Rank-determining sets of metric graphs

Details Rank-determining sets of metric graphs Rank-determining sets of metric graphs

2009-06-15

arxiv.org/abs/0906.2807v2

Mathematics

Combinatorics

20 pages, 6 figures; typos corrected; more results added to the last subsection

Scientific paper

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph $\Gamma$ is an element of the free abelian group on $\Gamma$. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a \emph{rank-determining set} of a metric graph $\Gamma$ to be a subset $A$ of $\Gamma$ such that the rank of a divisor $D$ on $\Gamma$ is always equal to the rank of $D$ restricted on $A$. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is a based on an algorithm to derive the $v_0$-reduced divisor from any effective divisor in the same linear system.

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