---
abstract: 'Let $M$ denote the moduli space of stable vector bundles of rank $n$ and fixed determinant of degree coprime to $n$ on a non-singular projective curve $X$ of genus $g \geq 2$. Denote by ${\mathcal{U}}$ a universal bundle on $X \times M$. We show that, for $x,y \in X,\; x \neq y$, the restrictions ${\mathcal{U}}|\{x\} \times M$ and ${\mathcal{U}}|\{y\} \times M$ are stable and non-isomorphic when considered as bundles on $X$.'
address:
- |
    H. Lange\
    Mathematisches Institut\
    Universität Erlangen-Nürnberg\
    Bismarckstraße $1\frac{ 1}{2}$\
    D-$91054$ Erlangen\
    Germany
- |
    P.E. Newstead\
    Department of Mathematical Sciences\
    University of Liverpool\
    Peach Street, Liverpool L69 7ZL, UK
author:
- 'H. Lange'
- 'P. E. Newstead'
title: On Poincaré bundles of vector bundles on curves
---

[^1]

Introduction
============

Let $X$ be a non-singular projective curve of genus $g \geq 2$ over the field of complex numbers. We denote by $M = M(n,L)$ the moduli space of stable vector bundles of rank $n$ with determinant $L$ of degree $d$ on $X$, where gcd$(n,d) = 1$. We denote by ${\mathcal{U}}$ a universal bundle on $X \times M$. For any $x \in X$ we denote by ${\mathcal{U}}_x$ the bundle ${\mathcal{U}}|\{x\} \times M$ considered as a bundle on $M$.

In a paper of M. S. Narasimhan and S. Ramanan [@nr] it was shown that ${\mathcal{U}}_x$ is a simple bundle and that the infinitesimal deformation map $$\label{eq1}
T_{X,x} {\rightarrow}H^1(M, \mbox{End} ({\mathcal{U}}_x))$$ is bijective for all $x \in X$. In [@bbn Proposition 2.4] it is shown that ${\mathcal{U}}_x$ is semistable with respect to the unique polarization of $M$. In fact, ${\mathcal{U}}_x$ is stable; since we could not locate a proof of this in the literature, we include one here.

Let ${\mathcal{M}}$ denote the moduli space of stable bundles on $M$ having the same Hilbert polynomial as ${\mathcal{U}}_x$. Then (\[eq1\]) implies that the natural morphism $$X {\rightarrow}{\mathcal{M}}$$ is étale and surjective onto a component ${\mathcal{M}}_0$ of ${\mathcal{M}}$.

It is stated in [@nr] that it can be easily deduced from the results of that paper that the map $X {\rightarrow}{\mathcal{M}}_0$ is also injective. This would imply that the curve $X$ can be identified with ${\mathcal{M}}_0$. However no proof of this fact seems to be given. There is a proof in a paper of A. N. Tyurin [@tyu Theorem 2], but this seems to us to be incomplete. We offer here a proof which is in the spirit of [@tyu]. To be more precise, our main result is the following theorem.\
[**Theorem**]{} [*Let $X$ be a non-singular projective curve of genus $g \geq 2$. If $x,y \in X, \; x \neq y$, then ${\mathcal{U}}_x \not\simeq {\mathcal{U}}_y$.*]{}\
Note that if $X$ is a general curve of genus $g \geq 3$ or any curve of genus 2, then $X$ does not admit étale coverings $X {\rightarrow}{\mathcal{M}}_0$ of degree $>1$. So for such curves the theorem is immediate. For the proof we can therefore assume that $g \geq 3$. In fact, our proof fails for $g=2$.

In Section 2 we prove the stability of ${\mathcal{U}}_x$. In Sections 3 and 4 we make some cohomological computations, from which a family of stable bundles on $X$ can be constructed. This construction is carried out in Section 5, where we also use the morphism to $M$ given by this family in order to prove the theorem.

Stability of ${\mathcal{U}}_x$
==============================

Let $X$ be a non-singular projective curve of genus $g \geq 2$. Let $n \geq 2$ and $d$ be integers with gcd$(n,d) = 1$. There are uniquely determined integers $l$ and $e$ with $0<l<n$ and $0 \leq e<d$ such that $$\label{eq2}
ld-en = 1.$$ The bundles ${\mathcal{U}}_x$ were shown to be semistable in [@bbn Proposition 2.4], but the proof does not seem to imply stability directly, even though we know also by [@nr] that ${\mathcal{U}}_x$ is simple.

\[propos2.1\] For all $x \in X$, the vector bundle ${\mathcal{U}}_x$ is stable with respect to the unique polarization of $M$.

