Rank of divisors on hyperelliptic curves and graphs under specialization

Abstract

Let $(G,\omega )$ be a hyperelliptic vertex-weighted graph of genus $g\ge 2$. 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.

34 pages. The proof of Theorem 1.13 has been significantly simplified

References in corpus (2)

• Rank of divisors on tropical curves
• Tropical varieties for non-archimedean analytic spaces

Cited by in corpus (2)

• Lifting divisors on a generic chain of loops
• Algebraic rank on hyperelliptic graphs and graphs of genus $3$