# 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 $$R$$.