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 \(\alpha_{L}\).