Algebraic rank on hyperelliptic graphs and graphs of genus 3

Abstract

Let $\overline{G}=(G,\omega )$ be a vertex-weighted graph, and let $\delta$ be a divisor class on $G$. Let $r_{\overline{G}}\left(\delta \right)$ denote the (combinatorial) rank of $\delta$. Caporaso has introduced the algebraic rank $r_{\overline{G}}^{\mathrm{alg}}\left(\delta \right)$ of $\delta$ by using nodal curves with dual graph $\overline{G}$. In this paper, when $\overline{G}$ is hyperelliptic or of genus $3$, we show that $r_{\overline{G}}^{\mathrm{alg}}\left(\delta \right)\ge r_{\overline{G}}\left(\delta \right)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a nonhyperelliptic curve of genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.