I am not familiar with the language of cohomology and definitely not fluent with the modern language of sheaves. I had learnt about Riemann Roch spaces and the associated theorem in the context of curves, but now I am trying to understand how divisors, rational functions and the Riemann Roch space looks like for a 2 dimensional algebraic variety.
How do we define the Riemann Roch space for an 2 dimensional algebraic variety? I know that a divisor will be a formal sum of curves ( co dimension one sub-variety). Let $X$ be the algebraic surface. If $f \in K(X)^*$, then the principal divisor is defined as follows $$\mbox{div}(f)=\sum_{C}\mbox{ord}_C(f)C \quad (\text{sum over irreducible curves}),$$ where \begin{equation} \mbox{ord}_C(f)=\begin{cases} \; \; \;n, & \text{if $f$ has a zero of order $n$ along $C$}.\\ -n, & \text{if $f$ has a pole of order $n$ along $C$}.\\ \; \; \;0, & \text{otherwise}. \end{cases} \end{equation}
Is it correct to write the Riemann Roch space of a divisor $D$ on $X$ as follows? $$ \mathcal{L}(D)=\{f\in K(X)^*: \mbox{div}(f)+D\geq 0\}\cup \{0\} $$
How do we define the degree of a rational function on a surface? For $f \in K(C)^*$, where $C$ is an irreducible curve, we define the degree of a rational function to be the degree of pole divisors of the rational function. Can I use a similar definition? Where does intersection theory come in?
Please correct me if my definitions are wrong, maybe it is foolish to try to understand this without the modern language, but any input would be useful! I am interested in using these definitions for surfaces defined over finite fields.