,L)$ with the corresponding twisted moduli spaces.

##  2\. Hecke correspondence

In this section, we recall the Hecke correspondence, introduced by Narasimhan
and Ramanan ([22]]). This was further investigated by Beauville-Laszlo-Sorger
([1]). We shall use the terminology and notations from ([1, p. 206]). We
recall the definition of the stability index $s(E)$ of a vector bundle $E$ on
$C$.

######  Definition 2.1.

The stability index $s(E)$ is defined to be the minimum of
$\mu(E^{\prime\prime})-\mu(E^{\prime})$ as $(E^{\prime},E^{\prime\prime})$
vary over pairs of locally free sheaves on $C$ which fit into an exact
sequence

$0\to E^{\prime}\to E\to E^{\prime\prime}\to 0.$

The bundle $E$ is stable if and only if $s(E)>0$ and semistable if and only if
$s(E)\geq 0$.

Let $C\to{\mathbb{P}}^{1}$ be a hyperelliptic curve of genus $g\geq 2$ defined
over a field $k$ of characteristic not 2. We assume that $C(k)\neq\emptyset$.
For any line bundle $L$, let $M_{C}(r,L)$ be the moduli space of stable rank
$r$ locally free sheaves of determinant $L$ and $M^{ss}_{C}(r,L)$ the moduli
space of semistable rank $r$ locally free sheaves of determinant $L$. Let $P$
be a closed point of $C(k)$. For $r\geq 2$, let

$\mathscr{E}\to C\times M_{C}(r,\mathscr{O}_{C}(P))$

be the Poincaré bundle ([36], [26]). Let $\mathscr{E}_{P}$ be the restriction
of $\mathscr{E}$ to $P\times M_{C}(r,\mathscr{O}_{C}(P))$. For $0<h<r$, let
$\mathscr{P}=Gr(h,\mathscr{E}_{P})$ be the Grassmannian bundle parametrizing
rank $h$ locally free quotients of $\mathscr{E}_{P}$. A closed point of
$\mathscr{P}$ is a pair $(E,F)$ of vector bundles on $C$ with $E\in
M_{C}(r,\mathscr{O}_{C}(P))$ and $E(-P)\subset F\subset E$ and
dim$(E_{P}/F_{P})=h$ ([1, Section 10]).

We have a morphism $s:\mathscr{P}\to M_{C}(r,\mathscr{O}_{C}(P)$ given by
$s(E,F)=E$. This map is a Grassmann bundle. We recall the following Lemma from
([1, Lemma 10.2]).

######  Lemma 2.2.

Let $(E,F)\in\mathscr{P}$. If $g\geq 2$ and $E$ is general, i.e., $s(E)=g-1$,
then $F$ is semistable with determinant $\mathscr{O}_{C}((1-h)P)$.

We thus have the Hecke correspondence

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss}(r,\mathscr{O}_{C}((1-h)P))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star)\\\
M_{C}(r,\mathscr{O}_{C}(P)).\end{array}$

where the rational map $s^{\prime}$ is defined by $s^{\prime}(E,F)=F$ if $E$
is general. Let V be the open subscheme of $M_{C}(r,\mathscr{O}_{C}(P)$
consisting of $E$ with $s(E)=g-1$ and $U=s^{-1}(V)$. Then $U$ is an open
subscheme of $\mathscr{P}$ and $s^{\prime}/U:U\to
M_{C}^{ss}(r,\mathscr{O}_{C}((1-h)P))$ is a morphism.

In the above Hecke correspondence we may replace $\mathscr{O}_{C}(P)$ by any
line bundle L of degree 1 and get a Hecke correspondence

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss}(r,L\otimes\mathscr{O}_{C}((-h)P))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star)\\\ M_{C}(r,L).\end{array}$

By tensoring with a suitable power of $\mathscr{O}_{C}(P)$, one also gets a
Hecke correspondence

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss}(r,L^{\prime}\otimes\mathscr{O}_{C}((-h)P))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star)\\\
M_{C}(r,L^{\prime}).\end{array}$

for any line bundle $L^{\prime}$ of odd degree.

######  Lemma 2.3.

Let $g=2$ and $L$ a line bundle of degree $1$ on $C$. Then for $E\in
M_{C}(2,L)$, $s(E)=1$.

###### Proof.

Since the stability index of a rank 2 vector bundle is an integer, $E$ being
stable, $S(E)\geq 1$.

Let

$0\to L^{\prime}\to E\to L^{\prime\prime}\to 0$

be an exact sequence with $L^{\prime},L"$ line bundles on $C$. Then,
$s(E)\geq\mu(L^{\prime\prime})-\mu(L^{\prime})=deg(L^{\prime\prime})-deg(L^{\prime})=deg(E)-2deg(L^{\prime})$.
Since $E$ is stable, for any subbundle $L^{\prime}$ of $E$,
$\mu(L^{\prime})<\mu(E)=1/2$ so that $deg(L^{\prime})\leq 0$. We now prove
that there is a line subbundle $L_{0}$ of $E$ of degree zero. This will prove
that $s(E)=1$.

In view of ([22, Lemma 10.2]), there is a line bundle $L_{0}$ on $C$ of degree
0 and an injective map $L_{0}\to E$. Indeed $L_{0}$ is a line subbundle of
$E$; otherwise it would generate a line subbundle $\tilde{L}_{0}$ of degree
$\geq 1$, contradicting $\mu(E)=\frac{1}{2}$. Thus $s(E)=1$. ∎

For the rest of the section, we assume that genus of $C$ is 2, $r=2$. In view
of (2.3), for a line bundle $L$ of degree 1, every element of $M_{C}(2,L)$ has
stability index $1$ so that the morphism $s^{\prime}$ is defined on the whole
of $\mathscr{P}$ and we have the Hecke diagram

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{\longrightarrow}&M_{C}^{ss}(2,L\otimes\mathscr{O}_{C}(-P))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star)\\\ M_{C}(2,L).\end{array}$

where $s$ and $s^{\prime}$ are both morphisms and $s$ is a ${\mathbb{P}}^{1}$
bundle. The map $s^{\prime}$ is a conic bundle. The following lemma asserts
that each fiber of $s^{\prime}$ at a point representing a stable bundle in
$M_{C}^{ss}(2,L\otimes\mathscr{O}_{C}(-P))$ is isomorphic to
${\mathbb{P}}^{1}$.

######  Lemma 2.4.

Let $x\in M_{C}^{s}(2,L\otimes\mathscr{O}_{C}(-P))(k)$ be a stable point
represented by rank 2 vector bundle $F$ on $C$. Then, $x$ belongs to the image
of $s^{\prime}$ and the fiber of $s^{\prime}$ at $x$ is isomorphic to
${\mathbb{P}}^{1}$.

###### Proof.

Let $E^{\prime}$ be a rank two vector bundle satisfying
$F\otimes\mathscr{O}_{C}(-P)\subset E^{\prime}\subset F$ and
$dim(F_{P}/E^{\prime}_{P})=1$. Then
$det(E^{\prime})=det(F)\otimes\mathscr{O}_{C}(-P))=L\otimes\mathscr{O}_{C}(-2P)$
so that $deg(E^{\prime})=-1$. We claim that $E^{\prime}$ is stable. Suppose
not. Then there is a line subbundle $L^{\prime}$ of $E^{\prime}$ such that
$deg(L^{\prime})\geq\mu(E^{\prime})=-1/2$. Thus $deg(L^{\prime})\geq 0$. Let
${\tilde{L}^{\prime}}$ be the line subbundle of $F$ spanned by $L^{\prime}$.
Then $deg({\tilde{L}^{\prime}})\geq deg(L^{\prime})\geq 0$. However $F$ is
stable and $deg({\tilde{L}^{\prime}})<\mu(F)=0$. This leads to a
contradiction. Thus $E^{\prime}$ is stable. We have $E^{\prime}\subset
F\subset E^{\prime}\otimes\mathscr{O}_{C}(P)$ with
$E^{\prime}\otimes\mathscr{O}_{C}(P)$ stable and
$det(E^{\prime}\otimes\mathscr{O}_{C}(P))=L$. The pair
$(E^{\prime}\otimes\mathscr{O}_{C}(P),F)\in\mathscr{P}$ and maps to $F$ under
$s^{\prime}$. The bundles $E^{\prime}$ satisfying
$F\otimes\mathscr{O}_{C}(-P)\subset E^{\prime}\subset F$ and
$dim(F_{P}/E^{\prime}_{P})=1$ are pametrized by the 1-dimensional quotients of
the vector space $F_{P}$. Thus the fiber at $x$ of the map $s^{\prime}$ is
isomorphic to $Gr(1,F_{P})\simeq\mathbb{P}^{1}$. ∎

##  3\. Twisted Hecke Correspondence

Let $C$ be a smooth projective hyperelliptic curve over a field $k$ with a
rational point $P$ . Let $L$ be a line bundle of degree 1 over $C$. Let

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss}(r,L\otimes\mathscr{O}_{C}((-h)P))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star)\\\ M_{C}(r,L).\end{array}$

be the Hecke Correspondence as in §2. Tensoring gives an action of
${}_{r}\mathrm{Pic}C$ on $M_{C}(r,L)$ and $M_{C}(r,{\mathscr{O}}_{C}(-hP))$.
There is also an action of ${}_{r}\mathrm{Pic}C$ rank r sheaves $(E,F)$ on $C$
with $E\in M_{C}(r,\mathscr{O}_{C}(P)),E(-P)\subset F\subset E$ and
dim($E_{P}/F_{P})=h$. The maps $s,s^{\prime}$ are compatible with the action
of ${}_{r}Pic(C)$.

Let $\tilde{\alpha}\in H^{2}(C,G_{m})$ be an $r$-torsion element with a lift
$\alpha\in H^{2}(C,\mu_{r})$ such that $\alpha_{k^{s}}=0$. Such a lift is
possible since $C(k)$ is not empty. Let $\beta:\Gamma_{k}\to~{}_{r}PicC$ be a
cocycle whose class $\tilde{\beta}\in H^{1}(k,_{r}PicC)$ is the image of
$\alpha$ under the projection in (**). Then $\beta$ gives twists
$\mathscr{P}^{\beta}$, $M_{C}(r,L)^{\beta}$ and
$M_{C}(r,L\otimes\mathscr{O}_{C}(-hP))^{\beta}$ together with morphisms

$\begin{array}[]{cccc}\mathscr{P}^{\beta}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss.\beta}(r,L\otimes\mathscr{O}_{C}(-h)P))^{\beta}\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star\star)\\\
M_{C}^{\beta}(r,L).\end{array}$

We call this diagram a twisted Hecke correspondence. We may think of this as a
Hecke correspondence between twisted sheaves of odd and even determinant in
the case $r=2$ and $h=1$.

######  Remark 3.1.

Suppose $r=2,h=1$ and $\alpha\in H^{2}(C,\mu_{2})$ is essentially trivial,
i.e., $\tilde{\alpha}=0$. There is a line bundle $N$ over $C$ such that under
$\delta:PicC/2\to H^{1}(k,{}_{2}\mathrm{Pic}C)$, the class of $N$ maps to
$\tilde{\beta}$. We note that $\alpha_{k^{s}}=0$ implies that the degree of
$N$ is even. Let ${\mathcal{N}}_{\alpha}$ be an invertible $\alpha$-twisted
sheaf such that $({\mathcal{N}}_{\alpha})^{2}=N$ (cf. §1). The map
$(M_{C}^{\beta}(2,\mathscr{O}_{C}(P))\to M_{C}(2,\mathscr{O}_{C}(P))\otimes
N)$ and the map $M_{C}^{ss,\beta}(2,\mathscr{O})\to M_{C}^{ss}(2,N)$ given by
tensoring with ${\mathcal{N}}_{\alpha}$ are isomorphisms which yield
untwisting of the twisted moduli spaces. The twisted Hecke correspondence
diagram

$\begin{array}[]{cccc}\mathscr{P}^{\beta}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss.\beta}(2,\mathscr{O}))\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star\star)\\\
M_{C}^{\beta}(r,\mathscr{O}_{C}(P)).\end{array}$

reduces to the usual Hecke correspondence

$\begin{array}[]{cccc}\mathscr{P}&\buildrel
s^{\prime}\over{-~{}-\longrightarrow}&M_{C}^{ss}(2,N)\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star\star)\\\
M_{C}(2,N\otimes\mathscr{O}_{C}(P))\end{array}$

with degree of $N\otimes\mathscr{O}_{C}(P)$ odd.

##  4\. Local global principle for conic fibrations

In this section we give a proof of a local global principle for rational
points on certain conic fibrations over global field of characteristic not 2
due to Colliot-Thélène. We begin with the following result of Suresh.

######  Theorem 4.1.

(Suresh) Let $k$ be a field of characteristic zero and
$K=k(t_{1},\cdots,t_{n})$. Let $\alpha$ be a quaternion algebra over $K$ which
is unramified on ${\mathbb{P}}^{n}_{k}$ except possibly at a unique
codimension one point of ${\mathbb{P}}^{n}_{k}$. Let $C$ be the conic over $K$
associated to $\alpha$. Then
$\operatorname{\mathrm{Br}}(k)\to\operatorname{\mathrm{Br}}_{nr}(K(C)/k)$ is
onto.

###### Proof.

Let $\tilde{\beta}\in\operatorname{\mathrm{Br}}_{nr}(K(C)/k)$. Since $C$ is a
conic and
$\operatorname{\mathrm{Br}}_{nr}(K(C)/k)\subseteq\operatorname{\mathrm{Br}}_{nr}(K(C)/K)$,
$\tilde{\beta}$ is in the image of $\operatorname{\mathrm{Br}}(K)$ ([8,
Proposition 6.2.1]). Let $\beta\in\operatorname{\mathrm{Br}}(K)$ which maps to
$\tilde{\beta}$.

Let $Z$ be a codimension one point of ${\mathbb{P}}^{n}_{k}$ and $\nu$ the
discrete valuation of $K$ given by $Z$. Let $\kappa(\nu)$ be the residue at
$\nu$. Let $\ell/k(\nu)$ be the cyclic extension given by the residue of
$\alpha$ at $\nu$. Then there exists a discrete valuation $w$ on $K(C)$
extending $\nu$ with $\ell$ as the algebraic closure of $\kappa(\nu)$ in the
residue field $\kappa(w)$ at $w$ ([30, Lemma 6.1]). We have
$\partial_{\nu}(\beta)_{\kappa(w)}=\partial_{w}(\tilde{\beta})=0$. Further if
$\ell\neq\kappa(\nu)$, then $\ell$ is the unique quadratic extension of
$\kappa(\nu)$ contained in $\kappa(w)$.

Suppose that $Z\neq Y$. Then $\ell=\kappa(\nu)$. Since $\kappa(\nu)$ is
algebraically closed in $\kappa(w)$,
$H^{1}(\kappa(\nu),{\mathbb{Q}}/{\mathbb{Z}})\to
H^{1}(\kappa(w),{\mathbb{Q}}/{\mathbb{Z}})$ is injective and
$\partial_{\nu}(\beta)=0$.

Suppose that $Z=Y$. Suppose $\partial_{\nu}(\beta)=0$. Then $\beta$ is
unramified on ${\mathbb{P}}^{n}$ and hence comes from a class in
$\operatorname{\mathrm{Br}}(k)$. Suppose $\partial_{\nu}(\beta)\neq 0$. Since
$\ell/\kappa(\nu)$ is a degree two extension and $\ell$ is the algebraic
closure of $\kappa(\nu)$ in $\kappa(w)$,
$\partial_{\nu}(\beta)=\ell=\partial_{\nu}(\alpha)$. Since $\alpha$ and
$\beta$ are unramified except at $Y$, $\beta-\alpha$ is unramified on
${\mathbb{P}}^{n}$. Hence $\beta-\alpha$ comes from
$\operatorname{\mathrm{Br}}(k)$. Since the image if $\alpha$ in
$\operatorname{\mathrm{Br}}(K(C))$ is zero, $\tilde{\beta}$ is in the image of
$\operatorname{\mathrm{Br}}(k)$. ∎

######  Remark 4.2.

If char$(k)=p>0$ and not equal 2, then the above proof gives a surjection
$\operatorname{\mathrm{Br}}^{\prime}(k)\to\operatorname{\mathrm{Br}}^{\prime}_{nr}(K(C)/k)$
where $\operatorname{\mathrm{Br}}^{\prime}$ denotes the subgroup of
$\operatorname{\mathrm{Br}}$ whose torsion is prime to $p$.

