Riemann-Roch theory on finite sets (Q2879065)

In [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)], \textit{M. Baker} and \textit{S. Norine} proved a Riemann-Roch theorem for finite graphs and convincingly explained why their theorem is a discrete analogue of the classical Riemann-Roch theorem for Riemann surfaces. In [Proc.\ Am.\ Math.\ Soc.~141, No.~11, 3793--3802 (2013; Zbl 1270.05053)]; Rocky Mt.\ J.\ Math.~46, No.~5, 1559--1574 (2016; Zbl 1351.05101)], the authors of the paper under review generalised the Baker-Norine theorem to weighted graphs and gave a new proof. After re-interpreting the dimension function for divisors and showing that a certain set of divisors is symmetric with respect to the canonical divisor, their new proof becomes surprisingly short, slick and straightforward.NEWLINENEWLINEIn the paper under review, the authors use the latter proof to establish a Riemann-Roch theorem in the following even more general setting. Rather than assuming that the divisors are defined on the set of vertices of a graph they now assume that the divisors are defined on an arbitrary finite set, they now use the re-interpretation as the definition of the dimension function and they make the symmetry result mentioned above an assumption. The paper concludes with small graph examples and one non-graph example.