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A note on Brill-Noether thoery 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 couterexamples 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.