######  Remark 4.3.

In fact in the above theorem we can replace ${\mathbb{P}}^{n}$ by any smooth
projective variety $X$ over $k$ with
$\operatorname{\mathrm{Br}}(k)\to\operatorname{\mathrm{Br}}_{nr}(k(X)/k)$
onto.

We recall the following.

######  Theorem 4.4.

[6, Théorème 2] Let $k$ be a number field and $X/k$ be a smooth surface with
$X\to{\mathbb{P}}^{1}$ be a conic bundle with at most 4 geometric degenerate
fibres. Then the Brauer-Manin obstruction is the only obstruction to the Hasse
principle for $X$. In particular, if
$\operatorname{\mathrm{Br}}(k)\to\operatorname{\mathrm{Br}}(X)$ is onto, then
the Hasse principle holds for $X$.

This theorem is an extension of the case of Châtelet surfaces ([7, Theorem
8.11]). See also Salberger [31, Cor 7.8].

######  Lemma 4.5.

Let $Y\subset{\mathbb{P}}^{n+1}$ be an integral hypersurface of degree $d$
over a number field $k$. Let $A$ be a $k$-point in ${\mathbb{P}}^{n+1}$
outside $Y$. Let $F_{A}$ be the Grassmannian of lines in ${\mathbb{P}}^{n+1}$
passing through $A$. Then there is a Hilbertian set $H\subset F_{A}$ of
$k$-rational points such that for each $\ell\in H$, $\ell\cap Y$ is integral.

###### Proof.

We fix a hypersurface $T\subset{\mathbb{P}}^{n+1}$ with $A\not\in T$. The
morphism ${\mathbb{P}}^{n+1}\setminus\\{A\\}\to T$ sending a point $B$ to the
intersection of the line $AB$ joining $A$ and $B$ with $T$ restricts to a
morphism $f:Y\to T$ of degree $d$. For $M\in H$, the fibre $f^{-1}(M)$ is
simply the line $AM$ intersected with $Y$. Since $T\simeq{\mathbb{P}}^{n}$ and
$k(Y)/k(T)$ is a finite extension of degree $d$, by Hilbert irreducibility
theorem, there is a Hilbertian set $\tilde{H}$ of $k$-rational points in $T$
such that for every $M\in\tilde{H}$, the fibre $f^{-1}(M)$ is integral. We
have an isomorphism $T\buildrel\psi\over{\to}F_{A}$ sending a point $M\in H$
to the line $AM$. The set $H=\psi(\tilde{H})\subset F_{A}$ is the required
Hilbertian set. ∎

######  Theorem 4.6.

(Colliot-Thélène) Let $k$ be a number field. Let $n\geq 2$ be an integer. Let
$Y\subset{\mathbb{P}}^{n}_{k}$ be a geometrically integral quartic
hypersurface. Let $U\subset{\mathbb{P}}^{n}_{k}$ be the complement of $Y$. Let
$f:V\to U$ be a smooth conic bundle. Then the Hasse principle holds for $V$.

###### Proof.

Suppose $\prod_{\nu}V(k_{\nu})\neq\emptyset$.

Let $A\in U(k)$ and $F_{A}$ be the Grassmannian of lines in ${\mathbb{P}}^{n}$
passing through $A$. Then, by (4.5), there is a Hilbertian set $H\subset
F_{A}$ of $k$-rational points such that for each $\ell\in H$, $\ell\cap Y$ is
a closed point of degree 4.

Let $V_{A}=f^{-1}(A)$ be the fibre of $f$ at $A$. Then $V_{A}$ is a smooth
conic and hence locally has rational points at all places outside a finite set
$S$ of places of $k$. For each $\nu\in S$, we pick a point $M_{\nu}\in
V(k_{\nu})$ and set $B_{\nu}=f(M_{\nu})\in U(k_{\nu})$. Since $f:V\to U$ is
smooth, by implicit function theorem, there is a $\nu$-adic neighborhood
$\Omega_{\nu}$ of $B_{\nu}$ and a local section of $f:V(k_{\nu})\to
U(k_{\nu})$ above $\Omega_{\nu}$. We consider for each $\nu\in S$, the line
$\ell_{\nu}\in F_{A}(k_{\nu})$ through $A$ and $B_{\nu}$ in
${\mathbb{P}}^{n}_{k_{\nu}}$. By a theorem of Ekedahl ([11]), since $Y$ is
geometrically integral, $F_{A}$ is rational and $H\subset F_{A}(k)$ is
Hilbertian, the image of $H$ in $\prod_{\nu\in S}F_{A}(k_{\nu})$ is dense. We
pick a line $L\in H$ such that the image of $L$ in $F_{A}(k_{\nu})$ is close
enough to $\ell_{\nu}$ such that $L(k_{\nu})\cap\Omega_{\nu}\neq\emptyset$ for
all $\nu\in S$. We then have the following properties for $L$ :-
1) $\tilde{f}:V_{L}\to L$ is a smooth conic fibration except at a closed point
of degree 4; $\tilde{f}$ is the restriction $f$ to $V_{L}$.
2) $A\in L(k)$ and the conic $\tilde{f}^{-1}(A)=V_{A}$ has a rational point
over $k_{\nu}$ for all $\nu\not\in S$. In particular,
$V_{L}(k_{\nu})\neq\emptyset$ for all $\nu\not\in S$
3) For $\nu\in S$, $L(k_{\nu})\cap\Omega_{\nu}\neq\emptyset$ so that the
section to $f$ over $\Omega_{\nu}$ yields a $k_{\nu}$-rational point on
$V_{L}$; in other words $V_{L}(k_{\nu})\neq\emptyset$ for all $\nu\in S$.
4) $L\cap Y$ is a closed point of degree 4 since $L\in H$.

Thus the conic fibration $V_{L}\to L$ has the property:
$\prod_{\nu}V_{L}(k_{\nu})\neq\emptyset$ and it is smooth outside a closed
point of degree 4 in $L$. By (4.1, 4.4), $V_{L}(k)\neq\emptyset$. In
particular $V(k)\neq\emptyset$. ∎

##  5\. Period-index questions for genus 2 curves over number fields

Let $C$ be a smooth projective geometrically integral curve over a global
field $k$ of characteristic not dividing $n$. We begin with the following
proposition relating the existence of rational points on the twisted moduli
stack to the existence of rational points on the twisted moduli space
associated to a $\mu_{n}$ gerbe on C. The proof of the proposition is due to
Lieblich.

######  Proposition 5.1.

(Lieblich) Let $k$ be a global field of characteristic not $n$ and $C$ a
smooth projective geometrically integral curve over $k$. Let $L$ be a line
bundle of degree one on C. Let $\alpha\in H^{2}(C,\mu_{n})$ be a gerbe which
is locally essentially trivial and $\alpha_{k^{s}}=0$. Let
$\mathscr{M}_{C}^{\alpha}(n,L)$ denote the moduli stack of rank $n$
$\alpha$-twisted stable locally free sheaves of determinant $L$ and
$M_{C}^{\alpha}(n,L)$ the corresponding coarse moduli space. Then
$\mathscr{M}_{C}^{\alpha}(n,L)(k)\neq\emptyset$ if and only if
$M_{C}^{\alpha}(n,L)(k)\neq\emptyset$.

###### Proof.

The map $\mathscr{M}_{C}^{\alpha}(n,L)\to M_{C}^{\alpha}(n,L)$ is a $G_{m}$
gerbe. Let $M=M_{C}^{\alpha}(n,L)$. Suppose $M(k)\neq\emptyset$. The variety
$M$ is geometrically rational with $PicM_{\bar{k}}\simeq{\mathbb{Z}}$ and
$\operatorname{\mathrm{Br}}(M_{\bar{k}})=0$ ([18]). The spectral sequence

$H^{i}(k,H^{j}(M_{\bar{k}},G_{m}))\implies H^{i+j}(M,G_{m}))$

yields a surjection
$\operatorname{\mathrm{Br}}(k)\to\operatorname{\mathrm{Br}}(M)$. Since
$M(k)\neq\emptyset$, this map is an isomorphissm. Thus the gerbe
$\mathscr{M}_{C}^{\alpha}(n,L)\to M$ is defined by a constant Brauer class
$\beta\in\operatorname{\mathrm{Br}}(k)=\operatorname{\mathrm{Br}}(M)$. Since
$\alpha$ is locally essentially trivial, $\alpha=\delta([N_{\nu}])$ for some
line bundle $N_{\nu}$ on $C$. Here $\delta:PicC/2\to
H^{1}(k,{}_{2}\mathrm{Pic}C)$ is the Kummer connecting homomorphism. By
(§1.1),
$\mathscr{M}_{C}^{\alpha_{k_{\nu}}}(n,L)\simeq\mathscr{M}_{C}(n,L\otimes
N_{\nu})$.

Since there exist rank $n$ stable vector bundles with a prescribed
determinant, $\mathscr{M}_{C}(n,L\otimes N_{\nu})$ has a $k_{v}$-rational
point. Hence the gerbe $\mathscr{M}_{C}^{\alpha_{k_{\nu}}}(2,L)\to
M_{C}^{\alpha_{k_{\nu}}}(n,L)$ is the trivial gerbe so that the class of
$\beta$ in $\operatorname{\mathrm{Br}}(k_{\nu})$ is zero. Since this is true
for all places $\nu$ of $k$, the class $\beta$ is zero in
$\operatorname{\mathrm{Br}}(k)=\operatorname{\mathrm{Br}}(M)$. Thus the gerbe
$\mathscr{M}_{C}^{\alpha}(n,L)\to M$ is the trivial gerbe. Since $M(k)$ is not
empty, $\mathscr{M}_{C}^{\alpha}(n,L)(k)$ is not empty. ∎

######  Theorem 5.2.

Let $k$ be a number field and $C$ a smooth projective geometrically integral
hyperelliptic curve over $k$ of genus 2. Suppose $C(k)$ is not empty. Let
$\tilde{\alpha}\in{}_{2}\mathrm{Br}(C)$ be a class which is zero in
$\operatorname{\mathrm{Br}}(C_{k_{\nu}})$ for all places $\nu$ of $k$. Let
$\alpha\in H^{2}(C,\mu_{2})$ be a lift of $\tilde{\alpha}$ with
$\alpha_{\bar{k}}=0$. Then $M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})$ is
isomorphic to ${\mathbb{P}}^{3}$.

###### Proof.

Since the genus of $C$ is two,
$M_{C}^{ss}(2,\mathscr{O}_{C})_{\overline{k}}\simeq{\mathbb{P}}^{3}_{\overline{k}}$
([22, §7, Theorem 2]). Since $M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})$ is a twist
of $M_{C}^{ss}(2,\mathscr{O}_{C})$, it is a Severi-Brauer variety associated
to a central simple algebra $A$ of degree $4$ over $k$. The variety admits a
rational point if and only if the algebra $A$ is a matrix algebra. For every
place $\nu$ of $k$, the element $\alpha_{k_{\nu}}$ is essentially trivial. By
(§1.1), $(M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})_{k_{\nu}}\simeq
M_{C}^{ss}(2,N_{\nu})$ where $N_{\nu}\in PicC_{k_{\nu}}$ and
$\delta([N_{\nu}])=[\alpha_{k_{\nu}}]$ in $H^{2}(C_{k_{\nu}},\mu_{2})$. Hence
$M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})(k_{\nu})\neq\emptyset$. This implies
that $A\otimes k_{\nu}$ is the matrix algebra for every place $\nu$ of $k$. By
Hasse-Brauer-Noether-Albert theoem, the algebra A is a matrix algebra over k.
Hence $M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})$ is isomorphic to
${\mathbb{P}}^{3}$ ∎

######  Theorem 5.3.

Let $k$ be a totally number field and $C$ a smooth projective geometrically
integral hyperelliptic curve over $k$ with a $k$-rational point. Let
$\alpha\in H^{2}(C,\mu_{2})$ be a locally essentially trivial element with
$\alpha_{\bar{k}}=0$. Then for any $P\in C(k)$,
$M_{C}^{\alpha}(2,\mathscr{O}_{C}(P))(k)\neq\emptyset$.

###### Proof.

We have the twisted Hecke correspondence:

$\begin{array}[]{cccc}\mathscr{P}^{\alpha}&\buildrel
s^{\prime}\over{\longrightarrow}&M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})\\\
s\Big{\downarrow}&&&\hskip 56.9055pt(\star\star)\\\
M_{C}^{\alpha}(2,\mathscr{O}_{C}(P)).\end{array}$

By (5.2), $M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})\simeq\mathbb{P}^{3}$. Further
the stable moduli space $U=M_{C}^{s,\alpha}(2,\mathscr{O}_{C})$ is a open
subscheme of $M_{C}^{ss,\alpha}(2,\mathscr{O}_{C})$ whose complement is a
geometrically integral quartic surface $Y$ which over the algebraic closure is
the Kummer surface associated to the canonical involution $x\mapsto-x$ on
$J_{C}$ ([23, Section 4, Remark 2]). Let $V=s^{\prime-1}(U)$. Then
$s^{\prime}:V\to U$ is a smooth conic fibration.

Let $\nu$ be a place of $k$. Since $\alpha_{k_{\nu}}$ is essentially trivial,
the Hecke diagram untwists over $k_{\nu}$. Since $C_{k_{\nu}}$ admits a rank 2
stable bundles of any determinant and the untwisted fibre over such points is
the projective line (2.4), it follows that $V(k_{\nu})\neq\emptyset$. Hence by
(4.6), $V(k)$ is not empty. Any rational point of $V$ maps to a rational point
of $M_{C}^{\alpha}(2,\mathscr{O}_{C}(P))$. ∎

######  Corollary 5.4.

Let $k$ be a number field and $C$ a smooth projective geometrically integral
hyperelliptic curve over $k$ of genus 2 with a zero-cycle of degree one. Then
for every element $\tilde{\alpha}$ in
${}_{2}\Sha\operatorname{\mathrm{Br}}(C)$, index of $\tilde{\alpha}$ divides
2.

###### Proof.

Since the index of elements in ${}_{2}\operatorname{\mathrm{Br}}(C)$ does not
change over an odd degree of $k$, we assume that $C(k)\neq$. If $\alpha\in
H^{2}(C,\mu_{2})$ is a lift of $\tilde{\alpha}$ with $\alpha_{\bar{k}}=0$,
then, by hypothesis, $\alpha$ is locally essentially trivial. Hence by (5.3),
$M_{C}^{\alpha}(2,\mathscr{O}_{C}(P))$ admits a $k$-rational point. By (5.1),
$\mathscr{M}_{C}^{\alpha}(n,L)(k)\neq\emptyset$. If $\mathscr{E}$ is an
$\alpha$ twisted sheaf of rank two, $End(\mathscr{E})$ is a degree two Azumaya
algebra representing the class of $\tilde{\alpha}$ in the Brauer group of $C$.
Thus the index of $\tilde{\alpha}$ divides 2.

∎

######  Corollary 5.5.

Let $k$ be a totally imaginary number field and $C$ a smooth projective
geometrically integral curve of genus 2 over $k$ with a zero-cycle of degree
one. Let $\mathcal{C}$ be a regular proper model of $C$ over the ring of
integers $\mathscr{O}$ in $k$. Then for an
$\tilde{\alpha}\in{}_{2}\mathrm{Br}(\mathcal{C})$, the index of
$\tilde{\alpha}$ divides 2.

