THE BRILL-NOETHER RANK OF A TROPICAL CURVE
Yoav Len · 2014
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 Brill–Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill–Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill–Noether rank of a tropical curve with the dimension of the Brill–Noether locus of an algebraic curve.