By [@bbn Proposition 2.4] the bundle ${\mathcal{U}}_x$ is semistable. By [@ram Remark 2.9] and possibly after tensoring ${\mathcal{U}}$ by a line bundle on $M$, $$c_1({\mathcal{U}}_x) = l \alpha,$$ where $\alpha$ is the positive generator of $H^2(M)$. By (\[eq2\]), $l$ and $n$ are coprime. It follows that ${\mathcal{U}}_x$ is stable.

Cohomological constructions
===========================

Let $l$ and $n$ be as in (\[eq2\]). Let $V$ be a semistable vector bundle of rank $l$ and degree $l(n-l)+e$ and $W$ a semistable bundle of rank $n-l$ and degree $d-e-l(n-l)$ on $X$. Then $$\deg (W^* \otimes V) = nl(n-l) -1.$$ Let $q_i, \; i=1,2,$ denote the projections of $X \times X$ on the two factors, $\Delta$ the diagonal of $X \times X$ and write for brevity $$U = q_1^*(W^* \otimes V).$$

\[lem2.1\] For $ n \geq 2$ and $1 \leq i \leq n$,\
[*(a)*]{} $h^0(U(-i\Delta)|\Delta) = (n+(2i-1)(g-1))l(n-l) -1$;\
[*(b)*]{} $h^1(U(-i\Delta)|\Delta) = 0$.

Identifying $\Delta$ with $X$, we have $U(-i\Delta)|\Delta = W^* \otimes V \otimes K_X^i$. Since $$\deg (W^* \otimes V \otimes K_X^i) = (n+(2g-2)i)l(n-l) - 1 > l(n-l)(2g-2)$$ and $W^* \otimes V$ is semistable, (b) holds and Riemann-Roch gives (a).

\[lem2.2\] For $n \geq 2$, $$h^1(U(-n\Delta)) = gh^0(W^* \otimes V) + l(n-l)(n-1)(g(n-1) + 1) -(n-1).$$

For $0 \leq i \leq n$, consider the exact sequence $$\label{eqn12}
0 {\rightarrow}U(-(i+1)\Delta) {\rightarrow}U(-i\Delta) {\rightarrow}U(-i\Delta)|\Delta {\rightarrow}0$$ on $X \times X$. For $i=0$, this sequence gives $$0 {\rightarrow}H^1(U(-\Delta)) {\rightarrow}H^1(U) \stackrel{\psi}{{\rightarrow}} H^1(U|\Delta),$$ since the restriction map $H^0(U) {\rightarrow}H^0(U|\Delta)$ is an isomorphism. The map $\psi$ is surjective, since its restriction to the Künneth component $H^1(W^* \otimes V) \otimes H^0({\mathcal{O}}) \subset H^1(U)$ is an isomorphism. Hence $$\begin{array}{ll}
h^1(U(-\Delta)) &= h^1(U) - h^1(U|\Delta)\\
&= h^1(W^* \otimes V) h^0({\mathcal{O}}) + h^0(W^* \otimes V)h^1({\mathcal{O}}) - h^1(W^* \otimes V)\\
&= g\cdot h^0(W^* \otimes V).
\end{array}$$ For $1 \leq i \leq n-1$, the sequence (\[eqn12\]) gives, by Lemma \[lem2.1\] (b), $$0 {\rightarrow}H^0(U(-i\Delta)|\Delta) {\rightarrow}H^1(U(-(i+1)\Delta)) {\rightarrow}H^1(U(-i\Delta)) {\rightarrow}0.$$ This gives, by Lemma \[lem2.1\] (a) and the above computation, $$\begin{array}{l}
h^1(U(-n\Delta)) = h^1(U(-\Delta)) + \sum_{i=1}^{n-1} h^0(U(-i\Delta)|\Delta)\\
\quad = g h^0(W^* \otimes V) + \sum_{i=1}^{n-1}((n+(2i-1)(g-1))l(n-l) -1)\\
\quad = g h^0(W^* \otimes V) + l(n-l)(n-1)(g(n-1) + 1) - (n-1).
\end{array}$$

\[lem2.3\] Let $n \geq 2$ and $x \in X$. Then, except in the case when $n=2$ and $W^* \otimes V \simeq {\mathcal{O}}(x)$, $$h^1(U(-n\Delta-X\times\{x\}) = h^1(U(-\Delta - X \times \{x\})) + l(n-l)(n-1)^2g -(n-1).$$