###### Proof.

We need only to verify that $\operatorname{\mathrm{Br}}\mathcal{C}$ is
contained in $\Sha\operatorname{\mathrm{Br}}(C)$. Over the valuation ring
${\mathscr{O}}_{\nu}$, $\mathcal{C}_{\mathscr{O}_{\nu}}$ is a relative curve
which is regular and the special fiber is a curve over a finite field. By a
theorem of Grothendieck (cf. [8, Thm 10.3.1]), the Brauer group of
$\mathcal{C}_{\mathscr{O}_{\nu}}$ is zero. ∎

##  6\. Pencils of quadrics and hyperelliptic curves

Let $k$ be a field of characteristic not 2. Let $Q_{1}$ and $Q_{2}$ be two
regular quadratic forms on a vector space $V$ of dimension $2n+2$, $n\geq 2$
over $k$. Let $Q=tQ_{1}-Q_{2}$ be the associated pencil. Let $f(t)=$
disc$(Q)=(-1)^{n+1}$det$(Q)\in k[t]$. The polynomial is determined up to a
square in $k^{*}$. We assume throughout that $f(t)$ is a product of distinct
linear factors over the separable closure $k^{s}$ of $k$. We call such a
pencil a nonsingular pencil. Let $X_{Q}$ be the base locus of the pencil $Q$,
namely the intersection of the quadrics defined by $Q_{1}$ and $Q_{2}$ in
$\mathbb{P}(V)$. The variety $X_{Q}$ is smooth. Let $T=Q_{2}Q_{1}^{-1}$.

For a nonsingular quadratic form $q$ of rank $2n+2$ over $k$, we recall the
definition of the following groups:

$\mathbf{Sim}(q)(k)=\\{\alpha\in\mathbf{GL}(V)\mid\alpha q\alpha^{t}=\lambda
q~{}\text{ for some }\lambda\in k^{*}\\}$
$\mathbf{Sim}^{+}(q)(k)=\\{\alpha\in\mathbf{GL}(V)\mid\alpha
q\alpha^{t}=\lambda q~{}\text{ for some }\lambda\in
k^{*}~{}\text{with}~{}\lambda^{n+1}=\det{\alpha}\\}$
$\mathbf{PSim}^{+}(q)=\mathbf{Sim}^{+}(q)/\mathbf{G}_{\text{m}}.$

We define

$\mathbf{Sim}_{Q}(k)=\\{\alpha\in\mathbf{GL}(V)\mid\alpha
Q_{1}\alpha^{t}=\lambda Q_{1},\alpha Q_{2}\alpha^{t}=\lambda Q_{2}~{}\text{
with }~{}\lambda\in k^{*}\\}.$

Let $\mu:\mathbf{Sim}_{Q}\to\mathbf{G}_{\text{m}}$ be the similarity map given
by $\mu(\alpha)=\lambda$. We also define

$\mathbf{Sim}^{+}_{Q}(k)=\\{\alpha\in\mathbf{GL}(V)\mid\alpha
Q_{1}\alpha^{t}=\lambda Q_{1},\alpha Q_{2}\alpha^{t}=\lambda Q_{2}~{}\text{
with }~{}\lambda\in k^{*},\lambda^{n+1}=\det(\alpha)\\}$

and
$\operatorname{\mathbf{Aut}}^{+}_{Q}=\mathbf{Sim}^{+}_{Q}/\mathbf{G}_{\text{m}}$.

Since $H^{1}(k,\mathbf{G}_{\text{m}})$ is trivial, we have a surjection
$\mathbf{Sim}^{+}_{Q}(k)\to\operatorname{\mathbf{Aut}}^{+}_{Q}(k)$.

We have
$\operatorname{\mathbf{Aut}}^{+}_{Q}\subset\mathbf{PSim}^{+}(Q_{1})\cap\mathbf{PSim}^{+}(Q_{2})\subset\mathbf{PGL}(V)$.

######  Proposition 6.1.

[28, Page 37]
$\operatorname{\mathbf{Aut}}^{+}_{Q}(k^{s})\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{2n}$.

###### Proof.

Let $\lambda_{1},\cdots,\lambda_{2n+2}$ be the distinct zeroes of $f(t)$ in
$k^{s}$. Let $v_{i}$ be a nonzero vector in the radical of the quadratic form
$\lambda_{i}Q_{1}-Q_{2}$, $1\leq i\leq 2n+2$. Then the set of vectors
$\\{v_{1},\cdots,v_{2n+2}\\}$ form an orthogonal basis for $Q_{1}$ and $Q_{2}$
simultaneously. Further after scaling $v_{i}$, $1\leq i\leq 2n+2$, we may
assume that $Q_{1}$ is represented by the identity matrix and $Q_{2}$ is the
diagonal matrix with diagonal entries $\lambda_{1},\cdots,\lambda_{2n+2}$ with
respect to the basis $\\{v_{1},\cdots,v_{2n+2}\\}$ of $V$.

Let $T\in\mathbf{SL}_{2n+2}(k^{s})$ with $TQ_{1}T^{t}=Q_{1}$ and
$TQ_{2}T^{t}=Q_{2}$. Then $T^{-1}=T^{t}$ and $TQ_{2}=Q_{2}T$. Since $Q_{2}$ is
a diagonal matrix with distinct entries on the diagonal, it follows that $T$
is a diagonal matrix with diagonal entries $\pm 1$. Since
$T\in\mathbf{SL}_{2n+2}(k^{s})$, the product of all the entries the diagonal
of $T$ is 1. Since every element in
$\operatorname{\mathbf{Aut}}^{+}_{Q}(k^{s})$ admits a lift
$T\in\mathbf{SL}_{2n+2}(k^{s})$ with $TQ_{1}T^{t}=Q_{2}$ and
$TQ_{2}T^{t}=Q_{2}$, it follows that

$\operatorname{\mathbf{Aut}}^{+}_{Q}(k^{s})=\\{\alpha\in\mathbf{SL}_{2n+2}(k^{s})\mid\alpha\text{
a diagonal matrix with diagonal entries }\pm 1\\}/\\{\pm 1\\}$
$\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{2n}.$

∎

The above proposition can also be deduced from a rational description of the
group scheme $\operatorname{\mathbf{Aut}}^{+}_{Q}$ which is due to J.-P.
Tignol.

######  Proposition 6.2.

Let $u=Q_{2}Q_{1}^{-1}\in GL(V)$. Then $k[u]$ is an étale $k$-algebra of
dimension $2n+2$ and there is an isomorphism of group schemes
$\operatorname{\mathbf{Aut}}^{+}_{Q}\simeq R^{1}_{k[u]/k}(\mu_{2})/\mu_{2}.$

###### Proof.

Since $f(t)$ is the characteristic polynomial of $u$ up to a non-zero scalar
and $f(t)$ is a product of distinct linear factors over $k^{s}$, $k[u]$ is an
étale algebra over $k$ of degree $2n+2$.

Let $\tau_{1}$ denote the adjoint involution on
$\operatorname{\mathrm{End}}(V)$ with respect to $Q_{1}$. Then $\tau_{1}$ is
identity on $k[u]$. Let $\alpha\in\mathbf{Sim}_{Q}(k)$. Then $\alpha$ commutes
with $u$ and $\alpha\tau_{1}(\alpha)=\alpha^{2}=\lambda$. Since $k[u]$ is a
maximal commutative subalgebra of $\operatorname{\mathrm{End}}(V)$, $\alpha\in
k[u]$. Thus

$\mathbf{Sim}_{Q}(k)=\\{\alpha\in k[u]^{*}\mid\alpha^{2}\in k\\}$

and

$\mathbf{Sim}^{+}_{Q}(k)=\\{\alpha\in k[u]^{*}\mid\alpha^{2}=\lambda\in
k,N_{k[u]/k}(\alpha)=\lambda^{n+1}\\}.$

We have exact sequences of group schemes

$1\to\mathbf{Sim}_{Q}/\mathbf{G}_{\text{m}}\to
R_{k[u]/k}(\mathbf{G}_{\text{m}})/\mathbf{G}_{\text{m}}\buildrel
2\over{\to}R_{k[u]/k}(\mathbf{G}_{\text{m}})/\mathbf{G}_{\text{m}}\to 1.$

This exact sequence yields an isomorphism

$R_{k[u]/k}(\mu_{2})/\mu_{2}\simeq\mathbf{Sim}_{Q}/\mathbf{G}_{\text{m}}.$

The homomorphism $\theta:\mathbf{Sim}_{Q}/\mathbf{G}_{\text{m}}\to\mu_{2}$
given by $\theta(\bar{\alpha})=\det(\alpha)\mu(\alpha)^{-n-1}$ makes the
following diagram commute

$\begin{array}[]{ccc}R_{k[u]/k}(\mu_{2})/\mu_{2}&\to&\mathbf{Sim}_{Q}/\mathbf{G}_{\text{m}}\\\
\Big{\downarrow}N_{k[u]/k}&&\Big{\downarrow}\theta\\\ \mu_{1}\hskip
34.1433pt&=&\mu_{2}\par{}{}\end{array}$

Hence we have an isomorphism
$R^{1}_{k[u]/k}(\mu_{2})/\mu_{2}\simeq\mathbf{Sim}^{+}_{Q}/\mathbf{G}_{\text{m}}=\operatorname{\mathbf{Aut}}^{+}_{Q}$.
∎

Let $C$ be the smooth projective hyperelliptic curve given by the affine
equation $y^{2}=f(t)$ and $\theta:C\to\mathbb{P}^{1}$ be the corresponding
fibration.

######  Proposition 6.3.

There is a natural isomorphism
$\theta_{Q}:{}_{2}\mathrm{Pic}C\simeq\operatorname{\mathbf{Aut}}^{+}_{Q}$.

###### Proof.

We shall define this map $\theta_{Q}$ over $k^{s}$ and prove that it descends
to an isomorphism over $k$. Let $\lambda_{1},\cdots,\lambda_{2n+2}$ be the
distinct zeroes of $f(t)$ in $k^{s}$. Over $k^{s}$, we choose a basis
$\\{v_{1},\cdots,v_{2n+2}\\}$ of $V$ for which $Q_{1}$ is represented by the
identity matrix and $Q_{2}$ represented by the diagonal matrix with diagonal
entries $\lambda_{1},\cdots,\lambda_{2n+2}$ (cf. proof of 6.1). The points
$\\{\lambda_{1},\cdots,\lambda_{2n+2}\\}\in\mathbb{A}^{1}_{k^{s}}$ corresponds
to the ramification points of $\theta:C\to\mathbb{P}^{1}$ over $k^{s}$. Let
$P_{1},\cdots,P_{2n+2}\in C(k^{s})$ mapping onto the points
$\lambda_{1},\cdots,\lambda_{2n+2}$ in $\mathbb{A}^{1}$ under $\theta$. Then
${}_{2}\mathrm{Pic}C$ is a free ${\mathbb{Z}}/2{\mathbb{Z}}$-module generated
by the divisor classes $\\{[P_{2n+2}-P_{i}]\mid 1\leq i\leq 2n+1\\}$ with
exactly one relation $\sum_{1}^{2n+1}[P_{2n+2}-P_{i}]=0$. Thus
${}_{2}\mathrm{Pic}C_{k^{s}}\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{2n}$. Let
$\theta_{Q}:{}_{2}\mathrm{Pic}C_{k^{s}}\to\operatorname{\mathbf{Aut}}_{Q_{k^{s}}}^{+}$
be the map
$\theta_{Q}([P_{2n+2}-P_{i}])=<\epsilon_{1},\cdots,\epsilon_{2n+2}>$ modulo
$\mu_{2n+2}$, with $\epsilon_{2n+2}=\epsilon_{i}=-1$ and $\epsilon_{j}=1$ for
$j\neq i,2n+2$. Then $\theta_{Q}$ is an isomorphism of groups. This map is
independent of the choice of basis $\\{v_{1},\cdots v_{2n+2}\\}$ of $V$ over
$k_{s}$. In fact any such choice of a basis $\\{w_{1},\cdots,w_{2n+2}\\}$ of
$V$ with $Q_{1}$ represented by the identity matrix and $Q_{2}$ is represented
by the diagonal matrix $<\lambda_{1},\cdots,\lambda_{2n+2}>$ necessarily is
given by $w_{j}=\epsilon_{j}v_{j}$, $\epsilon_{j}=\pm 1$, $1\leq j\leq 2n+2$.
Hence $\theta_{Q}([P_{2n+2}-P_{i}])$ is well defined, commutes with the Galois
action and descends to a $k$-isomorphism
$\theta_{Q}:{}_{2}\mathrm{Pic}C\simeq\operatorname{\mathbf{Aut}}^{+}_{Q}$. ∎

##  7\. Twisting of Pencils

Let $k$ be a field of characteristic not 2. Let $Q$ be a nonsingular quadratic
form of dimension $2n+2$ over $k$ represented by a symmetric matrix
$q\in\mathbf{GL}_{2n+2}(k)$. Let $\Gamma_{k}$ be the Galois group $k^{s}/k$
and $\gamma:\Gamma_{k}\to\mathbf{PSO}(q)(k^{s})$ be a 1-cocycle. Then the
twisted quadric $q^{\gamma}$ as a variety over $k$ is defined by Tao ([35]).
Let $A$ be a central simple algebra over $k$ of degree $2n+2$ with an
orthogonal involution $\tau$ associated to the cocycle $[\gamma]\in
H^{1}(k,\mathbf{PSO}(q))$ ([16, p. 409]). The twisted quadric $q^{\gamma}$ is
the variety of $(2n+2)$-dimensional totally isotropic ideals in $(A,\tau)$. If
$A=M_{2n+2}(k)$ and $\tau$ is adjoint to a quadratic form $\tilde{q}$ of
dimension $2n+2$, the twisted quadric $q^{\gamma}$ is indeed the quadric
defined by $\tilde{q}$.

Let $Q_{1}$ and $Q_{2}$ be two nonsingular quadratic forms defined on a vector
space $V$ of dimension $2n+2$. Let $Q=tQ_{1}-Q_{2}$ be the associated pencil
in $\mathbb{P}(V)$. We assume that $Q$ is nonsingular pencil. For a choice of
a basis of $V$, we assume that $Q_{1}$ and $Q_{2}$ are symmetric matrices.
Given a cocycle $\alpha:\Gamma_{k}\to\operatorname{\mathbf{Aut}}^{+}(Q)$, we
shall describe the twisted pencil $Q^{\alpha}$. For $\sigma\in\Gamma_{k}$, let
$\alpha_{\sigma}\in\mathbf{SL}_{2n+2}(k^{s})$ such that
$\alpha(\sigma)=\alpha_{\sigma}$ modulo $\mu_{2n+2}$. We have
$\alpha_{\sigma}Q_{1}\alpha_{\sigma}^{t}=\lambda_{\sigma}Q_{1}$ and
$\alpha_{\sigma}Q_{2}\alpha_{\sigma}^{t}=\lambda_{\sigma}Q_{2}$ with
$\lambda_{\sigma}\in k^{s*}$ and $\lambda_{\sigma}^{n+1}=1$. For
$\sigma,\tau\in\Gamma_{k}$, let
$\epsilon_{\sigma,\tau}=\alpha_{\sigma}\sigma(\alpha_{\tau})(\alpha_{\sigma\tau})^{-1}\in\mu_{2n+2}$.
Then $\epsilon_{\alpha}:\Gamma_{k}\to\mu_{2n+2}\subset\mathbf{GL}_{2n+2}$
given by $\epsilon_{\alpha}(\sigma,\tau)=\epsilon_{\sigma,\tau}$ is a
2-cocycle which defines a central simple algebra $A$ over $k$ of degree
$2n+2$.

