If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a semiabelian variety $X$ of dimension $g$ and torus rank $1$ is the same as the data of a $g-1$-dimensional abelian variety $A$ and a line bundle in $\operatorname{Pic}^0(A)$ (up to inversion).
In the case where the dual graph of $C$ is a cycle, so the normalisation of $C$ is a disjoint union of smooth curves $C_1, \ldots , C_k$ glued at points $p_i, q_i \in C_i$ (so that $p_i = q_{i+1}$ in $C$, and the subscripts are modulo $k$), I am almost sure that the Jacobian of $C$ is the one associated to the $g-1$ abelian variety $A = \operatorname{Jac}(C_1) \times \ldots \times \operatorname{Jac}(C_k)$ and the line bundle $$ (\mathcal O_{C_1}(p_1-q_1), \ldots ,\mathcal O_{C_k}(p_k-q_k)) $$ (I am using the principal polarisation on $A$ to identify it with $\operatorname{Pic}^0(A)$).
Is there any reference where this is treated?