For $1 \leq i \leq n-1$ consider the exact sequence $$\begin{array}{ll}
0 {\rightarrow}U(-(i+1)\Delta - X \times \{x\}) {\rightarrow}& U(-i\Delta-X \times \{x\})\\ 
   &{\rightarrow}U(-i\Delta-X \times \{x\})|\Delta {\rightarrow}0
\end{array}$$ on $X \times X$. Identifying $\Delta$ with $X$, we have $$U(-i\Delta-X \times \{x\})|\Delta \simeq K_X^i \otimes W^* \otimes V(-x).$$ If either $i \geq 2$ or $ n \geq 3$, $$\deg(K_X^i \otimes W^* \otimes V(-x)) > l(n-l)(2g-2).$$ So semistability implies $$\label{eqn2}
h^1(K_X^i \otimes W^* \otimes V(-x)) = 0.$$ If $n=2$ and $i=1$, then $W^* \otimes V$ has rank 1 and $$\deg(K_X \otimes W^* \otimes V(-x)) = 2g-2.$$ So (\[eqn2\]) is still true, unless $W^* \otimes V \simeq {\mathcal{O}}(x)$.

Now Riemann-Roch implies $$h^0(K_X^i \otimes W^* \otimes V(-x)) = ((2g-2)i + n-g)l(n-l) -1.$$ Hence applying the above sequence $n-1$ times, we get

$$\begin{array}{l}
h^1(U(-n\Delta-X \times \{x\}) = \\
\quad = h^1(U(-\Delta - X \times \{x\})) + \sum_{i=1}^{n-1} h^0(K_X^i \otimes W^* \otimes V(-x))\\
\quad = h^1(U(-\Delta -X \times \{x\})) + \sum_{i=1}^{n-1} \{((2g-2)i +n-g)l(n-l) -1\}\\
\quad = h^1(U(-\Delta -X \times \{x\})) + l(n-l)(n-1)^2g -(n-1).
\end{array}$$

Now suppose $(V,W)$ is a general pair of bundles on $X$ with the given ranks and degrees. Here by “general” we mean that the theorem of Hirschowitz (see [@hir]) is true, which says that either $H^0(W^* \otimes V) = 0$ or $H^1(W^* \otimes V) = 0$.

\[prop2.4\] For $n \geq 3$, $g \geq 3$ and $(V,W)$ general, there is a 2-dimensional vector subspace $T_0 \subset H^1(U(-n\Delta))$ such that the restriction map $$\label{eqn3}
H^1(U(-n\Delta)) {\rightarrow}H^1(W^* \otimes V(-nx))$$ is injective on $T_0$ for all $x \in X$.

Consider the exact sequence $$0 {\rightarrow}U(-n\Delta-X \times \{x\}) {\rightarrow}U(-n\Delta) {\rightarrow}U(-n\Delta)|X \times \{x\} {\rightarrow}0$$ on $X \times X$. Since $U(-n\Delta)|X \times \{x\} \simeq W^* \otimes V(-nx)$ is of degree $-1$ and $W^* \otimes V$ is semistable, this gives $h^0(W^* \otimes V(-nx)) = 0$ and thus $$0 {\rightarrow}H^1(U(-n\Delta-X\times \{x\}) {\rightarrow}H^1(U(-n\Delta)) {\rightarrow}H^1(W^* \otimes V(-nx)).$$ We claim that $$\label{eq3}
C := h^1(U(-n\Delta)) - h^1(U(-n\Delta - X \times \{x\})) \geq 3.$$

According to Lemmas \[lem2.2\] and \[lem2.3\], $$\begin{array}{ll}
C & = gh^0(W^* \otimes V) + l(n-l)(n-1) - h^1(U(-\Delta-X \times \{x\})).
\end{array}$$ Now the exact sequence $$0 {\rightarrow}U(-\Delta-X \times \{x\}) {\rightarrow}U(-X \times \{x\}) {\rightarrow}U(-X \times \{x\})|\Delta {\rightarrow}0$$ implies $$\begin{array}{ll}
h^1(U(-\Delta-X \times \{x\}) &\leq h^0(U(-X \times \{x\})|\Delta) + h^1(U(-X \times \{x\}))\\ 
&= h^0(W^* \otimes V(-x)) + gh^0(W^* \otimes V).
\end{array}$$ Hence $$C \geq l(n-l)(n-1) - h^0(W^* \otimes V(-x)).$$

