Generalized Jacobians of graphs

TL;DR

This paper introduces generalized Jacobians and Picard groups for graphs with respect to a modulus, establishing their properties, universal mapping, and connections to tropical geometry.

Contribution

It defines new generalized groups for graphs, proves their universal property, and links them to tropical geometry and sheaf theory.

Findings

Defined generalized Jacobians and Picard groups for graphs.

Proved a universal mapping property for these groups.

Connected the groups to tropical geometry and sheaf theory.

Abstract

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 N\'{e}ron 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.