Page 129 - Geometric Modeling and Algebraic Geometry

130 T. Beck and J. Schicho
We call D I a support divisor because the corresponding linear system L S (D I )
on the surface can be described by simple support conditions. To this end let

Π I := {(r, s) | a i r + b i s ≥ c i +1 − δ I (i)}.
i∈[1,n]
In other words Π I is constructed as the convex hull of the lattice points in Π(f) with
◦
edges e i removed for i ∈ I. In particular Π [1,n] = Π(f) and Π ∅ = Π(f) . With
r s
this notation we have L S (D I )= x y | (r, s) ∈ Π I ∩ Z by lemma 2 (see also
2
K
the Ansatz polynomials in 7.7.3 in the example section).
Note that since C is not a component of D I the intersection divisor
∗
D I := (π ◦ ι) (D I ) ∈ Div(C)

is also well defined. This is sometimes called the pullback along π◦ι, see [16, III.1.2].
Further we can explicitly give its degree. The integer length of an edge of a lattice
polygon is the number of lattice points on that edge minus 1.Let d I denote the sum
over the integer lengths of the edges i of Π(f) for i ∈ I.
Lemma 6. We have deg(D I )=2#(Π(f) )+ d I − 2.
◦

Proof. See [1, Lemma 12].

7.5.2 Twists of principal divisors and valuations

A part of the principal divisor (h) of a rational function h ∈ K(C) (or K(C) likewise)

is in a certain sense predetermined by the support. The following definitions are
meant to make this distinction precise.

Definition 7. Let h ∈ K(C) be a rational function. We define the twisted principal
divisor (h) I := (h)+ D I ∈ Div(C). With this definition the divisor (h) I has local

v
equation h i,I = u −c i−1 −1+δ I (i−1) −c i −1+δ I (i) h in π −1 (C ∩ U i ).
i i
If h is given by an element of L S (D I ), i.e. by a polynomial with support in Π I , then
the local equations h i,I are given by polynomials in u i and v i . Therefore in this case
(h) I is an effective divisor on C.

Definition 8. Let ν be a valuation of K(C). We define the twist of the valuation ν by
ν I (h):= ν(h i,I ) for all rational functions h ∈ K(C) if Center(ν) ∈ C ∩ U i and
h i,I as in the previous definition.

Taking into account the previous definition that means ν I (h)= ν(h) if Center(ν) ∈

C ∩ T and ν I (h)= ν(h) − ν v c i +1−δ I (i) if Center(ν) ∈ C ∩ E i . Note that
i
C ∩ E i ∩ E j = ∅ for i = j.