ag.algebraic geometry – The compactified Jacobian is birational to a $mathbb{P}^1$-fibration over the Jacobian of normalization


Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$pi:Xrightarrow Y$ 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 $F(L)in L_xoplus L_y$ is a $1$-dimensional subspace), take the kernel of the surjective map $pi_*Lrightarrow pi_*Lotimes k(y)(cong pi_*(L_xoplus L_y))rightarrow pi_*(L_xoplus L_y/F(L))$.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.