| 概要 | Let ) be a hyperelliptic vertex-weighted graph of genus  . We give a characterization of for which there exists a smooth projective curve  of genus  over a complete discrete valuation field with reduction graph such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph in general, how the existence of such relates the Riemann--Roch formulae for and , and also how the existence of such is related to a conjecture of Caporaso. This is a joint work with S. Kawaguchi. |