I am reading "Riemann-Roch and Abel-Jacobi Theory on a Finite Graph" by Baker and Norine (2007, arXiv 0608360). In this paper, the authors formulate abstract criterions for a set X, its set of divisors and an equivalence relation on the set of divisors for the Riemann-Roch (RR) formula to hold. I'm trying to prove the classical RR on Riemann surfaces by showing that in that setting, these conditions are satisfied. This should, in theory, be possible since by BN, RR holds "if and only if" these criterions are satisfied.
There is two conditions to check. The second should be fine, however I am having trouble with the first one. It reads: For all $D \in \textrm{Div}(X)$ there is $\nu \in \mathcal{N}$ such that exactly one of the following is true: Either $|D|\neq \emptyset$ and $|\nu - D|=\emptyset$ OR $|D|=\emptyset$ and $|\nu-D|\neq \emptyset$, where $\mathcal{N}=\{\nu \in \textrm{Div}(x): \textrm{deg}(\nu)=g-1 \,\,\textrm{and} \,\, |\nu|=\emptyset\}$.
I think that in the "theory of divisors", most of the notation should be clear, ex. $|D|$ denotes the set of all effective divisors that are linearly equivalent to $D$ and so on (if there are questions concerning the notation, please don't hesitate to ask).
My problem is the following: I have read quite a lot about classical RR in standard literature (Miranda, Fulton...) but nowhere do I find arguments based on considering this type of divisors $\mathcal{N}$. Doing more research, I found that publications dealing with BNs approach to RR call them "moderators".
In BN, these divisors are constructed explicitly using combinatorics on the graph. My approach was the following: Triangulate the (cpct) Riemann surface X such that for all points $p\in X$ where $D(p)\neq 0$, p corresponds to a 0-simplex in the triangulation. Consider the resulting (finite) graph. Use BNs combinatorical approach to define a divisor $\nu$ on this graph, transport it back to $X$ and show that it does what I need it to do. Unfortunately, I think this doesn't work: The combinatorical construction of $\nu$ in BN results in $\nu$ wich doesn't have the degree $g-1$ where $g$ ist the topological genus of $X$. This can be seen using $2-2g=V-E+F$ for a triangulation of $X$, I can elaborate if necessary.
Any hints, ideas, whatsoever on this problem are greatly appreciated! I realize that to understand in detail what I am talking about here, knowledge of the original paper is probably helpful, and I am sorry if that's too much to ask for. To put this all in a nutshell, this is a short version:
Do you know an approach of proving RR involving the set of divisors $\mathcal{N}=\{\nu \in \textrm{Div}(X): \textrm{deg}(\nu)=g-1 \,\,\textrm{and} \,\, |\nu|=\emptyset\}$?
Thanks!