Rank-determining sets of metric graphs

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:118

Article

Luo, Ye

关键词: Finite graph;    Metric graph;    Tropical curve;    Algebraic curve;    Rank-determining set;    Special open set;   

DOI  :  10.1016/j.jcta.2011.03.002

来源: Elsevier

【 摘 要 】

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 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 based on an algorithm to derive the nu(0)-reduced divisor from any effective divisor in the same linear system. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free