I'm reading through the chapter on Riemann Roch in Garrity at al's Algebraic: Geometry A Problem Solving Approach, and I have a question regarding a definition.
Let $V$ be a projective plane curve over $\mathbb{C}$ and let $D$ be a divisor on the curve. Garrity defines $L(D) \subseteq \mathcal{K}(V)$ to be the subspace of functions $\{f \mid f = 0 \text{ or } \text{div}(f) + D \ge 0\}$.
However, it seems somewhat unnatural to me that if $D = \text{div}(f)$, we don't get $f \in L(D)$. The way I am interpreting it, divisors act as a "specification" of function's roots and poles. It seems to me that if $\text{div}(f)$ is to denote the specification of $f$ and $L(D)$ to denote the set of functions meeting the specification of $D$, then any function $f$ meets its own specification and thus should belong to $L(\text{div}(f))$.
So I am wondering if $L(D)$ is ever defined the other way around: $L(D) = \{f \mid f = 0 \text{ or } \text{div}(f) \le D\}$.
I know that a bunch of theorems would have to be slightly tweaked. I believe we would have $L$ acting as an antitonic rather than monotonic function: $D_1 \le D_2 \Rightarrow L(D_1) \supseteq L(D_2)$, $l(D+p)\le l(D)+1$ would need to be changed to $l(D-p)\le l(D)+1$, and of course, the Riemann-Roch theorem itself would need to be reworded too.
So my question is, are there any authors who define $L(D)$ this way? Or perhaps, is there some reason it is defined the way it is that I don't see?
You are actually asking about two conventions:
Question 1. What is $\operatorname {div(f)}$ ?
Question 2. What is $L(D)$ ?
Answer 1.
Here the usual convention to count zeros positively and poles negatively is used universally in elementary holomorphic function theory, complex manifolds and algebraic geometry.
I know of no exception and it is very reasonable: you want the holomorphic function $(z-17)^6$ to have divisor $6\cdot[17]$ in $\mathbb C$ and the rational (or meromorphic) function $\frac {1}{(z-17)^6}$ to have divisor $-6\cdot[17]$.
Answer 2.
The usual convention nowadays is the following: given a Cartier divisor $D=(U_i, f_i)$ on the holomorphic manifold or algebraic variety $X$ ($U_i\subset X$ open, $f_i$ meromorphic or rational), associate to it the locally free sheaf $\mathcal O(D)$ such that $\mathcal O(D)\vert U_i=\frac 1 f_i \mathcal O_{U_i}$.
Your $L(D)$ is then $\Gamma(X,\mathcal O(D))$.
Here the choice of taking $\mathcal O(D)\vert U_i=\frac 1 f_i \mathcal O_{U_i}$ instead of $ f_i \mathcal O_{U_i}$ indeed looks a bit artificial and has indeed not always been adopted.
The rationale for this counter-intuitive convention is that the divisor $D$ gives a canonical meromorphic (or rational) section $s_D\in L(D)$ and that the divisor of that section is precisely $D$. The convention has other advantages: for example if $H$ is a smooth hypersurface in the smooth manifold (or algebraic variety) $X$ the normal bundle of $H$ in $X$ is $N_X(H)=\mathcal O(H)|D$.
This convention is always adopted nowadays in algebraic geometry and complex manifold theory.
The only books I know which take the apparently more intuitive convention $L(D) = \{f \mid f = 0 \text{ or } \text{div}(f) \geq D\}$ are George Springer's Introduction to Riemann Surfaces, and Farkas-Kra's Riemann Surfaces.
These books are respectively 59 years and 37 years old. Independently of the conventions we are discussing these books are completely obsolete and practically never quoted anymore.
Their main drawback is to not incorporate the powerful sheaf-theoretic methods invented during World War II.