Algebraic rank on hyperelliptic graphs and graphs of genus 3

Abstract

Let G¯=(G,ω) be a vertex-weighted graph, and let δ be a divisor class on G. Let rG¯(δ) denote the (combinatorial) rank of δ. Caporaso has introduced the algebraic rank rG¯alg(δ) of δ by using nodal curves with dual graph G¯. In this paper, when G¯ is hyperelliptic or of genus 3, we show that rG¯alg(δ)≥rG¯(δ) 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.