The Brill-Noether rank of a tropical curve
arXiv:1209.6309 · doi:10.1007/s10801-014-0510-0
Abstract
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 algerbaic curve.
17 pages, 4 figures; v2: changed title, updated references, minor improvements. To appear in Journal of Algebraic Combinatorics