Rank of divisors on hyperelliptic curves and graphs under specialization

TL;DR

This paper characterizes hyperelliptic vertex-weighted graphs that can be realized as reduction graphs of algebraic curves with preserved divisor ranks, linking graph theory, algebraic geometry, and the Caporaso conjecture.

Contribution

It provides a characterization of graphs for which divisor ranks are preserved under specialization, connecting algebraic curves and combinatorial graph invariants.

Findings

Identifies conditions for the existence of algebraic curves with given reduction graphs.

Relates Riemann--Roch formulae for curves and graphs.

Connects the existence of such curves to Caporaso's conjecture.

Abstract

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