Rank of Divisors on Hyperelliptic Curves and Graphs Under Specialization

Abstract

Let (G,ω) be a hyperelliptic vertex-weighted graph of genus g≥2. We give a characterization of (G,ω) for which there exists a smooth projective curve X of genus g over a complete discrete valuation field with reduction graph (G,ω) such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph (G,ω) in general, how the existence of such X relates the Riemann–Roch formulae for X and (G,ω), and also how the existence of such X is related to a problem by Caporaso.