Case 1: Suppose $[\epsilon_{\alpha}]\in H^{2}(k,\mu_{2n+2})$ is zero. Then
$A=M_{2n+2}(k)$. There exists a function $g:\Gamma_{k}\to\mu_{2n+2}$ such that
$\epsilon_{\alpha}(\sigma,\tau)=g(\sigma)~\\!{}^{\sigma}g(\tau)g(\sigma\tau)^{-1}$.
Then $g(\sigma)^{-1}\alpha_{\sigma}$ is a 1-cocycle with values in
$\mathbf{Sim}(Q_{1})\cap\mathbf{Sim}(Q_{2})$. We have

$(g(\sigma)^{-1}\alpha_{\sigma})Q_{1}(g(\sigma)^{-1}\alpha_{\sigma})^{t}=g(\sigma)^{-2}\lambda_{\sigma}Q_{1}$

and

$(g(\sigma)^{-1}\alpha_{\sigma})Q_{2}(g(\sigma)^{-1}\alpha_{\sigma})^{t}=g(\sigma)^{-2}\lambda_{\sigma}Q_{2}$

with $g(\sigma)^{-2}\lambda_{\sigma}\in\mu_{2n+2}$ is a 1-cocycle. Since
$H^{1}(k,G_{m})$ is zero, there is $\theta\in k^{s*}$ such that
$g(\sigma)^{-2}\lambda_{\sigma}=\theta^{-1}\sigma(\theta)$ for
$\sigma\in\Gamma_{k}$. Let $\theta=\delta^{2}$ with $\delta\in k^{s*}$. Then
$(\delta^{-1}\sigma(\delta))^{2n+2}=(\theta^{-1}\sigma(\theta))^{n+1}=1$. The
map $\Gamma_{k}\to\mathbf{SL}_{2n+2}(k^{s})$ given by
$\sigma\mapsto(\delta\sigma(\delta)^{-1})g(\sigma)^{-1}\alpha_{\sigma}$ is a
1-cocycle with values in $\mathbf{SO}(Q_{1})\cap\mathbf{SO}(Q_{2})$. This
cocycle yields descents $\tilde{Q}_{1}$ and $\tilde{Q}_{2}$ of the quadratic
forms $Q_{1}$ and $Q_{2}$ with det$(\tilde{Q}_{1})=$ det$(Q_{1})$ and
det$(\tilde{Q}_{2})=$ det$(Q_{2})$. Let
$\tilde{Q}=t\tilde{Q}_{1}-\tilde{Q}_{2}$.

There is an isomorphism

$\psi:Q_{k^{s}}\to\tilde{Q}_{k^{s}}$

such that the associated cocycle
$\Gamma_{k}\to\operatorname{\mathbf{Aut}}^{+}_{Q}(k^{s})$ defined by
$\sigma\mapsto\psi^{-1}\sigma\psi\sigma^{-1}$ is precisely
$(\delta\sigma(\delta))^{-1}g(\sigma)^{-1}\alpha_{\sigma}$. Here $\sigma$
stands for the Galois action on $Q_{k^{s}}$ as well $\tilde{Q}_{k^{s}}$. We
call $t\tilde{Q}_{1}-\tilde{Q}_{2}$ the twist of $tQ_{1}-Q_{2}$ with respect
to the cocycle $\alpha$.

Case 2: Suppose $[\epsilon_{\alpha}]\in H^{2}(k,\mu_{2n+2})$ is non-zero. The
map $\Gamma_{k}\to\mathbf{PSO}(Q_{1})$, $\sigma\mapsto$int$(\alpha_{\sigma})$
is a 1-cocycle. This yields a descent $(A,\tau_{1})$ of
$(M_{2n+2}(k^{s}),\tau_{Q_{1}})$ where $A$ is a central simple algebra of
degree $2n+2$ and $\tau_{1}$ an orthogonal involution on $A$. Here
$\tau_{Q_{1}}$ is the adjoint involution $X\mapsto Q_{1}X^{t}Q_{1}^{-1}$ on
the matrix algebra. We note that $A$ is the descent of the cocycle
$\alpha_{\sigma}\in\mathbf{PGL}_{2n+2}(k^{s})$. There is an isomorphism of
algebras with involutions

$\phi:(M_{2n+2}(k^{s}),\tau_{Q_{1}})_{k^{s}}\to(A,\tau_{1})_{k^{s}}$

such that int$(\alpha_{\sigma})=\phi^{-1}\sigma\phi\sigma^{-1}$.

Since

$\tau_{Q_{1}}(Q_{2}Q_{1}^{-1})=Q_{1}(Q_{2}Q_{1}^{-1})^{t}Q_{1}^{-1}=Q_{2}Q_{1}^{-1},$

$Q_{2}Q_{1}^{-1}$ is a symmetric element in $(M_{2n+2}(k),\tau_{Q_{1}})$. For
$\sigma\in\Gamma_{k}$, we prove that
$\sigma\phi(Q_{2}Q_{1}^{-1})=\phi(Q_{2}Q_{1}^{-1})$ thereby proving that
$\phi(Q_{2}Q_{1}^{-1})\in A$. We have

$\begin{array}[]{rcl}\sigma(\phi(Q_{2}Q_{1}^{-1}))&=&\phi~{}int(\alpha_{\sigma})\sigma(Q_{2}Q_{1}^{-1})\\\
&=&\phi((\alpha_{\sigma}Q_{2}\alpha_{\sigma}^{t})((\alpha_{\sigma}^{t})^{-1}Q_{1}^{-1}\alpha_{\sigma}^{-1}))\\\
&=&\phi(\lambda_{\sigma}Q_{2})\lambda_{\sigma}^{-1}Q_{1}^{-1})\\\
&=&\phi(Q_{2}Q_{1}^{-1}).\end{array}$

Thus $u=\phi(Q_{2}Q_{1}^{-1})$ is a symmetric element in $(A,\tau_{1})$. The
pencil $t-u$ on $(A,\tau_{1})\otimes_{k}k[t]$ is called the twisted pencil
associated to $\alpha$.

Suppose $A=M_{2n+2}(k)$ and $\tau_{1}=\tau_{\tilde{Q}_{1}}$. Let
$\tilde{Q}=t\tilde{Q}_{1}-\tilde{Q}_{2}$ be the descent of $tQ_{1}-Q_{2}=Q$
and $\psi:Q_{k^{s}}\to\tilde{Q}_{k^{s}}$ an isomorphism satisfying
$\psi^{-1}\sigma\psi\sigma^{-1}=\alpha_{\sigma}\in\mathbf{PSO}(Q_{1})\cap\mathbf{PSO}(Q_{2})$.
Then

$int(\psi)=\phi:(M_{2n+2}(k^{s}),\tau_{Q_{1}})\to(A,\tau_{\tilde{Q}_{1}})_{k^{s}}$

is an isomorphism with

$\phi^{-1}\sigma\phi\sigma^{-1}=int(\alpha_{\sigma}).$

It is easy to verify that

$\phi(Q_{2}Q_{1}^{-1})=\tilde{Q}_{2}\tilde{Q}_{1}^{-1}=u$

and the pencil $t-u$ over $(M_{2n+2}(k),\tau_{\tilde{Q}_{1}})$ corresponds
under Morita equivalence to the pencil
$(t-u)(\tilde{Q}_{1})=t\tilde{Q}_{1}-\tilde{Q}_{2}$ over $k$.

######  Definition 7.1.

The Grassmannian $I_{s}(Q^{\alpha})$ of $s$-dimensional zero subspaces of the
twisted pencil $Q^{\alpha}$ is the Grassmannian of $s(2n+2)$-dimensional
totally isotropic ideals in $A$ for both the involutions $\tau_{1}$ and
$Int(u)\circ\tau_{1}$.

If $A$ is split and $Q^{\alpha}=t\tilde{Q}_{1}-\tilde{Q}_{2}$,
$I_{1}(Q^{\alpha})$ is simply the base locus $X$ of the intersection of
$\tilde{Q}_{1}$ and $\tilde{Q}_{2}$.

######  Proposition 7.2.

Let $[\alpha_{\sigma}]\in H^{1}(k,\operatorname{\mathbf{Aut}}^{+}(Q))$ and
$Q^{\alpha}$ the twisted pencil associated to $[\alpha_{\sigma}]$. Then
$Q^{\alpha}$ is a pencil of quadrics if and only if the image of
$[\alpha_{\sigma}]$ in $H^{1}(k,\mathbf{PGL}_{2n+2})$ under the composite
$\operatorname{\mathbf{Aut}}^{+}(Q)\to\mathbf{PSO}(Q_{1})\to\mathbf{PGL}_{2n+2}$
is zero. Further if $Q^{\alpha}$ is a pencil of quadrics, then
disc$(Q^{\alpha})=$ disc$(Q)$.

###### Proof.

The only remark is that for a symmetric unit $u$ in the algebra with
involution $(A,\tau)$ over $k$, the pencil $(t-u)$ on the central simple
algebra $A\otimes k[t]$ with the involution $\tau\otimes 1$ is a pencil of
quadrics if and only if $A$ is split. Since the cocycle $\alpha_{\sigma}$ has
values in $\operatorname{\mathbf{Aut}}^{+}(Q)$, it follows that
disc$(Q^{\alpha})=$ disc$(Q)$. ∎

##  8\. Twisted moduli spaces on hyperelliptic curves

Let $C$ be a smooth projective geometrically integral hyperelliptic curve of
genus $g\geq 2$ defined over a field $k$ of characteristic not 2 and
$\theta:C\to{\mathbb{P}}^{1}$ be the hyperelliptic covering. We assume that
there is a rational point $P\in C(k)$ unramified for $\theta$. Let
$Q=\theta(P)\in\mathbb{P}^{1}(k)$. Let $\eta={\mathscr{O}}((2g-1)P)$ and
$M_{2}(C,\eta)$ the moduli space of rank two stable locally free sheaves with
determinant $\eta$ on $C$. We recall a construction of Ramanan (§A)
identifying the moduli space $M_{C}(2,\eta)$ of rank two stable locally free
sheaves of determinant $\eta$ with a certain quadratic Grassmannian. The above
construction over an algebraically closed field goes back to ([10]).

Let $D$ be the divisor on ${\mathbb{P}}^{1}$ of degree $2g+2$ with support the
ramification points of $\theta$ and $\tilde{D}$ the divisor on $C$ of degree
$2g+2$ mapping isomorphically onto $D$. Let $s$ be a general section in
$H^{0}(\mathbb{P}^{1},D)$. Restricted to
$\mathbb{A}^{1}=\mathbb{P}^{1}-\\{Q\\}$, $s$ is given by a polynomial $f(t)$
of degree $2g+2$. Further, $Q$ being a rational point of $\mathbb{P}^{1}$
which splits into two rational points $P$ and $P^{\prime}$ in $C(k)$, leading
coefficient of $f(t)$ is a square which we assume to be one after scaling $s$.
The affine equation of $\theta$ restricted to
$\mathbb{A}^{1}=\mathbb{P}^{1}-\\{Q\\}$ is given by $y^{2}=f(t)$ with $f(t)$
monic and has distinct zeroes $\lambda_{1},\cdots,\lambda_{2n+2}$ in $k^{s}$
corresponding to the support of $D$ over $k^{s}$.

Let $\eta={\mathscr{O}}((2g-1)P)$. Then
$V=H^{0}(\tilde{D},\eta\mid_{\tilde{D}})$ is a vector space of dimension
$2g+2$. Associated to $\eta$, there is a nonsingular pencil $Q_{\eta}$ in $V$
(A.5) with disc$(Q_{\eta})=\lambda f(t)$ for some scalar $\lambda$. The pencil
$Q_{\eta}$ at the point $Q$ is split so that disc$(Q_{\eta})_{t=\infty}=1$.
Thus $\lambda$ is a square and after scaling we assume disc$(Q_{\eta})=f(t)$
with $f(t)$ monic.

Let $M_{C}(2,\eta)$ be the moduli space of rank two stable locally free
sheaves with determinant $\eta$ on $C$. Let $I_{g-1}(Q_{\eta})$ be the
Grassmannian of $(g-1)$ dimensional linear subspaces contained in the base
locus $X_{\eta}$ of $Q_{\eta}$. Then an isomorphism

$\phi_{\eta}:M_{C}(2,\eta)\to I_{g-1}(Q_{\eta})$ $(\star\star\star)$

over $k$ is constructed in (A.14). In fact this isomorphism commutes with the
action of ${}_{2}\mathrm{Pic}(C)$ on both sides (A.15). We shall now discuss a
twisted version of this isomorphism.

Let $\alpha\in H^{2}(C,\mu_{2})$ be a $\mu_{2}$-gerbe. Let
$M_{C}^{\alpha}(2,\eta)$ be the moduli space of $\alpha$-twisted rank 2 stable
locally free sheaves with determinant $\eta$ over $C$. Let
$\beta:\Gamma_{k}\to\operatorname{\mathrm{{}_{2}Pic}}C(k^{s})$ be a 1-cocycle
with cocycle class $[\beta]$ being the image of $\alpha$ under projection in
$(\star\star)$ of §3. By §1.3, $M_{C}^{\alpha}(2,\eta)$ is the Galois twist of
$M_{C}(2,\eta)^{\beta}$ of the moduli space $M_{C}(2,\eta)$ by the cocycle
$\beta$.

On the other hand, the isomorphism
$\theta_{Q_{\eta}}:\operatorname{\mathrm{{}_{2}Pic}}(C)\to\operatorname{\mathbf{Aut}}^{+}(Q_{\eta})$
defined in (6.3) gives a cocycle $\theta_{Q_{\eta}}\circ\beta$ with values in
$\operatorname{\mathbf{Aut}}^{+}(Q_{\eta})$. We denote by $Q_{\eta}^{\alpha}$
the twisted pencil associated to the 1-cocycle $\theta_{Q_{\eta}}\circ\beta$.
Then $I_{g-1}(Q_{\eta}^{\alpha})$ is simply a descent of $I_{g-1}(Q_{\eta})$
with respect to $\theta_{Q_{\eta}}\circ\beta$; i.e.

$I_{g-1}(Q_{\eta}^{\alpha})=I_{g-1}(Q_{\eta})^{\beta}.$

Twisting both sides of $(\star\star\star)$ by $\alpha$, we get a
$k$-isomorphism

$\phi_{\eta}^{\alpha}:M_{C}^{\alpha}(2,\eta)\to I_{g-1}((Q_{\eta})^{\alpha}).$

We record this in the following

######  Theorem 8.1.

Let $C$ be a smooth projective geometrically integral hyperelliptic curve of
genus $g\geq 2$ defined over a field $k$ of characteristic not 2 and
$\theta:C\to\mathbb{P}^{1}$ be a double cover. Suppose that there exists $P\in
C(k)$ unramified for $\theta$. Let $\eta={\mathscr{O}}((2g-1)P)$ and
$\alpha\in H^{2}(C,\mu_{2})$ a $\mu_{2}$-gerbe. Then there is a
$k$-isomorphism

$\phi_{\eta}^{\alpha}:M_{C}^{\alpha}(2,\eta)\to I_{g-1}(Q_{\eta}^{\alpha})$

which descends the isomorphism $\phi_{\eta}:M_{C}(2,\eta)\to
I_{g-1}(Q_{\eta})$.

##  9\. Soluble pencils

Let $k$ be a field of characteristic not 2. A nonsingular pencil
$Q=tQ_{1}-Q_{2}$ on a vector space $V$ of dimension $2n+2$ is soluble if there
is subspace of $V$ of dimension $n$ on which both $Q_{1}$ and $Q_{2}$ are
identically zero; in other words, the base locus $X_{Q}$ contains an
$n$-dimensional linear subspace.