According to the above mentioned theorem of Hirschowitz, either $H^0(W^* \otimes V) = 0$ or $H^1(W^* \otimes V) = 0$. In the first case also $H^0(W^* \otimes V(-x)) = 0$ and thus $$C \geq l(n-l)(n-1) \geq 3.$$ In the second case Riemann-Roch implies $$h^0(W^* \otimes V(-x)) \leq h^0(W^* \otimes V) = (n+1-g)l(n-l) -1$$ and thus, for $g \geq 3$, $$C \geq l(n-l)(g-2) + 1 \geq 3.$$ We have thus proved (\[eq3\]) in all cases. This implies that the codimension of the union of the kernels of (\[eqn3\]) for $x \in X$ is at least 2. Hence there is a vector subspace $T_0$ of dimension 2 meeting this union in 0 only.

The case $n=2$
==============

Now suppose $n=2$, which implies $l=1$. So $V$ and $W$ are line bundles with $\deg(W^* \otimes V) = 1$. In this case the proof of Proposition \[prop2.4\] fails. In fact, we have to choose $V$ and $W$ such that $$W^* \otimes V \simeq {\mathcal{O}}(x_0)$$ for some fixed $x_0 \in X$. Then Lemmas \[lem2.1\] and \[lem2.2\] remain true and so does Lemma \[lem2.3\] except when $x=x_0$.

\[prop3.1\] For $n=2$, there is a $(g-1)$-dimensional vector subspace $T_1 \subset H^1(U(-2\Delta))$ such that the restriction map $$H^1(U(-2\Delta)) {\rightarrow}H^1(W^* \otimes V(-2x))$$ is injective on $T_1$ for all $x \in X$.

Since $h^0(W^* \otimes V) = 1$, Lemma \[lem2.2\] says that $$h^1(U(-2\Delta)) = 2g.$$ Lemma \[lem2.3\] implies that, if $x \neq x_0$, then $$\label{eqn4}
h^1(U(-2\Delta - X \times \{x\})) = h^1(U(-\Delta - X \times \{x\})) + g-1.$$ If $x=x_0$, then the same proof gives $$\label{eqn5}
h^1(U(-2\Delta - X \times \{x\})) \leq h^1(U(-\Delta - X \times \{x\})) + g.$$ Now consider the exact sequence $$\label{eqn6}
0 {\rightarrow}U(-\Delta-X \times \{x\}) {\rightarrow}U(-X \times \{x\}) {\rightarrow}U(-X \times \{x\})|\Delta {\rightarrow}0$$ on $X \times X$. Since under the identification of $\Delta$ with $X$, $$U(-X \times \{x\})|\Delta \simeq {\mathcal{O}}(x_0-x),$$ we get, for $x \neq x_0$, $$0 {\rightarrow}H^1(U(-\Delta - X \times \{x\})) {\rightarrow}H^1(U(- X \times \{x\})) \stackrel{\varphi}{{\rightarrow}} H^1({\mathcal{O}}(x_0-x)).$$ The map $\varphi$ is surjective, since its dual is the canonical injection $$H^0(K_X(x-x_0)) {\rightarrow}\mbox{Hom}(H^0({\mathcal{O}}(x_0)),H^0(K_X(x))) = H^0(K_X(x)).$$ Hence $$\begin{array}{ll}
h^1(U(-\Delta - X \times \{x\})) & = h^1(U(-X \times \{x\})) - h^1({\mathcal{O}}(x_0-x))\\
& = h^0({\mathcal{O}}(x_0))h^1({\mathcal{O}}(-x)) - h^1({\mathcal{O}}(x_0-x))\\
& = g - (g-1) = 1.
\end{array}$$ If $x = x_0$, the map $\varphi$ is still surjective and thus an isomorphism. So (\[eqn6\]) implies $$h^1(U(-\Delta - X \times \{x\})) = h^0({\mathcal{O}}(x_0-x)) = 1.$$ Now (\[eqn4\]) and (\[eqn5\]) give $$\label{eqn7}
h^1(U(-2\Delta-X \times \{x\}))  \left\{ \begin{array}{lll}
                                        \leq g+1 & if & x=x_0,\\
                                        = g& if & x \neq x_0. 
                                         \end{array} \right.$$ Now $$0 {\rightarrow}U(-2\Delta-X \times \{x\}) {\rightarrow}U(-2\Delta) {\rightarrow}U(-2\Delta)|X \times \{x\} {\rightarrow}0$$ gives $$0 {\rightarrow}H^1(U(-2\Delta-X \times \{x\})) {\rightarrow}H^1(U(-2\Delta)) {\rightarrow}H^1(W^* \otimes V(-2x)).$$ So the kernel of the restriction map is $H^1(U(-2\Delta-X \times \{x\}))$ which, together with (\[eqn7\]), implies the assertion as in the proof of Proposition \[prop2.4\].

