Riemann-Roch
Let D =
P
ne
P
e
P formal Z-linear combination of points of e
C.
The attached Riemann-Roch space is
L(D) = {h ∈ K(C) | orde
P h ⩾ −ne
P for all e
P}.
This is a finite-dimensional K-vector space. We want a basis.
Represent points e
P ∈ e
C either as nonsingular points P ∈ C, or
as local parametrisations.
Strategy:
Find d(x) ∈ K[x] such that
h(x, y) ∈ L(D) =⇒ d(x)h(x, y) ∈ OC .
Use local parametrisations to find combinations vanishing
at appropriate order at relevant points.
Nicolas Mascot Algebraic curves