The BrillNoether rank of a tropical curve
We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the BrillNoether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the BrillNoether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the BrillNoether rank of a tropical curve with the dimension of the BrillNoether locus of an algebraic curve.
tropical curves specialization lemma tropical geometry brillnoether theory