Proof of the Theorem for $g \geq 3$
===================================

We want to consider extensions of the form $$0 {\rightarrow}q_1^*V(-(n-l)\Delta) {\rightarrow}E {\rightarrow}q_1^*W(l\Delta) {\rightarrow}0   \eqno(e)$$ on $X \times X$. The extension $(e)$ is classified by an element $e \in H^1(U(-n\Delta))$. The restriction of $(e)$ to $X \times \{x\}$ is the extension $$0 {\rightarrow}V(-(n-l)x) {\rightarrow}E_x {\rightarrow}W(lx) {\rightarrow}0$$ corresponding to the image of $e$ in $H^1(W^* \otimes V(-nx))$. We can therefore choose a vector subspace $T_0$ of $H^1(U(-n\Delta))$ of dimension 2 such that, for all $0 \neq e \in T_0$, the image of $e$ in $H^1(W^* \otimes V(-nx))$ is non-zero. Note that $$\begin{array}{ll}
\det E_x & = \det (V(-(n-l)x)) \otimes \det (W(lx))\\
& = \det V \otimes {\mathcal{O}}(-l(n-l)x) \otimes \det W \otimes  {\mathcal{O}}(l(n-l)x)\\
& = \det V \otimes \det W
\end{array}$$ for all $x$. On the other hand, by [@ram Lemma 2.1], provided $V$ and $W$ are stable, the bundle $E_x$ is stable for all $0 \neq e \in T_0$ and all $x \in X$.\
Let ${\mathbb P}^1 = P(T_0)$ and consider the product variety $X \times X \times {\mathbb P}^1$. Let $p_i$ and $p_{ij}$ denote the projections of $X \times X \times {\mathbb P}^1$. The non-trivial extensions of the form $(e)$ with $e \in T_0$ form a family parametrized by ${\mathbb P}^1$ which has the form (see for example [@ram Lemma 2.4]) $$\label{eqn8}
0 {\rightarrow}p_1^*V \otimes p_{12}^*{\mathcal{O}}(-(n-l)\Delta)) {\rightarrow}{\mathcal{E}}{\rightarrow}p_1^*W \otimes p_{12}^*{\mathcal{O}}(l\Delta) \otimes p_3^*(\tau^*) {\rightarrow}0,$$ where $\tau$ is the tautological hyperplane bundle on ${\mathbb P}^1$.\
[*Proof of the Theorem*]{}. By what we have said above, ${\mathcal{E}}$ is a family of stable bundles on $X$ of fixed determinant $L = \det V \otimes \det W$ parametrized by $X \times {\mathbb P}^1$. This gives a morphism $$f: X \times {\mathbb P}^1 {\rightarrow}M$$ such that $$(\mbox{id} \times f)^* {\mathcal{U}}\simeq {\mathcal{E}}\otimes p_{23}^*(N)$$ for some line bundle $N \in \mbox{Pic}(X \times {\mathbb P}^1)$. Considering $${\mathcal{E}}_x = {\mathcal{E}}|\{x\} \times X \times {\mathbb P}^1$$ as a bundle on $X \times {\mathbb P}^1$, we have $$f^*{\mathcal{U}}_x \simeq {\mathcal{E}}_x \otimes N.$$ Hence, in order to complete the proof of the theorem, it suffices to show that the bundle ${\mathcal{E}}_x \otimes N$ determines the point $x$.

For this we compute the Chern class $c_2({\mathcal{E}}_x \otimes N)$ in the Chow group $\mbox{CH}^2(X \times {\mathbb P}^1)$.