Let $C$ be a smooth projective geometrically integral hyperelliptic curve of
genus $g\geq 2$ defined over $k$ and $\theta:C\to\mathbb{P}^{1}$ be a double
cover. Suppose that there exists $P\in C(k)$ unramified for $\theta$. Let
$Q=\theta(P)\in\mathbb{P}^{1}(k)$. Choosing
$\mathbb{A}^{1}=\mathbb{P}^{1}-\\{Q\\}$, we may write the affine equation of
$C$ by

$y^{2}=f(t)$

with $f(t)$ monic of degree $2g+2$. Let $\eta$ be the line bundle on $C$ given
by $\eta={\mathscr{O}}((2g-1)P)$. Let $\tilde{D}$ the divisor on $C$ mapping
isomorphically onto the divisor $D$ on $\mathbb{P}^{1}$ which is the
ramification divisor for $\theta$. Then $D\subset\mathbb{A}^{1}$ and the
support of $D$ is precisely the set zeros of $f(t)$. Let
$V=H^{0}(\tilde{D},\eta\mid_{\tilde{D}})$. The dimension of $V$ is $2g+2$ and
there is a pencil of quadrics $Q_{\eta}$ in $\mathbb{P}(V)$ (§A) satisfying
the following properties.

######  Proposition 9.1.

1) The pencil $Q_{\eta}$ is soluble.
2) Suppose $Q_{\eta}=tQ_{\eta}^{1}-Q_{\eta}^{2}$. Then $Q^{1}_{\eta}$ is
hyperbolic and disc$(Q_{\eta})=f(t)$.

###### Proof.

1) By (§A.5) there is a $g$-dimensional linear subspace in $V$ on which
$Q_{\eta}^{1}$ and $Q_{\eta}^{2}$ vanish identically. Thus $Q_{\eta}$ is a
soluble pencil.

2) By (§A.5) disc$(Q_{\eta})=\lambda f(t)$ for some $\lambda\in k^{*}$ and
$(Q_{\eta})_{Q}$ is hyperbolic. Since $(Q_{\eta})_{Q}=Q^{1}_{\eta}$,
disc$(Q_{\eta}^{1})=$ square class of leading coefficient of $f(t)$ $=1$. Thus
$\lambda$ is a square and disc$(Q_{\eta})=f(t)$. ∎

There is another example of a soluble pencil constructed in ([37]). We recall
this construction. Let $f(t)\in k[t]$ be a monic polynomial of degree $2n+2$
with no multiple zeros. Then $L=k[t]/(f(t))$ is an étale $k$-algebra of
dimension $2n+2$. Let $\beta\in L$ be the class of $t$. Then
$\\{1,\beta,\cdots,\beta^{2n+1}\\}$ is a basis of $L$ over $k$. We define a
bilinear form $Q_{1}^{0}:L\times L\to k$ by

$Q_{1}^{0}(\lambda,\mu)={\rm~{}coefficient~{}of~{}}\beta^{2n+1}{\rm~{}in~{}}\lambda\mu.$

Then $Q_{1}^{0}$ is hyperbolic. Let $T_{0}:L\to L$ denote the linear map given
by multiplication by $\beta$. Then $T_{0}$ is a self adjoint operator with
respect to $Q_{1}^{0}$ and the characteristic polynomial of $T_{0}$ is $f(t)$.
Let $Q_{2}^{0}=Q_{1}^{0}T_{0}$. Then $Q_{0}=tQ_{1}^{0}-Q_{2}^{0}$ is a
nonsingular pencil with discriminant $f(t)$. Since $Q_{1}^{0}$ and $Q_{2}^{0}$
are identically zero on the $n$-dimensional subspace generated by
$\\{1,\beta,\cdots\beta^{n-1}\\}$ of $L$, $Q_{0}$ is a soluble pencil.

Let $C\to\mathbb{P}^{1}$ be the smooth projective hyperelliptic curve given by
the affine equation $y^{2}=f(t)$ and
$\theta_{Q_{0}}:{}_{2}\mathrm{Pic}C\simeq\operatorname{\mathbf{Aut}}^{+}_{Q_{0}}$
the isomorphism given in (6.3). Let
$c:\operatorname{\mathbf{Aut}}^{+}(Q_{0})\to\mathbf{PSO}(Q_{1}^{0})$ be the
inclusion. Then

$\frac{\operatorname{\mathrm{Pic}}C}{2}\buildrel\delta\over{\to}H^{1}(k,{}_{2}\mathrm{Pic}C)\buildrel
c\theta_{Q_{0}}\over{\to}H^{1}(k,\mathbf{PSO}(Q_{1}^{0}))$ $(\\#)$

is a complex ([34, Theorem 10]).

######  Proposition 9.2.

Let $f(t)\in k[t]$ be a monic polynomial of degree $2n+2$ with no multiple
zeros. Let $C\to\mathbb{P}^{1}$ be the smooth projective hyperelliptic curve
given by the affine equation $y^{2}=f(t)$. Let $Q=tQ_{1}-Q_{2}$ be a soluble
pencil with disc$(Q)=f(t)$. Let
$\theta_{Q}:{}_{2}\mathrm{Pic}C\simeq\operatorname{\mathbf{Aut}}^{+}_{Q}$ be
the isomorphism given in (6.3). Then the following sequence

$\frac{\operatorname{\mathrm{Pic}}C}{2}\buildrel\delta\over{\to}H^{1}(k,{}_{2}\mathrm{Pic}C)\buildrel
c\theta_{Q}\over{\to}H^{1}(k,\mathbf{PSO}(Q_{1}))$ $(\star\star\star\star)$

is a complex.

We begin with the following

######  Lemma 9.3.

Let $Q=tQ_{1}-Q_{2}$ be a soluble pencil with disc$(Q)=f(t)$ a monic
polynomial with no multiple zeros. With the notation as in (9.2), there is a
1-cocycle $\beta:\Gamma_{k}\to{}_{2}\mathrm{Pic}C$ whose class in
$H^{1}(k,{}_{2}\mathrm{Pic}C)$ belongs to the image of
$\delta:\frac{\operatorname{\mathrm{Pic}}C}{2}\to
H^{1}(k,{}_{2}\mathrm{Pic}C)$ such that $Q=Q_{0}^{\beta}$.

###### Proof.

Since $Q$ is soluble, $Q_{1}$ contains an $n$-dimensional linear subspace. We
have disc$(Q)=$ disc$(tQ_{1}-Q_{2})=f(t)$ with $f(t)$ monic. Hence
disc$(Q_{1})=$ leading coefficient of $f(t)$ modulo squares = 1. Hence $Q_{1}$
is hyperbolic and $Q_{1}$ is isomorphic to $Q_{1}^{0}$. Hence $Q$ is
isomorphic to $tQ_{1}^{0}-Q_{1}^{\prime}$ for some regular quadratic form
$Q_{1}^{\prime}$. The pencil $tQ_{1}^{0}-Q_{1}^{\prime}$ is soluble and in
view of ([34, Theorem 10]), there is
$[N]\in\frac{\operatorname{\mathrm{Pic}}C}{2}$ whose image $\tilde{\beta}\in
H^{1}(k,{}_{2}\mathrm{Pic}C)$ under $\delta$ gives the twist
$(Q_{0})^{\beta}\simeq tQ_{1}^{0}-Q_{1}^{\prime}$; here $\beta$ is a cocycle
representing $\tilde{\beta}$. Since $Q\simeq tQ_{1}^{0}-Q_{1}^{\prime}$, we
get $Q\simeq(Q_{0})^{\beta}$. ∎

We now recall the twisting bijection in Galois cohomology ([33]). Let $G$ be a
discrete $\Gamma_{k}$-group and $a:\Gamma_{k}\to G$ be a cocycle. Let
${}_{a}G$ be the twist of $G$ by the cocycle
$\tilde{a}:\Gamma_{k}\to\operatorname{\mathrm{Aut}}(G)$ given by
$\tilde{a}(\sigma)(g)=a(\sigma)\sigma(g)a(\sigma)^{-1}$. The twisting mp
${}_{a}T:H^{1}(k,~{}_{a}G)\to H^{1}(k,G)$ is defined by ${}_{a}T(b)=ba$ at the
level of cocycles. Indeed one verifies that if $b:\Gamma_{k}\to~{}_{a}G$ is a
1-cocycle, the map $\Gamma_{k}\to G$, $\sigma\mapsto b(\sigma)a(\sigma)$ is a
1-cocycle. The twisting map ${}_{a}T$ is a bijection which sends $[1]$ to
$[a]\in H^{1}(k,G)$.

Proof of the proposition: Since $Q=tQ_{1}-Q_{2}$ is a soluble pencil with
disc$(Q)=f(t)$, by (9.3), there is
$\alpha\in\frac{\operatorname{\mathrm{Pic}}C}{2}$ such that if
$\tilde{\beta}=\theta_{Q_{0}}\circ\delta(\alpha)\in
H^{1}(k,\operatorname{\mathbf{Aut}}^{+}(Q_{0}))$, a cocycle $\beta$ associated
to $\tilde{\beta}$ gives an isomorphism $(Q_{0})^{\beta}\simeq Q$. Hence
$\operatorname{\mathbf{Aut}}(Q_{0})^{\beta}=\operatorname{\mathbf{Aut}}(Q)$,
$(\mathbf{PSO}(Q_{1}^{0}))^{\beta}=\mathbf{PSO}(Q_{1})$. We have a commutative
diagram of maps

$\begin{array}[]{cccccc}H^{1}(k,{}_{2}\mathrm{Pic}C)&\buildrel\theta_{Q}\over{\to}&H^{1}(k,\operatorname{\mathbf{Aut}}^{+}(Q))&\buildrel
c\over{\to}&H^{1}(k,\mathbf{PSO}(Q_{1}))\\\
{}_{\delta(\alpha)}T\Big{\downarrow}&&{}_{\beta}T\Big{\downarrow}&&{}_{c\beta}T\Big{\downarrow}\\\
H^{1}(k,{}_{2}\mathrm{Pic}C)&\buildrel\theta_{Q_{0}}\over{\to}&H^{1}(k,\operatorname{\mathbf{Aut}}^{+}(Q_{0}))&\buildrel
c\over{\to}&H^{1}(k,\mathbf{PSO}(Q^{0}_{1})).\end{array}$

Here $({}_{2}\mathrm{Pic}C)^{\delta}(\alpha)={}_{2}\mathrm{Pic}C$ and
${}_{\delta(\alpha)}T(f)=f+\delta(\alpha)$ for all $f\in
H^{1}(k,{}_{2}\mathrm{Pic}C)$.

We now prove that the sequence

$\frac{\operatorname{\mathrm{Pic}}C}{2}\buildrel\delta\over{\to}H^{1}(k,{}_{2}\mathrm{Pic}C)\buildrel
c\theta_{Q}\over{\to}H^{1}(k,\mathbf{PSO}(Q_{1}))$

is a complex.

Let $\gamma\in\frac{\operatorname{\mathrm{Pic}}C}{2}$. Then

${}_{c\beta}Tc\theta_{Q}\delta(\gamma)=c\theta_{Q_{0}}~{}_{\delta(\alpha)}T(\delta(\gamma))=c\theta_{Q_{0}}(\delta(\alpha+\gamma))=0$

in view of $(\\#)$.

Since $Q$ is a soluble pencil with disc$(Q)=f(t)$ monic, $Q_{1}$ is hyperbolic
and $Q_{1}\simeq Q_{1}^{0}$. Since $Q_{1}=(Q_{1}^{0})^{c\beta}$, $c\beta=0\in
H^{1}(k,\mathbf{PSO}(Q_{0}))$ and ${}_{c\beta}T$ is a bijection. Thus,
$(c\theta_{Q})\circ\delta$ is zero and $(\star\star\star\star)$ is a complex.
$\Box$.

######  Corollary 9.4.

Let $k$ be a number field and $C/k$ a smooth projective geometrically integral
hyperelliptic curve of genus $g\geq 2$. Suppose that $C$ admits a rational
point $P$ which is unramified for the covering $C\to\mathbb{P}^{1}$. Let
$\eta=\mathscr{O}((2g-1)P)$ and $Q_{\eta}$ the pencil of quadrics in (§A). Let
$\alpha\in H^{2}(C,\mu_{2})$ be a gerbe which is locally essentially trivial.
Then $Q_{\eta}^{\alpha}$ is a pencil of quadrics and the twisted moduli space
$M_{C}^{\alpha}(2,\eta)$ is isomorphic to the Grassmannian
$I_{g-1}(Q_{\eta}^{\alpha})$ of the pencil of quadrics $Q_{\eta}^{\alpha}$.

###### Proof.

Since $\alpha$ is locally essentially trivial, $\alpha\in
Im(\delta_{k_{\nu}}:\frac{\operatorname{\mathrm{Pic}}C_{k_{\nu}}}{2}\to
H^{2}(C_{k_{\nu}},\mu_{2}))$ for all places $\nu$ of $k$. Let $\beta\in
H^{1}(k,{}_{2}\mathrm{Pic}C)$ be the image of $\alpha$. Then $\beta\in
Im(\frac{\operatorname{\mathrm{Pic}}C_{k_{\nu}}}{2}\to
H^{1}(k_{\nu},{}_{2}\mathrm{Pic}C))$ for all places $\nu$ of $k$. By (9.2), we
have $c\theta_{Q_{\eta}}(\beta)=0\in
H^{1}(k_{\nu},\mathbf{PSO}(Q_{\eta}^{1}))$ for all places $\nu$ of $k$. In
particular the image of $c\theta_{Q_{\eta}}(\beta)$ in
$H^{1}(k_{\nu},\mathbf{PGL}_{2g+2})$ is trivial for all places $\nu$ of $k$.
Let $A$ be a central simple algebra representing the image of
$c\theta_{Q_{\eta}}(\beta)$ in $H^{1}(k,\mathbf{PGL}_{2g+2})$. Then
$A\otimes{k_{\nu}}$ is a matrix algebra for all places $\nu$ of $k$. By Hasse-
Brauer-Noether-Albert theorem ([2]), $A$ is a matrix algebra. Hence by (7.2),
$Q_{\eta}^{\alpha}$ is a pencil. ∎

##  10\. Pencils of quadrics in $\mathbb{P}^{5}$ over number fields

In this section we prove the following weak form of a Hasse principle for a
smooth intersection of two quadrics in $\mathbb{P}^{5}$ over a number field.

######  Theorem 10.1.

Let $k$ be a global field of characteristic not $2$. Let $X$ be a smooth
intersection of two quadrics $Q_{1}$ and $Q_{2}$ in $\mathbb{P}^{5}$. Suppose
$disc(Q_{1})=1$. If $X(k_{\nu})$ contains a line for all places $\nu$ of $k$,
then, $X(k)$ is nonempty.

###### Proof.

Let $Q=tQ_{1}-Q_{2}$ and $f(t)=disc(Q)$. Since $X=Q_{1}\cap Q_{2}$ is smooth,
$f(t)$ has distinct linear factors. Since $disc(Q_{1})=1$, $f(t)$ has leading
coefficient a square, which we assume to be 1 after scaling. Further the
degree of $f(t)$ is 6. Let $C$ be the smooth projective hyperelliptic curve
given by the affine equation $y^{2}=f(t)$ and $\gamma:C\to{\mathbb{P}}^{1}$ be
the corresponding covering. Since $f(t)$ is monic, the point $Q$ at infinity
of $\mathbb{P}^{1}$ splits into two rational points in $C(k)$. Let $P$ be a
point of $C(k)$ mapping to $Q$ under $\gamma$.

Let $Q_{\eta}$ be the pencil of quadrics in $\mathbb{P}^{5}$ constructed by
Ramanan in (§A) with reference to the choice of the line bundle
$\eta=\mathscr{O}((2g-1)P)$ (cf. §5). Then $Q_{\eta}$ is a soluble pencil with
disc($Q_{\eta}$)$=f(t)$. Hence there is an isomorphism $Q_{\eta,k^{s}}\simeq
Q_{k^{s}}$. Let $\beta:\Gamma_{k}\to{}_{2}\mathrm{Pic}C$ be a cocycle such
that $Q=Q_{\eta}^{\beta}$. Let the class $\tilde{\beta}\in
H^{1}(k,{}_{2}\mathrm{Pic}C)$ of $\beta$ be the image of $\alpha\in
H^{2}(C,\mu_{2})$ with $\alpha_{k^{s}}=0$. We claim that $\alpha$ is locally
essentially trivial.

