  Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in $\mathbb{P}^{2g+1}$ and let
$Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace
in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the
left adjoint functor of $\Phi:D^b(C)\rightarrow D^b(Y)$. As an application, we
give a different proof of the classification of line bundles and stable bundles
of rank $2$ on hyperelliptic curves given by Desale and Ramanan. When $g=3$, we
show that the projection functor induces a closed embedding
$\alpha:Y\rightarrow SU^s_C(4,h)$ into the moduli space of stable bundles on
$C$ of rank $4$ of fixed determinant.
