Page 130 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 131
7.5.3 Adjoint divisors and a canonical divisor

Now we want to define the adjoint order at a point P ∈ C which is given by a

valuation ν P of K(C). In sections 7.3.3 and 7.3.4 we have seen that giving ν P is
the same as giving a (primitive) injective homomorphism ϕ P : K(C) → K((t)).
Assume that Center(ν P ) ∈ U i . We define the adjoint order (see also [6, Remark
2.5] and 7.7.1 in the example section) as follows:

∂f d ϕ P (u i )
α ν P := ν P − ord t
∂v i d t
It can be shown that this definition is indeed independent of the choice of ϕ P and if
=0. Further the adjoint order at conjugate points
Q is a smooth point of C then α ν P
for σ ∈ Gal(K | K).So
is the same, i.e. α ν P = α ν σP

A := α ν P ν ∈ Div(C)

ν
where ν runs over all discrete valuations of K(C) is a well defined K-rational divisor.

It is actually given by the finite sum ν∈V Sing(C) α ν P ν .
Definition 9. With the above notation we define the shifted adjoint divisor
K I := D I − A.

Note that K I is the difference of two K-rational divisors and hence K-rational as

well. We know that deg(A)=2 δ Q (see for example [8, p. 1620]). Together

Q∈C
with lemma 6 and theorem 4 this implies:
Corollary 10. The degree of the divisor K I is deg(K I )= d I − 2.

∼
We have C = P and so a divisor on C is canonical if it has degree −2.In

1
K
particular K ∅ is a canonical divisor. The importance of the divisors K I stems from

the following theorem:
(K I ) = ∅ and we can compute a

Theorem 11. If I =[1,n] and d I ≥ 2 then L C
basis in K(C).
Proof. See [1, Theorem 17].
Let us briefly recall the algorithm within the constructive proof. We start with the
K-vector space L S (D I ) (for a basis of that space see lemma 2 above) and com-
pute the subspace V := {h ∈L S (D I ) | ν I (h) ≥ α ν for all ν ∈V Sing(C) }. Then
dim K (V )= d I − 1. A priori V is a space of rational functions on the surface S,but
it can be considered a space of rational functions over C or C as well. It turns out

∼
that via this identification V = L C (K I ) as K-vector spaces. We will execute this
algorithm several times on an example in section 7.7. There it will also become clear
that a basis in K(C) can be computed.