By hypothesis, the pencil $Q_{k_{\nu}}$ is soluble with disc ($Q_{1})$ = 1. By
(9.3), there is a 1-cocycle
$\theta_{\nu}:\Gamma_{k_{\nu}}\to{}_{2}\mathrm{Pic}C_{k_{\nu}}$ whose class
belongs to the image of $\delta:PicC_{k_{\nu}}/2\to
H^{1}(k_{\nu},{}_{2}\mathrm{Pic}C)$ such that
$Q_{k_{\nu}}=Q_{0}^{\theta_{\nu}}$, where $Q_{0}$ is the pencil of Arul-Wang
(cf. §5). Since $Q_{\eta}$ is also a soluble pencil with $Q_{\eta}^{1}$
hyperbolic, there is a 1-cocycle $\phi:\Gamma_{k}\to{}_{2}\mathrm{Pic}C$ whose
class belongs to the image of $\delta$ and $Q_{0}=Q_{\eta}^{\phi}$. Thus over
$k_{\nu}$, $Q=Q_{\eta}^{\theta_{\nu}+\phi}$. Since $Q=Q_{\eta}^{\beta}$, we
have, locally, $[\beta]=[\theta_{v}+\phi]$ which is in the image of
$\delta:PicC_{k_{\nu}}/2\to H^{1}(k_{\nu},{}_{2}\mathrm{Pic}C)$. Thus $\alpha$
maps to zero in $H^{2}(C_{k_{\nu}},G_{m})$ and $\alpha$ is locally essentially
trivial.

We have an isomorphism (9.4)

$\phi_{\eta}^{\alpha}:M_{C}^{\alpha}(2,\eta)\to
I_{1}(Q_{\eta}^{\alpha})=I_{1}(Q)=X.$

Since $\alpha$ is locally essentially trivial and degree($\eta$) is odd, by
(5.3), $M_{C}^{\alpha}(2,\eta)(k)$ is nonempty. Since $\phi_{\eta}^{\alpha}$
is an isomorphism, $X(k)$ is not empty and this completes the proof of the
theorem. ∎

Let $Q=tQ_{1}-Q_{2}$ be a pencil of quadrics in $\mathbb{P}^{5}$ with
disc$(Q)=f(t)$. Let $C$ be the smooth projective hyperelliptic curve $C$ given
by $y^{2}=f(t)$. The next corollary, as pointed out by Colliot-Thélène,
replaces the condition disc$(Q_{1})=1$ in (10.1) with $C(k)\neq\emptyset$.

######  Corollary 10.2.

Let $k$ be a global field of characteristic not $2$. Let $Q=tQ_{1}-Q_{2}$ be a
pencil of quadrics in $\mathbb{P}^{5}$ with disc$(Q)=f(t)$ and $X=Q_{1}\cap
Q_{2}$ smooth. Suppose the hyperelliptic curve $C$ given by $y^{2}=f(t)$ has a
zero-cycle of degree 1. If $X(k_{\nu})$ contains a line for all places $\nu$
of $k$, then, $X(k)$ is nonempty.

###### Proof.

The curve $C$ having a rational point which is unramified for the
hyperelliptic covering $C\to{\mathbb{P}}^{1}$ is equivalent to writing the
pencil $Q$ as $tQ_{1}-Q_{2}$ with disc$(Q_{1})=1$ for a suitable rational
point of ${\mathbb{P}}^{1}$ at infinity.

Since $C$ has a zero-cycle of degree 1, by ([5, p. 599]), $C$ has a zero cycle
$\sum n_{i}x_{i}$ of degree one with $x_{i}$ closed points of $C$ which are
unramified over ${\mathbb{P}}^{1}$. Let $L_{i}=\kappa(x_{i})$ be the residue
field at $x_{i}$. Then there exists $P_{i}\in C(L_{i})$ which is unramified
over ${\mathbb{P}}^{1}_{L_{i}}$. By (10.1), $X(L_{i})\neq\emptyset$ for all
$i$. Since $\sum n_{i}[L_{i}:k]=1$, $[L_{j}:k]$ is odd for some $j$. Since
$X(L_{j})\neq\emptyset$, the quadratic form $tQ_{1}-Q_{2}$ is isotropic over
$L_{j}(t)$ and by Springer’s theorem, $tQ_{1}-Q_{2}$ is isotropic over $k(t)$.
Using Amer Brumer theorem, one concludes that $X(k)\neq\emptyset$. ∎

##  Appendix A A note on hyperelliptic curves and pencils of quadrics

S. Ramanan

###  A.1. Introduction

There is a deep relationship between a hyper-elliptic curve of genus $g>0$,
and a pencil of quadrics in $\mathbb{C}^{2g+2}$. This goes back to Gauthier
([12]). The space of $(g+1)$-dimensional (maximal) isotropic subspaces for a
non-singular quadric has two connected components. If the quadric has rank 1
less, maximal isotropic subspaces are still $(g+1)$-dimensional but the space
of such subspaces is connected. Thus pairs consisting of a quadric of the
pencil (assuming it is generic) and a component of the space of maximal
isotropic subspaces in that quadric, constitute a 2-sheeted covering over
$\mathbb{P}^{1}$ or what is the same, a hyper-elliptic curve. Hence it is not
surprising that many constructions over the hyper-elliptic curve have
corresponding geometric interpretations in terms of the pencil.

In particular, initially the Jacobian of the curve was described in terms of
the variety of $g$-dimensional isotropic spaces for all quadrics of the
pencil. Narasimhan and I in the case $g=2$, and Desale and I ([10]) in the
general case, gave a similar description for the moduli varieties of vector
bundles of rank $2$ over the curve, in terms of the geometry of the pencil.
Still later, I gave such a meaning ([27]) to the variety of isotropic
subspaces of other dimensions in terms of suitable moduli on the curve.

More recently, Manjul Bhargava [3]) considered similar properties over a
number field. I am giving below a proof of the results mentioned above in such
a way that they are valid for curves over any field of characteristic $\neq
2$. This requires an assumption that the curve itself admits a rational point.
I have given full details of the construction in proving the isomorphism of
the Jacobian with the variety of $g$-dimensional isotropic subspaces for the
pencil and have sketched how similar considerations in the case of vector
bundles of rank 2 are also valid over such fields.

Parimala Raman and Jaya Iyer have been studying Selmer groups and Hasse
principle etc. and wished for such a proof. I am thankful to Parimala for
encouraging me to re-visit what I had done half a century ago.

###  A.2. Preliminaries and Notation

All considerations below are over any field $k$ of characteristic $\neq 2$.

Let $U$ be a 2-dimensional vector space and $P=P(U)$ the corresponding
projective line. A general element $s$ (with all roots nonzero and distinct
over the algebraic closure) of
$\Gamma(P,{\mathscr{O}}(2g+2))=S^{2g+2}(U^{*}),g\geq 2$ defines an hyper-
elliptic curve $C$ of genus $g$, namely
$Spec(\mathscr{O}\oplus\mathscr{O}(-g-1))$ over $P$, with the obvious algebra
structure on this $\mathscr{O}_{P}$-Module, which uses the map
$Sym^{2}\mathscr{O}(-g-1)=\mathscr{O}(-2g-2)\to\mathscr{O}$ given rise to by
$s$. This realises $C$ as a $2$-sheeted covering $\pi:C\to P$.

The automorphism of the sheaf of algebras
$Spec(\mathscr{O}\oplus\mathscr{O}(-g-1))$ acting as $+1$ on $\mathscr{O}$ and
$-1$ on $\mathscr{O}(-g-1)$ gives rise to an involution $\iota$ of $C$ and the
quotient is $P$. The $\iota$-fixed divisor $W$ (of degree $2g+2$) in $C$ maps
isomorphically on the ramification divisor $D$ (which is the divisor in $P$
defined by $s$). The hyperplane bundle $\mathscr{O}(1)$ on $P$ has as inverse
image, a line bundle $h$ on $C$ of degree $2$. The divisor class of $D$ (resp.
$W$) is $\mathscr{O}(2g+2)$ (resp. $h^{g+1}$). The section $s$ lifts to a
section of $h^{2g+2}$ and is the square of a section of $h^{g+1}$ (unique up
to $\pm 1$) with $W$ as the divisor defined by it.

For any line bundle $L$ on $C$ of degree $d$, we have an isomorphism (unique
up to a scalar multiple) $L\otimes{\iota^{*}(L)}\simeq h^{d}$. In particular,
any line bundle $\alpha$ of degree $0$ gives rise to a trivialisation of
$\alpha\otimes\iota^{*}(\alpha^{-1})$ unique up to $\pm 1$.

In what follows we will adopt the following notation. For any vector bundle
$E$ on $C$ (over $k$ or any extension of $k$), we denote the vector space
$\Gamma(W,E|W)$ by $W(E)$ and the image under the restriction map of
$\Gamma(C,E)$ to $W(E)$, by $C(E)$.

We denote the top exterior power of any (finite dimensional) vector space (or
vector bundle) $E$ by $det(E)$. We assume that the $1$-dimensional vector
space $det(U)$ is identified with the field. Consequently we may identify $U$
and $U^{*}$.

###  A.3.  Duality in $W$

The sheaf $\mathscr{O}(-2)$ is naturally the dualising sheaf $K_{P}$ (or what
is the same, canonical bundle) of $P$. The dualising sheaf $K_{C}$ of $C$ is
consequently $\pi^{*}(K_{P})\otimes\mathscr{O}(W)\simeq h^{-2}\otimes
h^{g+1}=h^{g-1}$. Also, the dualising sheaf $K_{W}$ of the divisor $W$ is the
tensor product of the restriction to $W$ of $K_{C}$ and $\mathscr{O}(W)$,
namely $(h^{g-1}\otimes h^{g+1})|W=h^{2g}|W$.

The duality in $C$ and that in $W$ are compatible in the following sense. If
$E$ is any vector bundle on $C$ of degree $d$, we have $\Gamma(C,E)$ is dual
to $H^{1}(C,h^{g-1}\otimes E^{*})$ on the one hand and $W(E)$ is dual to
$W(h^{2g}\otimes E^{*})$ on the other. The transpose of the restriction map
$\Gamma(C,E)\to W(E)$ is given by the connecting homomorphism of the exact
sequence

$0\to h^{g-1}\otimes E^{*}\to h^{2g}\otimes E^{*}\to(h^{2g}\otimes E^{*})|W\to
0$

namely, $W(E)^{*}=W(h^{2g}\otimes E^{*})\to H^{1}(C,h^{g-1}\otimes
E^{*})=\Gamma(C,E)^{*}$. The duality in $W$ can also be described by the map
$W(E)\times W(h^{2g}\otimes E^{*})\to W(h^{2g})$, followed by the connecting
homomorphism of the exact sequence

$0\to h^{g-1}\to h^{2g}\to h^{2g}|W\to 0.$

Also, if $f:E\to F$ is any homomorphism, it gives a map
$C(f):\Gamma(C,E)\to\Gamma(C,F)$. One knows that the transpose of $C(f)$ is
the induced homomorphism $H^{1}(C,h^{g-1}\otimes F^{*})\to
H^{1}(C,h^{g-1}\otimes F^{*})$. Similarly we have a map $W(f):W(E)\to W(F)$
whose transpose is given by the natural map $W(h^{2g}\otimes F^{*})\to
W(h^{2g}\otimes E^{*})$.

###  A.4. Properties of a generic pencil

We wish to study properties of generic pencils of quadrics. We recall first
some well-known properties of a quadric defined by a quadratic form $Q$. If
$V$ is a vector space of dimension $n$, and $Q$ is a quadratic form on $V$,
often we will denote the corresponding quadric in $P(V)$ also by $Q$. We can
also think of $Q$ as a symmetric linear map $V\to V^{*}$. The kernel of the
above map is called the nul-space of $Q$. A subspace $A$ of $V$ is isotropic
for $Q$ if the restriction of the quadratic form to $A$ is zero. If $n=2g+2$
and $Q$ is non-degenerate or has nul-space of dimension 1, then the maximal
isotropic subspaces have dimension $g+1$. The subvariety of the Grassmannian
of $(g+1)$-dimensional subspaces of $V$, consisting of isotropic subspaces,
has two irreducible components if $Q$ is non-degenerate and is itself
irreducible if the nullspace of $Q$ is of dimension 1. Orthogonal
transformations of $Q$ act on the space of isotropic subspaces of fixed
dimension $d$. If $Q$ is non-degenerate, and $d=g+1$, an orthogonal
transformation preserves each component if and only if its determinant is 1.

Let $P=P(U)$ be a projective line and $V\to V^{*}\otimes\mathscr{O}(1)$ be a
family of quadrics parametrised by $P$. We call it a pencil of quadrics in
$V$. We identify $det(V)$ with the field $k$. Its determinant $\delta$ (called
the discriminant) is a section of $\mathscr{O}(n)$. In the following, we
always assume that $\delta$ is non-zero, which is the same as saying that the
generic quadric of the pencil is non-degenerate. If $\delta$ has $n$ zeros (in
the algebraic closure) with multiplicity $1$ each, we call it a generic
pencil. The divisor in $P$ defined by $\delta$ will be denoted by $D$. Those
quadrics which correspond to the zeros of $\delta$ are then singular quadrics
with nullspaces $N_{i}$ of dimension 1 each, and all other quadrics are
smooth.

If $Q$ and $Q^{\prime}$ are two singular quadrics in a generic pencil with
nullspaces $N$ and $N^{\prime}$ respectively, then $N$ and $N^{\prime}$ are
orthogonal for both $Q$ and $Q^{\prime}$ and hence for all quadrics of the
pencil. Thus $V$ is the direct sum of the nullspaces $N_{i}$ of singular
quadrics of the pencil and therefore has a basis which is orthogonal for all
the quadrics of the pencil.

We will now recall some properties of a generic pencil over an algebraically
closed field of characteristic $0$ in a vector space $V$ of even dimension,
say $2g+2$.

####  A.4.1.  Dual Family of quadrics

######  Lemma A.1.

Assume given a generic pencil on $V$, and that $H$ is any hyperplane in $V$.
Then the generic quadric in the induced pencil on $H$ is non-degenerate.

###### Proof.

Since $H$ cannot contain all the nullspaces of the pencil, let us assume that
some nullspace $N$ is not a subspace of $H$. The corresponding singular
quadric factors through a non-degenerate quadric in $V/N$. Then $H$, along
with the restriction of this quadric to $H$, maps isomorphically on the
quadratic space $V/N$. Hence it, and hence the generic quadric in $H$ of the
induced pencil, are non-degenerate ∎

######  Remark A.2.

In Equivalently, the restriction of the pencil to any hyperplane $H$ has as
its discriminant a non-zero section of $\mathscr{O}(2g+1)$. On identifying
$V^{*}$ with $\Lambda^{2g+1}(V)$, the $(2g+1)$-th exterior map
$\Lambda^{2g+1}(V)\to\Lambda^{2g+1}(V)^{*}\otimes\mathscr{O}(2g+1)$ gives rise
to a quadratic map $V^{*}\to\Gamma(P,\mathscr{O}(2g+1))$. In other words, this
gives a dual family of quadrics, parametrised by
$\Gamma(P,\mathscr{O}(2g+1))^{*}$. The restriction map
$\Gamma(P,\mathscr{O}(2g+1))\to\Gamma(D,\mathscr{O}(2g+1))$ is an isomorphism
since both have dimension $2g+2$ and the kernel is a section of
$\mathscr{O}(-1)=0$ Thus $\Gamma(P,\mathscr{O}(2g+1))^{*}$ can be identified
with the direct sum of the fibres at $\pi w$ of the line bundle
$\mathscr{O}(2g+1)$ on evaluation. Hence we have a family of diagonal quadrics
with respect to the dual decomposition $V^{*}\simeq\Sigma N_{w}^{*}$. The
space $\otimes^{2}(N_{w})^{*}$ is mapped as above isomorphically on the
1-dimensional subspace of sections of $\mathscr{O}(2g+1)$ which vanish at all
points of $W$ other than $w$, or what is the same, the fibre at $w$ of
$\mathscr{O}(2g+1)$. If $s\in V^{*}$ the corresponding section of
$\mathscr{O}(2g+1)$ evaluated at $w$, is simply the image of $s|N_{w}$ under
the above map.In other words, we have a canonical quadratic map
$V^{*}\to\mathscr{O}(2g+1))|{\pi w}$ for every $w\in W$.

