Page 137 - Geometric Modeling and Algebraic Geometry

138 T. Beck and J. Schicho

0= c 1 − c 2
1 1 1
0= − c 5 + c 4 + c 2 − c 3
4 4 4
0= c 1 − c 3 + c 6
5
0= c 7 − c 4 + c 2 +5c 6 − c 3
2
0= c 1 − c 2 − 16c 3 +16c 4 − 16c 5 + 256c 6 − 256c 7 + 256c 8
Solving the system with respect to c 6 , c 7 and c 8 , we compute a basis of the vector
space V of theorem 12:

b 1 =4 + 4x +5y − yx + 15 yx + y 2
2
7
2 2
b 2 = −4 − 4x − 4y +7yx − 5yx + y x
2
2
b 3 =4 + 4x +4y − 6yx +6yx + y x
2 2
2
!
In other words g 1 g 2 , g 1 g 2 , g 1 g 2 is a basis of L C (−K ∅ ), the linear system of our

b 3
b 2
b 1
anticanonical divisor.
7.7.5 Birational equivalence to a conic
b i
The rational functions g 1 g 2 are the coordinates of a map from C to the projective
plane P . We get the same map if we multiply all coordinates by their common
2
Q
denominator g 1 g 2 :
⎛ ⎞

4+4x +5y − yx + 15 yx + y 2
7
2
x 2 2
C P : → ⎝ −4 − 4x − 4y +7yx − 5yx + y x
2
2 ⎠
2
Q y
4+4x +4y − 6yx +6yx + y x
2 2
2
Dividing by the last coordinate, we get a map to A :
2
Q
" #

8+8x+10y−7yx+15yx +2y 2
2
x
ψ 1 : C A : → 2(4+4x+4y−6yx+6yx 2 +y 2 x 2 )
2
Q y −4−4x−4y+7yx−5yx +y x
2
2
4+4x+4y−6yx+6yx 2 +y 2 x 2
The image of this map is a conic C ⊂ A . To avoid confusion, we use the

2
Q
coordinates x and y in the image domain. We can compute the implicit equation

f =12x y + y +9x of C by eliminating the variables x and y using Gr¨ obner

2
bases techniques. Then ψ 1 is a birational morphism with inverse
" #

(4y +3)(4y +36y +27)
2
x
ψ −1 : C C : → y (2y +3)(34y +27) .

2
1 y −(2y +3)(14y +45y +27)
4y 2 (4y +3)
In other words ψ −1 is a parametrization of C by the conic C .

1