TITLE: Importance of Riemann-Roch theorem
QUESTION [8 upvotes]: I read yesterday the statement of Riemann-Roch theorem and I didn't actually detect the huge importance that anyone tells me it has... So, can anyone provide me with some examples or reasons for being considered one of the most important theorems in algebraic geometry-algebraic curves?

REPLY [3 votes]: As Benjamin mentions, Serre duality plays a key role in understanding the meat of Riemann-Roch for curves. However, there are more general forms of Riemann-Roch, all of which can somehow be understood as the "equality"
$$\text{topological data} = \text{algebraic data}.$$
In the case of curves, this is straightforward. By rewriting the Riemann-Roch formula as 
$$ g=l(D)-l(D-K)-\text{deg}(D)-1, $$ 
we can express "topological data" (the genus of the curve) as "algebraic data" (the sum of the dimension of the cohomology groups and the degree of the divisor). This can allow for ease of computation in some cases, but more importantly, it has powerful consequences. For example: 
Theorem (Hurwitz). For $f:X \rightarrow Y$ a finite seperable morphism of curves, degree $n$, 
$$2g(X)-2=n\left(2g(Y)-2 \right) + \text{deg}(R) $$
where $R$ is the ramification divisor of $f$. 
In other words, we can compute the genus of $X$ just by knowing the genus of $Y$ is and what $f$ does.