A Note on Brill–Noether Theory and Rank-Determining Sets for Metric Graphs

We produce open subsets of the moduli space of metric graphs without separating edges where the dimensions of Brill–Noether loci are larger than the corresponding Brill–Noether numbers. These graphs also have minimal rank-determining sets that are larger than expected, giving counterexamples to a conjecture of Luo. Furthermore, limits of these graphs have Brill–Noether loci of the expected dimension, so dimensions of Brill–Noether loci of metric graphs do not vary upper semicontinuously in families. Motivated by these examples, we study a notion of rank for the Brill–Noether locus of a metric graph, closely analogous to the Baker–Norine definition of the rank of a divisor. We show that ranks of Brill–Noether loci vary upper semicontinuously in families of metric graphs and are related to dimensions of Brill–Noether loci of algebraic curves by a specialization inequality.