Take the 2-minute tour ×  This question aims to pin down what
Riemann-Roch can tell us about a divisor on a curve, without any
"geometric thinking". It can be annoying to wonder if there is
some clever trick you're missing out on in a problem, and it
would be nice to know the limits of Riemann-Roch formalism the
way we know we can't solve a system of 2 linear equations in 3
variables.  If someone prefers a different formalization of this
question, I'd be happy to get an answer to that one instead.
Here is mine:   S1) Div is a free abelian group generated by an
infinite set of letters P. [like points] S2) DDiv is "effective"
if all its coefficients are non-negative. S3) deg: Div -> Z is
the sum of coefficients map. S4) Prin is a distinguished
subgroup of ker(deg). [like principal divisors] S5) l: Div/Prin
-> N is a function to the non-negative integers. [like the
dimension of global sections] S6) K is an element of Div. [like
a cannonical divisor]  Relations (g:=l(K)):  R1) l(D)-l(K-D) =
deg(D) + 1 - g. [Riema