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:

Situation:

S1) `Div`

is a free abelian group generated by an infinite set of letters `P`

. [*like points*]

S2) `D`

∈ `Div`

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`

. [*Riemann Roch*]

R2) `l(D+P) = l(D)`

or `l(D)+1`

for any generator P.

R3) `l(D)>0`

iff `D`

is equivalent mod `Prin`

to an effective divisor.

R4) If `l(D)>0`

and `deg(D)=0`

then `D`

∈ `Prin`

.

Question A (hopefully manageable): What

exactlycan be inferred here about one of`l(D)`

or`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...

Optional definitions:

O1) `D`

is "free" if `l(D-P) = l(D)-1`

for any generator `P`

.

O2) `D`

is "very ample" if `l(D-P-Q) = l(D)-2`

for any generators `P,Q`

(not nececesarily distinct)

O3) `D`

is "ample" if `nD`

is very ample for some `n>0`

.

O4) `D`

is "big" if for some `c>0`

and all large `n`

, `l(nD) ≥ cm^n`

Question B (partial answers welcome): What

exactlycan be inferred here about`l(D)`

,`deg(D)`

, effectiveness, freeness, (very) ampleness, and bigness of`D`

given information about the others?

Some examples (see Hartshorne chapter IV):

`l(0)=1`

`deg(K) = 2g-2`

- If
`D`

is very ample then`deg(D)>0`

- If
`deg(D) ≥ 2g`

then`D`

is free - If
`deg(D) ≥ 2g+1`

then`D`

is very ample `D`

is ample iff`deg(D)>0`

So, yeah! What's the deal with Riemann-Roch?