####  A.4.2.  Isotropic subspaces for the pencil

A subspace of $V$ is said to be isotropic for the pencil if it is isotropic
for all quadrics of the pencil. The variety of $r$-dimensional subspaces of
$V$ isotropic to the pencil, will be denoted $I_{r}$. As stated above, all
quadrics of the pencil are linear combinations of any two quadrics $Q_{1}$ and
$Q_{2}$ (both of which we may assume non-degenerate). Obviously, a subspace is
isotropic for the pencil if and only if it is so with respect to $Q_{1}$ and
$Q_{2}$. With reference to a basis $S=\\{e_{i}\\}$, as mentioned above, he two
quadrics may be put in the form $\Sigma x_{i}^{2}=0$ and $\Sigma
a_{i}x_{i}^{2}=0$, where all the $a_{i}$ are distinct and non-zero.

The space consisting of pairs $(Q,c)$ where $Q$ is the quadric of the pencil
and $c$ is a connected component of the space of maximal isotropic spaces for
$Q$, is a 2-sheeted covering of $P$ ramified over $D$ and may be identified
with an hyper-elliptic curve $C$ of genus $g$. The set $W$ in $C$ corresponds
to the set of singular quadrics, and hence to the set of their nullspaces, and
we may as well write $V=\sum N_{w},w\in W$. Moreover, if $V^{\prime}=\sum
N_{w}^{2}$, the quadratic forms of the pencil are obtained as linear forms on
$V^{\prime}$ composed with the natural quadratic map $V\to V^{\prime}$.

There does not exist any $(g+1)$-dimensional subspace isotropic for the
pencil. Such a subspace, being a maximal isotropic space for every singular
quadric of the pencil, has as its image modulo the nullspace of the quadric,
an isotropic subspace of a non-degenerate quadric in a $(2g+1)$-dimensional
space and hence has dimension $\leq g$. In other words, it contains the
nullspace of every singular quadric. Thus it has to contain all the basis
elements, which is absurd.

####  A.4.3.  Automorphisms of the pencil

The group $G$ of automorphisms which leave both the quadratic forms invariant,
modulo $\pm Id$ is $(\mathbb{Z}/2)^{2g+1}$, namely those which act on the
spaces $N_{w}$ by $\pm 1$.

Let $W^{\prime}$ be a proper subset of the basis set $W$, of even cardinality
$2p$, and $\sigma$ an involution of $W(\eta)$ which acts as $+Id$ on all the
$N_{i},i\in W^{\prime}$ and as $-Id$ on all $N_{j},j\in W\setminus
W^{\prime}$. Then $\sigma$ acts on the set of $g$-dimensional isotropic
subspaces for the pencil, without fixed points. For, if $L$ is a subspace
isotropic for the pencil and invariant under $\sigma$, then $L$ is the direct
sum of $L^{+}$ and $L^{-}$, namely the intersection of $L$ with the
eigenspaces of $\sigma$. Since $L^{+}$ and $L^{-}$ are isotropic subspaces for
generic pencils in spaces of dimensions $2p$ and $2g+2-2p$ respectively, we
have $dimL^{+}\leq p-1$ and $dimL^{-}\leq g-p$ proving that $dimL\leq g-1$.

####  A.4.4.  Morphism $I_{g}\to J^{2g+1}$

If $A$ is a $g$-dimensional subspace of $V$ isotropic for the pencil, then for
every $c\in C$, we have a unique $(g+1)$-dimensional subspace $E_{c}$, which
is isotropic for the quadric $Q$ defined by $\pi c\in P$, contains $A$ and
belongs to the component $(Q,c)$. This defines a morphism of $C$ into
$P(V/A)$. The image of $C$ is not contained in any hyperplane of $V/A$,
namely, any hyperplane $H$ in $V$ containing $A$. For, the restriction of the
pencil to the hyperplane $H$ in $V$ has smooth quadrics in general, and
equivalently the restriction $H\to H^{*}\otimes\mathscr{O}(1)$ has a nonzero
section of $\mathscr{O}(2g+1)$ as its discriminant as we have seen above. Thus
$c\in C$ is a point of this hyperplane if and only if the quadric $Q_{c}$
corresponding to $c$ intersects $H$ in a quadric which contains a
$g+1$-dimensional isotropic subspace containing $A$, namely the inverse image
of $\\{c\\}$ under the map $H\to H/A$. This is possible if and only if
$Q_{c}\cap H$ is singular. Hence the mapping $C\to P(V/A)$ pulls back the line
bundle $\mathscr{O}(1)$ to a line bundle $\zeta$ on $C$ of degree $2g+1$. The
induced linear map $(V/A)^{*}\to\Gamma(C,\zeta)$ is injective and hence (both
being of dimension $g+2$), an isomorphism. Thus to any $g$-dimensional
subspace $V$ isotropic for the pencil, we have associated a line bundle
$\zeta$ of degree $2g+1$ on $C$ with an isomorphism of $\Gamma(C,\zeta)$ with
$(V/A)^{*}$. We denote this morphism $I_{g}\to J^{2g+1}$ by $f$.

Since the inverse image of $w\in C$ in $V$ under the map $V\to V/A$ is the
(direct) sum of $N_{w}$ and $A$, it follows that $N_{w}$ maps isomorphically
to $(\zeta^{*})_{w}$ under the natural map $V\to V/A$. These isomorphisms
determine the surjection $V\to V/A$.

We have seen above that there is a natural quadratic map
$q:V^{*}\to\Gamma(P,{\mathscr{O}}(2g+1))$. This takes non-zero vectors to non-
zero vectors and so gives a morphism of the corresponding projective spaces.
Indeed, it is clear that it induces an isomorphism of the quotient of $P(V)$
by the finite group $(\mathbb{Z}/2)^{2g+2}$ which acts as $\pm 1$ on $N_{i}$,
with $P\Gamma(P,\mathscr{O}(2g+1))$.

######  Remark A.3.

We have essentially shown that there is a natural isomorphism $\sum
N_{w}^{2}\simeq(\Gamma(P,\mathscr{O}(2g+1)))^{*}$. Besides, it is easy to
check that the pencil is given by the linear map
$U=\Gamma(P,\mathscr{O}(1))^{*}\to\Gamma(P,\mathscr{O}(2g+1))$ given rise to,
by $s$.

The map $q$ composed with the inclusion $\Gamma(C,\zeta)=(V/A)^{*}\to V^{*}$,
yields a quadratic map $\Gamma(C,\zeta)\to\Gamma(P,\mathscr{O}(2g+1))$. On the
other hand, we have also another natural quadratic map
$\Gamma(C,\zeta)\to\Gamma(P,\mathscr{O}(2g+1))$ given by
$\sigma\mapsto\sigma\otimes\iota^{*}(\sigma)$, noting that this image is an
$\iota$-invariant section of $h^{2g+1}$, and hence gives rise to a section
$\pi s$ of $\mathscr{O}(2g+1)$. Both of these quadratic maps take non-zero
vectors into non-zero vectors. The morphism $q$ maps any $s\in V^{*}\neq 0$,
to a section 0f $\mathscr{O}(2g+1)$ whose zeros correspond to the singular
quadrics of the pencil restricted to the hyperplane $s=0$. If $s$ belongs to
$\Gamma(C,\zeta)$, then $qs$ is the same as the section of
$\mathscr{O}(2g+1)$, the divisor of whose zeros is the image in $P$ of the
divisor of zeros of $s$ in $C$. We have just seen that these correspond to the
singular quadrics in the hyperplane $s=0$ in $V$. Hence $q$ restricted to
$\Gamma(C,\zeta)$ is the map $s\to\pi(s)$, up to a scalar. Modifying the map
$V\to\Gamma(C,\zeta)^{*}$ by a scalar, we may assume that the restriction of
$q$ to $\Gamma(C,\zeta)$ is $\pi$. In particular, the isomorphism
$N_{w}\to\zeta_{w}^{*}$ induces an isomorphism of $N_{w}^{2}$ with
$\zeta_{w}^{-2}$, which depends only on $\zeta$.

Let $A_{1}$ and $A_{2}$ be two elements of $I_{g}$, with
$f(A_{1})=f(A_{2})=\zeta$. We have then two surjections of $V$ onto
$(\Gamma(\zeta))^{*}$ or transposing, two inclusions $\Gamma(C,\zeta)$ in
$V^{*}$, whose compositions with $q$ are the same. This implies that we have
an automorphism $\varphi$ of $V$ which act as $\pm 1$ on $N_{w}$’s taking
$A_{1}$ to $A_{2}$. Moreover, its determinant is $1$ since it preserves the
component corresponding to $c$ on any non-degenerate quadric of the pencil.

######  Proposition A.4.

Any two elements of $I_{g}$ with the same image under $f$ are taken to one
another by a transformation of $V$ which is $Id$ or $-Id$ on an even number of
$N_{i}$’s.

###  A.5. Pencil of quadrics on $\Gamma(W,\eta)$

Let $C$ be defined over any field $k$ of characteristic $\neq 2$. We will
assume that $C$ has a rational point $x$. Hence we have a $k$-rational line
bundle $\eta$ of degree $2g-1$, for example $h^{g}\otimes\mathscr{O}(-x)$. Let
$\xi=h\otimes\iota^{*}(\eta)$.

We denote $\Gamma(W,h^{2g-1})$ by $V^{\prime}$. There is a natural tensor
product map $V^{\prime}\times\Gamma(C,h)\to\Gamma(W,h^{2g})$. Composing this
with the boundary homomorphism $\Gamma(W,h^{2g})\to H^{1}(C,K_{C})=k$, we get
a homomorphism $\Gamma(C,h)\to(V^{\prime})^{*}$. The pencil we construct is as
follows. We will give a $2g+2$-dimensional $k$-vector space $V$ with a
quadratic map $V\to V^{\prime}$. Composing this with the the above family of
linear forms on $V^{\prime}$, parametrised by $\Gamma(C,h)$, we get a pencil
of quadratic forms on $V$. If $\eta$ is a rational line bundle of degree
$2g-1$ as above, we take $V$ to be $\Gamma(W,\eta)\simeq\Gamma(W,\xi)^{*}$ and
the quadratic map $V\to V^{\prime}$ to be $s\mapsto s\otimes\iota^{*}(s)$.

By going over to the algebraic closure, one sees that the discriminant of the
above pencil vanishes only for those sections of $h$ which vanish at a point
of $W$ with multiplicity $1$. Thus we have a generic pencil whose discriminant
is $s$ (up to a non-zero scalar).

We now have the rather obvious

######  Remark A.5.

  1. 1)

If the degree $d$ of any line bundle $\zeta$ on $C$ is less than $2g+2$, the
restriction map $\Gamma(C,\zeta)\to W(\zeta)$ is injective, since its kernel
is $\Gamma(C,\zeta\otimes{\mathscr{O}}(-W))=\Gamma(C,\zeta\otimes
h^{-g-1})=0$, the degree of $\zeta\otimes h^{-g-1}=d-2g-2$ being negative by
assumption.

  2. 2)

Also, if $d$ is greater than $2g-2$, then $H^{1}(C,\zeta)=0$. Hence if
$2g-2<deg(\zeta)<2g+2$, we may associate to $\zeta$, a $(d+1-g)$-dimensional
subspace $C(\zeta)$ of $W(\zeta)$, namely the image of $\Gamma(C,\zeta)$ in
$\Gamma(W,\zeta)$. Moreover, if $d>2g-1$, then $\zeta$ is the pull-back of the
bundle ${\mathscr{O}}(1)$ by the natural map of $C$ into
$P\Gamma(C,\zeta)^{*}=PC(\zeta)^{*}$.

######  Remark A.6.

  1. 1)

The assumption that there is a rational point in $C$ implies that all the
Jacobians $J^{d}$ for all $d$, may be identified using tensorisation by $\eta$
and/or a suitable power of $h$.

  2. 2)

There is a canonical Poincaré bundle on $J^{g}$, namely the inverse image of
the $\Theta$ divisor in $J^{g-1}$ by the map
$(\zeta,x)\to\zeta\otimes{\mathscr{O}}(-x)$ of $J^{g}\times C\to J^{g-1}$.
Using the above remark, we may assume that there is a Poincaré bundle in any
degree $d$.

The quadratic map $W(\eta)\to W(h^{2g-1})$ when restricted to $C(\eta)$
factors through $\Gamma(C,h^{2g-1})$. Hence on tensoring with $\Gamma(h)$ the
restriction of the pencil to $W(\eta)$ factors through
$\Gamma(C,h^{2g})\to\Gamma(W,h^{2g})$ and hence goes to 0 under the boundary
homomorphism $W(h^{2g})\to H^{1}(C,h^{g-1})$ by exactness.

Thus we have associated to each line bundle $\eta$ of degree $2g-1$ over $k$,

  1. 1)

a pencil of quadrics on $W(\eta)$ parametrised by $P$, and

  2. 2)

a $k$-subspace $C(\eta)$ of $W(\eta)$ of dimension $g$ to which the
restrictions of all quadrics of the pencil are 0.

  3. 3)

To any rational point $c\in C$ corresponds a maximal isotropic $k$-subspace
for the quadric determined by a section of $h$ which vanishes at $c$ and
$\iota c$ and contains the above subspace $C(\eta)$.

###  A.6.  Jacobian as the space of isotropic subspaces of the pencil

Let $\eta$ be a $k$-rational line bundle of degree $2g-1$ and
$\xi=h\otimes\iota^{*}(\eta)$. We will denote by $J$, the Jacobian of any
degree and by $P$, a Poincaré bundle on $J\times C$. Let $Q$ be the line
bundle $P\otimes(1\times\iota)^{*}(P)^{-1}$ on $J\times C$. Clearly the
restriction of $Q$ to $J\times W$ is trivial. The direct image $E$ on $J$ of
$Q\otimes p_{C}^{*}(\eta)$ (resp. $F$ of $Q\otimes p_{C}^{*}(\xi)$) is a
vector bundle of rank $g$ (resp. $g+2)$, while the direct image on $J$ of the
restriction of $Q\otimes p_{C}^{*}(\eta)$ to $J\times W$, is a trivial vector
bundle of dimension $2g+2$ with fibre $\Gamma(W,\eta)$. We have now (denoting
$\alpha\otimes\iota^{*}(\alpha)^{-1}\otimes\eta$ by $L(\alpha)$, $\alpha\in
J$) the exact sequence

$0\to\Gamma(C,L(\alpha))\to\Gamma(W,\eta)\to H^{1}(C,h^{-g-1}\otimes
L(\alpha))\to 0.$

Since $L(\alpha)\otimes\iota^{*}(L(\alpha))$ is canonically isomorphic to
$h^{2g-1}$, we have the exact sequence

$0\to E\to\Gamma(W,\eta)\otimes{\mathscr{O}}\to F^{*}\to 0$

on $J$ with $E_{\alpha}=\Gamma(C,L(\alpha))$ and
$F_{\alpha}=\Gamma(C,h\otimes\iota^{*}L({\alpha}))$.

In particular, this defines a morphism $S$ of $J$ into the $g$-dimensional
Grassmannian of $W(\eta)$. Moreover, the group $J_{2}$ consisting of of line
bundles $\alpha$ with $\alpha^{2}$ trivial, acts on $J$. Since, for each such
element we have an isomorphism $\alpha\simeq\iota^{*}\alpha$ (which, we may
assume has Id as its square) it associates $\pm 1$ for every $w\in W$ (in the
algebraic closure). This gives an isomorphism of $J_{2}$ with the group of
automorphisms of the pencil modulo $\pm Id$.

