Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\overline{J}(Y)$ is the compactified Jacobian, parametrizing torsion free sheaves of rank $1$ on $Y$.
I want to show $\overline{J}(Y)$ is birational to a $\mathbb{P}^1$-fibration over $J(X)$.
Take Poincare bundle $\mathcal{P}\rightarrow J(X)\times X$, and set $\mathcal{P}_{+}=\mathcal{P}{\mid}_{J(X)\times\{x\}}$, $\mathcal{P}_{-}=\mathcal{P}{\mid}_{J(X)\times\{z\}}$.
Then, I get $\mathbb{P}^1$-bundle $\mathbb{P}(\mathcal{P}_+ \oplus \mathcal{P}_-)\rightarrow J(X)$.
There are two sections $S_+$ and $S_-$ corresponding to $\mathcal{P}_+$ and $\mathcal{P}_-$.
It suffices to show that there is a following commutative diagram,
$\require{AMScd}$
\begin{CD}
\overline{J}(Y) @>\displaystyle \cong >> \mathbb{P}(\mathcal{P}_+ \oplus \mathcal{P}_-)/S_+ \sim S_-\\
@V \displaystyle \pi^* V V\ @VV
\displaystyle{R} V\\
J(X) @>> id \displaystyle > J(X)
\end{CD}
Given $(L,F(L))\in \mathbb{P}(\mathcal{P}_+ \oplus \mathcal{P}_-)$($L$ is a line bundle on $X$ and $F(L)\in L_x\oplus L_y$ is a $1$-dimensional subspace), take the kernel of the surjective map $\pi_*L\rightarrow \pi_*L\otimes k(y)(\cong \pi_*(L_x\oplus L_y))\rightarrow \pi_*(L_x\oplus L_y/F(L))$.Then, I get $h:\mathbb{P}(\mathcal{P}_+ \oplus \mathcal{P}_-)\rightarrow \overline{J}(Y)$.
But I don't understand how to show it is an isomorphism and how to construct $R$.
Any help and comments would be appreciated.Thanks in advance.
