Brill–Noether existence conjecture for once-marked graphs

Let $(G,v)$ be a genus-$g$ graph with a marked vertex $v$. For a divisor $D$ and $i\geq 0$, define

$$s_i(D,v)=\operatorname{min}\{\ell\in\mathbb{Z}:r(D+\ell v)\geq i\},$$

and define its Weierstrass partition by

$$\lambda(D,v)=(\lambda_0(D,v),\lambda_1(D,v),\ldots),\qquad \lambda_i(D,v)=i-s_i(D,v)+g-\deg D.$$

Write $|\lambda|=\sum_{i=0}^{\infty}\lambda_i$. The divisor census of $(G,v)$ is the set of partitions $\lambda$ for which some divisor $D$ satisfies $\lambda_i(D,v)\geq\lambda_i$ for every $i\geq 0$.

Brill–Noether existence conjecture for once-marked graphs. For any once-marked graph $(G,u)$ of genus $g$, every partition $\lambda$ with $|\lambda|\leq g$ belongs to the divisor census.

This is the marked analogue of the Brill–Noether existence statement for graphs, with $g-|\lambda|$ playing the role of the Brill–Noether number. Under a suitable genericity hypothesis, a chain of loops with a vertex marked at one end is Brill–Noether general in this sense, so the conjecture is resolved.