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:
Div is a free abelian group generated by an infinite set of letters
P. [like points]
Div is "effective" if all its coefficients are non-negative.
deg: Div -> Z is the sum of coefficients map.
Prin is a distinguished subgroup of ker(deg). [like principal divisors]
l: Div/Prin -> N is a function to the non-negative integers. [like the dimension of global sections]
K is an element of Div. [like a cannonical divisor]
l(D)-l(K-D) = deg(D) + 1 - g. [Riemann Roch]
l(D+P) = l(D) or
l(D)+1 for any generator P.
D is equivalent mod
Prin to an effective divisor.
Question A (hopefully manageable): What exactly can be inferred here about one of
deg(D), given the other?
Maybe someone already knows the answer to this, from experience with solving RR-related problems, or from literature.
Awesomely, many other concepts can be reformulated in this context, and we can ask more...
D is "free" if
l(D-P) = l(D)-1 for any generator
D is "very ample" if
l(D-P-Q) = l(D)-2 for any generators
P,Q (not nececesarily distinct)
D is "ample" if
nD is very ample for some
D is "big" if for some
c>0 and all large
l(nD) ≥ cm^n
Question B (partial answers welcome): What exactly can be inferred here about
deg(D), effectiveness, freeness, (very) ampleness, and bigness of
Dgiven information about the others?
Some examples (see Hartshorne chapter IV):
deg(K) = 2g-2
Dis very ample then
deg(D) ≥ 2gthen
deg(D) ≥ 2g+1then
Dis very ample
Dis ample iff
So, yeah! What's the deal with Riemann-Roch?