The locus of Brill–Noether general graphs is not dense

Abstract

Abstract. We provide an example of a trivalent, 3-vertex connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.

Degeneration of Linear Series from the Tropical Point of View and Applications

“…If we fix a 2-coloring of the vertices of G, then the vertices of one color are a rank-determining set. This is the key observation in [Jen14], in which the second author shows that the Heawood graph admits a divisor of degree 7 and rank 2, regardless of the choice of edge lengths. The interest in this example arises because it shows that there is a non-empty open subset of the (highest-dimensional component of the) moduli space M …”

A note on algebraic rank, matroids, and metrized complexes

“…The rest of this section is devoted to proving the following result: The graph G M from the corollary is known as the Heawood graph. A similar graph but with generic edge lengths was used by Jensen to show that the locus of Brill-Noether general graphs is not dense in the moduli space of tropical curves [8]. Let (L, H) be a linear series on a nodal curve X.…”

Brill–Noether theory of maximally symmetric graphs

“…Remark 17. In recent work, Jensen [9] showed that the Haewood graph is a Brill-Noether special metric graph for arbitrary edge lengths, thus showing that the locus of Brill-Noether general metric graphs is not dense in the tropical moduli space. Figure 6.…”