######  Lemma A.7.

The map $S:\alpha\mapsto\Gamma(W,L(\alpha))\subset W(\eta)$ of $J$ into the
Grassmannian of $g$-dimensional subspaces of $W(\eta)$ is compatible with the
action of $J_{2}$ on $J$ and the action of the automorphism group of the
pencil on the Grassmannian.

###### Proof.

In fact, it is a consequence of the isomorphism of $J_{2}$ with the
automorphism group, which uses the fact that each $\alpha\in J_{2}$
corresponds to an automorphism of $W(\eta)$ which acts by $\pm 1$ on the basis
of $W(\eta)$. ∎

We claim that this map $S$ is actually into the space $I_{g}$ of
$g$-dimensional subspaces which are isotropic for all the quadrics of the
pencil defined above. In order to show this, for simplicity we will extend the
above consideration to the algebraic closure of $k$ and prove it there.

Now we restate our construction in this set-up. If $\alpha$ is a line bundle,
associate to it the line bundle
$L(\alpha)=\alpha\otimes\iota^{*}(\alpha)^{-1}\otimes\eta$ on $C$. Then
$L(\alpha)=L(\beta)$ if and only if $j=\alpha^{-1}\otimes\beta$ satisfies
$j=\iota^{*}(j)$, or what is the same, $j\in J_{2}$. In other words, $L:J\to
J^{2g-1}$ is a Galois covering with $J_{2}$ as Galois group. Consider then the
restriction $\Gamma(C,L(\alpha))\to W(L(\alpha))$, which is injective by 5.1.
Since $\iota$ acts as Id on $W$, it follows that $W(L(\alpha))$ is the same as
$W(\eta)$. Hence $C(L(\alpha))$ is a $g$-dimensional subspace $S(\alpha)$ of
$V=W(\eta)$.

Even if $L(\alpha)$ is isomorphic to $L(\beta)$, it does not imply that the
two subspaces $S(\alpha)$ and $S(\beta)$ are the same, since the isomorphism
of $L(\alpha)$ with $L(\beta)$ need not restrict to identity on $W$. This
restriction is the same as the restriction of the isomorphism $j\to\iota(j)$
to $W$. Assuming the isomorphism has square $Id$, it is unique up to $\pm 1$.
It acts on $W$ as $+1$ on a subset of $W$ of even cardinality and as $-1$ on
the complement. See [3, Lemma 2.1]. Thus we obtain an isomorphism of $J_{2}$
with a subgroup of automorphisms of $W(\eta)$ respecting the pencil of
quadrics, modulo $\pm Id$. Consequently, $J_{2}$ acts on $I_{d}$, for all $d$,
in particular on $I_{g}$. Moreover, the morphism $S$ commutes with the action
of $J_{2}$ on $J$ and that on $I_{g}$.

We have the tensor product map

$\Gamma(C,h)\otimes\Gamma(C,L(\alpha))\otimes\Gamma(C,\iota^{*}L(\alpha))\to\Gamma(C,h^{2g})$

on the one hand, and

$\Gamma(C,h)\otimes W(\eta)\otimes W(\eta)\to W(h^{2g})$

on the other. Note that the induced pencil on $S(\alpha)$ is given by
$\Gamma(C,h)\otimes S(\alpha)\otimes S(\alpha)\to k$. It is therefore clear
that this factors through $C(h^{2g})$. Thanks to the exact sequence mentioned
above, this is therefore 0 and hence $S(\alpha)$ is isotropic for all quadrics
of the pencil. Thus we have associated to each $\alpha\in J$, a
$g$-dimensional subspace $S(\alpha)$ of $W(\eta)$ isotropic for the pencil and
thus given a morphism of $J^{g}$ into $I_{g}$. Hence we have the

######  Proposition A.8.

The map $\alpha\to S(\alpha)$ is a $k$-rational morphism of $J$ into $I_{g}$
which respects the action of $J_{2}$.

######  Theorem A.9.

The morphism $S$ of $J^{g}$ into $I_{g}$ is an isomorphism.

###### Proof.

We have given the map over $k$ assuming the existence of a rational line
bundle of degree $2g-1$. It is enough to show that it is an isomorphism over
the algebraic closure. We have already given a morphism $f$ of $I_{g}$ into
$J^{2g+1}$. In the following we will show that the composite $f\circ S:J\to
J^{2g+1}$ is the map $\alpha\mapsto h\otimes\iota^{*}L(\alpha)$. ∎

In the following $\eta$ is any line bundle of degree $2g-1$ and
$\xi=h\otimes\iota^{*}(\eta)$.

######  Lemma A.10.

The space $C(\xi)$ is the orthogonal complement of $C(\eta)$ under the
duality.

###### Proof.

In fact, the duality restricted to $C(\eta)\otimes C(\xi)$ is given by the
natural map $\Gamma(C,\eta)\otimes\Gamma(C,\xi)\to\Gamma(C,h^{2g})$, then
restricting it to $W$ and finally composing it with the connecting
homomorphism $W(h^{2g})\to H^{1}(C,h^{g-1})$ of the exact sequence

$0\to h^{g-1}\to h^{2g}\to h^{2g}|W\to 0.$

This composite is clearly $0$ in view of the exactness of the associated
cohomology exact sequence. Thus $C(\xi)$ is contained in the orthogonal
complement of $C(\eta)$. Since $C(\eta)$ is of dimension $g+2$, our assertion
is proved. ∎

Thus we have the exact sequence

$0\to C(\xi)\to W(\xi)\to C(\eta)^{*}\to 0$

as well as the dual exact sequence

$0\to C(\eta)\to W(\eta)\to C(\xi)^{*}\to 0.$

Using the natural imbedding of $C$ in $P\Gamma(C,\xi)^{*}$, we get a morphism
of $C$ into the subvariety of the Grassmannian of $(g+1)$-dimensional
subspaces in $W(\eta)$ consisting of subspaces that contain $C(\eta)$.

######  Lemma A.11.

For every $x\in C$, the corresponding element $G_{x}$ of the Grassmannian has
as its orthogonal complement in $W(\xi)$ the subspace of $C(\xi)$ given by the
image of the subspace of $\Gamma(C,\xi)$ which vanish at $x$.

###### Proof.

This follows from the duality of the above two exact sequences. ∎

Given $x\in C$, we have the quadric $Q_{x}$ corresponding to the point in
$P\Gamma(C,h)$ which vanishes at $x$. We will assume first that $x\notin W$.
Then $W(\eta)\to W(\eta\otimes{\mathscr{O}}(x))$ is an isomorphism. It is
clear that $G_{x}$ is the same as $C(\eta\otimes{\mathscr{O}}(x))$ imbedded in
$W(\eta\otimes{\mathscr{O}}(x))=W(\eta)$. It gives rise to the map given as
follows, as remarked above. Consider the tensor product map
$\Gamma(\eta\otimes{\mathscr{O}}(x))\times\Gamma(\iota^{*}(\eta)\otimes{\mathscr{O}}(\iota
x))\to\Gamma(h^{2g})$ and compose this with $\Gamma(h^{2g})\to W(h^{2g})\to
H^{1}(h^{g-1})=k$. This is clearly $0$. Now
$\iota^{*}(\eta)\otimes{\mathscr{O}}(\iota x)\simeq\xi\otimes
h^{-1}\otimes{\mathscr{O}}(\iota x)\simeq\xi\otimes{\mathscr{O}}(-x)$. The
subspace $\Gamma(\eta\otimes{\mathscr{O}}(x))$ is thus isomorphic to the space
$H_{x}$ of sections of $\xi$ vanishing at $x$. We conclude that with respect
to the quadric corresponding to $x$, the spaces $G_{x}$ is a maximal isotropic
subspace for the quadric $Q_{x}$ and contains $C(\eta)$ which is isotropic to
all quadrics of the pencil.

This also gives every $x\in C$, one of the (two) components of the variety of
maximal isotropic subspaces in the quadric $Q_{x}$.

######  Remark A.12.

If $x$ belongs to $W$, $Q_{x}$ is of rank $2g+1$ and it is easy to check that
$G_{x}$ is in this case the $(g+1)$-dimensional (that is, maximal) subspace
contained in $Q_{x}$ and containing $C(\eta)$. It is the only such space in
this case.

To sum up, we have the diagram:

$\begin{array}[]{ccccccccc}&&&&0&&0\cr&&&&\downarrow&&\downarrow\cr&&&&N_{x}&\to&\xi_{x}&&\cr&&&&\downarrow&&\downarrow&&\cr
0&\to&C(\eta)&\to&W(\eta)&\to&\Gamma(\xi)^{*}&\to&0\cr&&\downarrow&&\downarrow&&\downarrow&&\cr
0&\to&C(\eta\otimes{\mathscr{O}}(x))&\to&W(\eta\otimes{\mathscr{O}}(x))&\to&\Gamma(\xi\otimes{\mathscr{O}}(-x))^{*}&\to&0\end{array}$

If $x\notin W,N_{x}=0$. If $x\in W,N_{x}$ is 1-dimensional. In any case, the
inverse image of $\xi_{x}$ in $W(\eta)$, namely $G_{x}$ is the same as the
inverse image of $C(\eta\otimes{\mathscr{O}}(x))$ by the map $W(\eta)\to
W(\eta\otimes{\mathscr{O}}(x))$. Hence the imbedding of $C$ in the linear
system of $\xi$ is determined by $\\{G_{x},x\in C\\}$. But then $G_{x}$ is the
space determined by $x$ from among the two subspaces which are isotropic for
$Q_{x}$, if $x\notin W$, and is the unique maximal isotropic space in $Q_{x}$
if $x\in W$. This implies that the space $\Gamma(\xi)^{*}$ as well as the
natural imbedding of $C$ in its projective space are determined by $S(\eta)$.
It is clear that we may replace $\eta$ in this argument by any line bundle
$L(\alpha)$.

This leads to the

######  Proposition A.13.

For every $\alpha\in J$, the image under $f$ of $S(\alpha)$ is
$h\otimes\iota^{*}(L(\alpha))$.

This essentially proves the Main theorem. Firstly, the map $S:J^{2g-1}\to
I_{g}$ is surjective, for given any $B\in I_{g}$, let $\alpha$ be any element
of $J^{2g-1}$ with $f(B)=h\otimes\iota^{*}(L(\alpha))$. Then $S(\alpha)$ and
and $B$ have the same image under $f$. We have shown that in that case,
$S(\alpha)$ and $B$ are taken one to another by an element of $J_{2}$. Thus
modifying $\alpha$ by this element of $J_{2}$, we see that $B$ is in the image
of $S$. In particular, this shows that the morphism $J^{g}\to I_{g}$ is onto,
and hence $I_{g}$ is irreducible. Finally, since the quotient under the free
action of $J_{2}$ on both $J^{g}$ and $I_{g}$ is $J^{2g-1}$, the map
$S:J^{g}\to I_{g}$ is an isomorphism.

###  A.7. Moduli of rank 2 vector bundles and the space of isotropic subspaces
of the pencil

The interpretation of the moduli space of vector bundles of rank $2$ and
determinant $\eta$, is similar. If $E$ is a stable vector bundle on $C$ of
rank 2 and determinant $\eta$, then $E\otimes\iota^{*}(E)$ is semistable with
slope $>2g-2$ and hence its first cohomology vanishes. Also the natural
restriction map $r$ of $\Gamma(C,E\otimes\iota^{*}E)$ (of dimension
$8g-4+4(1-g)=4g$) to $W(E\otimes E)$ (of dimension $8g+8$) is injective.
Similarly, we have a restriction map $r^{\prime}:\Gamma(C,h\otimes
E\otimes\iota^{*}E)$ (of dimension $8g+4+4(1-g)=4g+8$ into $W(h\otimes
E\otimes E)$, which is also injective.

The co-kernels of $r$ and $r^{\prime}$ are respectively
$H^{1}(C,h^{-g-1}\otimes E\otimes\iota^{*}E)$ and $H^{1}(C,h^{-g}\otimes
E\otimes\iota^{*}E)$. Since the dual of $E$ is isomorphic to
$E\otimes\eta^{-1}$, it follows that the dual of $h^{-g-1}\otimes
E\otimes\iota^{*}E$ is $h^{g+1}\otimes
E\otimes\iota^{*}(E)\otimes\eta^{-1}\otimes\iota^{*}(\eta^{-1})=h^{-g+2}\otimes
E\otimes\iota^{*}(E)$. Hence we conclude that the cokernel of $r$ is dual to
$\Gamma(C,h\otimes E\otimes\iota^{*}(E))$. Similarly the cokernel of
$r^{\prime}$ is $H^{1}(C,E\otimes\iota^{*}(E)\otimes h^{-g})$ which is dual to
$\Gamma(C,E\otimes\iota^{*}(E))$.

Note that all these vector spaces are acted on by $\iota$ and the maps respect
the $\iota$-action so that we may replace them by the respective
$-1$-eigenspaces to get two dual short exact sequences:

  1. 1)

$0\to\Gamma(C,E\otimes\iota^{*}E)^{-}\to W(\eta)\to(\Gamma(C,h\otimes
E\otimes\iota^{*}E)^{-})^{*}\to 0$.

  2. 2)

$0\to\Gamma(C,h\otimes E\otimes\iota^{*}E)^{-}\to
W(\xi)\to(\Gamma(C,E\otimes\iota^{*}E)^{-})^{*}\to 0$.

By Atiyah-Bott fixed point theorem, the trace of the action $\iota$ on
$\Gamma(X,E\otimes\iota^{*}E)$ is $(2g+2)$. Hence dimension of
$\Gamma(C,E\otimes\iota^{*}E)^{-}$ is $1/2(4g-2g-2)=g-1$.

The pencil maps $\Gamma(C,E\otimes\iota^{*}E)^{-}$ into $\Gamma(C,h\otimes
E\otimes\iota^{*}E)^{-}$ and hence maps to 0 into its dual. In other words, we
have associated to every stable vector bundle $E$ of rank 2 and determinant
$\eta$, a subspace of $W(\eta)$ of dimension $g-1$, which is isotropic to all
the quadrics of the pencil.

######  Theorem A.14.

There is a natural isomorphism of the moduli space $M(2,\eta)$ of stable
bundles of rank 2 and determinant $\eta$ with the space $I_{g-1}$ of
$(g-1)$-dimensional isotropic subspaces of the associated pencil, under the
assumption that $C$ has a rational point.s a rational point.

###### Proof.

It is proved in [10] that over the algebraic closure of $k$, this association
gives an isomorphism of the moduli space of vector bundles of rank $2$ and
determinant $\eta$ with the space of $(g-1)$-dimensional subspaces isotropic
to the pencil. Hence the above map, defined over $k$, is also an isomorphism.
∎

If $\alpha\in J_{2}$ and we replace $E$ by $\alpha\otimes E$, the action of
$\iota$ on $(\alpha\otimes E)\otimes\iota^{*}(\alpha\otimes
E)=\alpha\otimes\iota^{*}\alpha\otimes E\otimes\iota^{*}E$ is the same as on
$E\otimes\iota^{*}E$ since its action on $\alpha\otimes\iota^{*}\alpha$ is
trivial. Hence $\Gamma(E\otimes\iota^{*}E)^{-}$ and $\Gamma((\alpha\otimes
E)\otimes\iota^{*}(\alpha\otimes E))^{-}$ are isomorphic. On the other hand,
as we have seen in A.8, the trivialisation of $\alpha\otimes\iota^{*}\alpha$
restricts (over the algebraic closure) to $-1$ on a subset of $W$ of even
cardinality and $+1$ in the complement. Hence we have

######  Proposition A.15.

The isomorphism in A.14 is compatible with the action of $J_{2}$ on
$M(2,\eta)$ and that on $I_{g-1}$.

######  Remark A.16.

In a similar manner, one can interpret $I_{d}$ for other $d\leq g-2$ in terms
of bundles on $C$, on the lines of the results of [27], over the field $k$
itself.

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