On the degree-1 Abel map for nodal curves

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right]$ in the Jacobian of $C$. In this work we extend $\alpha_{L}$ to a morphism $\overline{\alpha}_{L}:C\rightarrow\overline{J}_{E}^{P}$ taking values on Esteves' compactified Jacobian for any given polarization $E$. The maps $\overline{\alpha}_{L}$ are limits of Abel maps of smooth curves of the type $\alpha_{L}$.