Page 128 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 129
7.4.2 Parametrizing a curve

Now assume that C is an irreducible curve of genus(C)=0 and π : C → C

a resolution. Assume that D ∈ Div(C) is a divisor with d := deg(D) ≥ 1 and

(D) is a basis of the corresponding linear system. Using the
{s 0 ,...,s d }⊂L C
isomorphism K(C) = K(C) we can define a birational map
∼

d
C P : Q → [s 0 (Q): ··· : s d (Q)]
K
d
which sends C to a rational normal curve C ⊆ P .

K
Now if D is a K-rational divisor then we can assume {s 0 ,...,s d }⊂ K(C) by
lemma 1. In other words the corresponding map and also its rational inverse do not
require any field extension. The best thing would thus be to find a K-rational divisor
of degree 1 because that would result immediately in a parametrization of C without
field extension. The existence of such a divisor however cannot be guaranteed and is
equivalent to the existence of a rational point on C.
On the other hand the existence of a K-rational anticanonical divisor is always
guaranteed and in the case of a rational curve it has degree 2 (see section 7.5). The
idea of the parametrization algorithms in [10] and [18] can therefore be explained as
follows: Compute the linear system associated to a K-rational anticanonical divisor.

This system defines a birational map from C to a conic C ⊂ P (see 7.7.5 in the
2
K
example section). The parametrization of C is an easy task once a point on C is

known. The algebraic degree of the resulting parametrization obviously depends on
the degree of the field extension needed to define that point. Hence it is at most two
and the problem of parametrizing a rational curve using a minimal field extension is
reduced to the problem of finding a rational point on a conic if it exists. This task
is not straight forward. For example if K = Q we refer to [11] and [17]. In this
case rational points may also be found by the function RationalPoint of the
computer algebra system Magma [2].

7.5 An anticanoncial divisor

π ι
We are in the situation C → C → S where C is an irreducible curve, embedded in

the smooth toric surface S, and C is a resolution. Throughout this section we further

assume that genus(C)=genus(C)=0.

7.5.1 Support divisors
For l, k ∈ Z/nZ we write [l, k]= {l, l +1,...,k}⊂ Z/nZ for the set of cyclically
consecutive indices between l and k.Fromnow on I will always denote an “interval”,
i.e. I =[l, k] or I = ∅, and δ I will be its characteristic function, i.e. δ I (i)=1 if
i ∈ I and δ I (i)=0 else. Set

D I := (−c i − 1+ δ I (i))E i ∈ Div(S).
i∈[1,n]