Page 125 - Geometric Modeling and Algebraic Geometry

126 T. Beck and J. Schicho
7.3.1 General and K-rational divisors
Let X be a smooth, irreducible K-variety with field of definition K. By this we mean
that X is locally defined by equations with coefficients in K. Let K(X) denote the
function field of X. Then the Galois group Gal(K | K) acts on K(X). For any field
L s.t. K ⊂ L ⊂ K we define the restricted function field L(X):= {g ∈ K(X) |
σ(g)= g for all σ ∈ Gal(K | L)}.
Irreducible closed K-subvarieties of X of codimension 1 are also called prime
(Weil) divisors.A general divisor D is defined to be a finite formal sum D =

n i D i with n i ∈ Z and D i prime divisors. For curves one defines the degree
i
of the divisor as deg(D)= n i . The set of divisors thus forms a free Abelian
i
group Div(X).An effective divisor is a non-negative linear combination of prime
divisors.
One associates to a nonzero rational function g ∈ K(X) its principal divisor
(g). Roughly speaking g has poles and zeroes with certain multiplicities on X along
subvarieties of codimension 1; then (g) is the divisor of zeroes minus the divisor of
poles (with multiplicities). We say that two divisors are equivalent if and only if their
difference is principal. For a more precise elaboration see [16, III.1.1].
Let D ∈ Div(X). The Galois group Gal(K | K) also acts on the divisors
Div(X). For any field extension L s.t. K ⊂ L ⊂ K we say that D is an L-rational
divisor if and only if it is invariant under the Galois group Gal(K | L). For example
if g ∈ L(X) then the principal divisor (g) is an L-rational divisor. From the defini-
tion it follows that also Gal(L | K) acts on the set of L-rational divisors if L | K is
itself Galois.
The linear system L X (D) of rational functions on X associated to a divisor D
is the K-vector space of rational functions g ∈ K(X) s.t. D +(g) is effective. K-
rational divisors are of particular interest because the corresponding linear systems
can be represented without introducing field extensions.
Lemma 1. Let X be a smooth, projective K-variety with field of definition K.If D is
a K-rational divisor then L X (D) has a basis in K(X) (or L X (D)= ∅).

Proof. Since X is projective the vector space L X (D) is finite-dimensional (or
empty, see [9, Theorem II.5.19]). Therefore we can assume without loss of generality
that b 1 ,...,b m = L X (D) with b i ∈ L(X) for some Galois extension L | K. Set
K
V := b 1 ,...,b m L = L(X) ∩L X (D).
Let σ ∈ Gal(L | K) and assume that g ∈ V , i.e. g ∈ L(X) and (g) ≥−D. Then
(σg)= σ(g) ≥−σD = −D, because D is K-rational. In other words σg ∈ V and
the Galois group Gal(L | K) acts semi-linearly on V . Let now V 0 := {g ∈ V | σg =
g for all σ ∈ Gal(L | K)}.
Then by [12, Lemma 2.13.1] the canonical map V 0 ⊗ K L → V is an isomorphism.
Since V 0 is fixed by Gal(L | K) we have V 0 ⊂ K(X). Choose a basis of V 0 .

7.3.2 Divisors on toric surfaces

For surfaces, prime divisors correspond to irreducible closed curves on the surface.
If S is a toric surface as constructed in section 7.2 then it is locally isomorphic to