Theorem: Let $X$ be a complete, non-singular algebraic curve of genus $2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector bundles of rank $2$ and degree $\Theta$. The group $\Gamma$ of elements of order $2$ in $J$ acts on $PH^0(J^1, L_\Theta^2)$ in a natural way; let $A$ be the associated projective bundle on $J/\Gamma \cong J$. Then $U(2, \Theta)$ is canonically isomorphic to $A$. In other words, $U(2, \Theta)$ is canonically isomorphic to the space of positive divisors on $J^1$ algebraically equivalent to $2\Theta$.

Moduli of Vector Bundles on a Compact Riemann Surface

Author(s): M. S. Narasimhan and S. Ramanan

Source: Annals of Mathematics, Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 14-51

Question: Is the projective bundle in theorem comes from some vector bundle on $J$. Means is it $P(E)$ of some bundle $E$ over $J$.

Theorem: Let $X$ be a complete, non-singular algebraic curve of genus $2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector bundles of rank $2$ and degree $\Theta$. The group $\Gamma$ of elements of order $2$ in $J$ acts on $PH^0(J^1, L_\Theta^2)$ in a natural way; let $A$ be the associated projective bundle on $J/\Gamma \cong J$. Then $U(2, 0)$ is canonically isomorphic to $A$. In other words, $U(2, 0)$ is canonically isomorphic to the space of positive divisors on $J^1$ algebraically equivalent to $2\Theta$.

Moduli of Vector Bundles on a Compact Riemann Surface

Author(s): M. S. Narasimhan and S. Ramanan

Source: Annals of Mathematics, Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 14-51

Question: Is the projective bundle in theorem comes from some vector bundle on $J$. Means is it $P(E)$ of some bundle $E$ over $J$.