From (\[eqn8\]) we get $$\label{eqn9}
c_1({\mathcal{E}}) = p_1^*\beta - (n-l)p_3^*h$$ where $\beta$ is the class of $\det V \otimes \det W$ in $\mbox{CH}^1(X)$ and $h$ is the positive generator of $\mbox{CH}^1({\mathbb P}^1)$.

For the computation of $c_2({\mathcal{E}})$ we use the formula $$c_2({\mathcal{F}}\otimes {\mathcal{L}}) = c_2({\mathcal{F}}) + (r-1)c_1({\mathcal{F}})c_1({\mathcal{L}}) + {r \choose 2} c_1({\mathcal{L}})^2$$ for any vector bundle ${\mathcal{F}}$ of rank $r$ and any line bundle ${\mathcal{L}}$.

The only terms in $c_2({\mathcal{E}})$ which can possibly survive in $c_2({\mathcal{E}}_x)$ when restricting are those involving $[\Delta]h$. So $c_2(p_1^*V \otimes p_{12}^* {\mathcal{O}}(-(n-l)\Delta))$ does not contribute. The coefficient of $[\Delta]h$ in $c_2(p_1^*W \otimes p_{12}^*{\mathcal{O}}(l\Delta) \otimes p_3^*(\tau^*))$ is ${n-l \choose 2}(-2l) $ and the coefficient of $[\Delta]h$ in $$c_1(p_1^*V \otimes p_{12}^*{\mathcal{O}}(-(n-l)\Delta))\cdot c_1(p_1^*W \otimes p_{12}^*{\mathcal{O}}(l\Delta) \otimes p_3^*(\tau^*))$$ is $-l(n-l)(-(n-l)) = l(n-l)^2$. This implies $$c_2({\mathcal{E}}_x) = l(n-l)(-(n-l-1) + n-l)(x \times p) = l(n-l)(x \times p),$$ where $p$ is the class of a point in ${\mathbb P}^1$.

Hence, using (\[eqn9\]), we get that $$c_2({\mathcal{E}}_x \otimes N) = l(n-l)(x \times p) + \gamma$$ with $\gamma \in \mbox{CH}^2(X \times {\mathbb P}^1)$ independent of $x$.

If ${\mathcal{U}}_x \simeq {\mathcal{U}}_y$, then $l(n-l)((x-y) \times p) = 0$ in $\mbox{CH}^2(X \times {\mathbb P}^1)$. This is equivalent to $$l(n-l)(x-y) = 0 \quad \mbox{in} \quad \mbox{CH}^1(X) = \mbox{Pic}(X).$$ Hence $x-y$ is a point of finite order dividing $l(n-l)$ in $\mbox{Pic}^0(X)$. But there are only finitely many such points in $\mbox{Pic}^0(X)$ and any such point has at most 2 representations of the form $x-y$ ( 2 occurs only if $X$ is hyperelliptic). So, for general $x \in X$, there is no $y \in X$ such that $x-y$ is of finite order dividing $l(n-l)$ in $\mbox{Pic}^0(X)$.

Now, as stated in the introduction, the natural morphism $X {\rightarrow}{\mathcal{M}}_0,  \; x \mapsto {\mathcal{U}}_x$ is étale and surjective. We have now proved that this étale morphism has degree 1. Hence it is an isomorphism, which completes the proof of the theorem. $\square$

[CAV]{}

V. Balaji, L. Brambila-Paz and P. E. Newstead: *Stability of the Poincaré Bundle*. Math. Nachr. 188 (1997), 5-15.

A. Hirschowitz: *Problème de Brill-Noether de Rang Supérieur*. Université de Nice, Prépublication Mathématiques No 91 (1986).

M. S. Narasimhan and S. Ramanan: *Deformations of the moduli space of vector bundles over an algebraic curve*. Ann. of Math. 101 (1975), 391-417.

S. Ramanan: *The moduli space of vector bundles over an algebraic curve*. Math. Ann. 200 (1973), 69-84.

A. N. Tyurin: *The geometry of moduli of vector bundles*. Usp. Mat. Nauk 29:6 (1974). 59-88. Russian Math. Surv. 29:6 (1974), 57-88.

[^1]: Both authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). The second author acknowledges support from EPSRC Grant No. EP/C515064, and would like to thank the Mathematisches Institut der Universität Erlangen-Nürnberg for its hospitality
