Generalized Jacobians of graphs

We define a generalized Jacobian $\mathrm{J}\mathfrak{m}(\mathit{Gr})$ and a generalized Picard group $\mathrm{P}\mathfrak{m}(\mathit{Gr})$ of a graph $\mathit{Gr}$ with respect to a modulus $ \mathfrak{m}=\sum_{i=1}^s m_iw_i$ with $w_i$ vertices of $\mathit{Gr}$ and $m_i\geq 1$. These groups occur as the component groups of Néron models of generalized Jacobians. We prove a universal mapping property for $\mathrm{J}\mathfrak{m}(\mathit{Gr})$ and show that an Abel-Jacobi map in this context induces an isomorphism from $\mathrm{P}\frak{m}(\mathit{Gr})$ to $\mathrm{J}\mathfrak{m}(\mathit{Gr})$. We also reinterpret $\mathrm{P}\mathfrak{m}(\mathit{Gr})$ in terms of sheaves on the geometric realization $\left| \mathit{Gr}\right|$ of $\mathit{Gr}$, making a connection with tropical geometry.

💡 Research Summary

The paper develops a comprehensive analogue of the theory of Jacobians and Picard groups for finite (and locally finite infinite) graphs, extending it to incorporate a modulus, a concept originally from the theory of algebraic curves.

The authors begin by recalling the classical picture for a smooth projective curve (X) over (\mathbb C): the divisor class group (\mathrm{Cl}(X)) and its degree‑zero part (\mathrm{Cl}^0(X)), the Picard group (\mathrm{Pic}(X)) and its degree‑zero part (\mathrm{Pic}^0(X)), the Jacobian variety (J(X)) and the Picard variety (P(X)). They review the four fundamental results: (a) Abel–Jacobi theorem (the map (\mathrm{Cl}^0(X)\to J(X)) is an isomorphism), (b) the universal mapping property of the Jacobian, (c) the exponential exact sequence giving (\mathrm{Pic}^0(X)\cong P(X)), and (d) the self‑duality of (J(X)).

Next they introduce a modulus (\mathfrak m=\sum_{i=1}^s m_iP_i) on a curve, recalling Rosenlicht‑Lang‑Serre’s construction of the generalized divisor class group (\mathrm{Cl}{\mathfrak m}(X)), the generalized Picard group (\mathrm{Pic}{\mathfrak m}(X)), the generalized Jacobian (J_{\mathfrak m}(X)) and the generalized Picard variety (P_{\mathfrak m}(X)). The generalized Jacobian fits into an exact sequence \