Page 122 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 123
what “inserting an edge” meant for Π 1 in figure 7.2. It basically corresponds to the
resolution of a toric surface, see e.g. [4].
T
T
The values det((a i−1 ,b i−1 ) , (a i ,b i ) )= a i−1 b i − a i b i−1 are invariant under
unimodular transformations (i.e. linear transformations of the vectors by an integral
b
− a i 0 i 0 −1 > 1 for some i 0 .By
matrix with determinant 1). Assume that a i 0 −1 b i 0
)=(0, 1). It follows
,b i 0
a suitable unimodular transformation we may assume (a i 0
that a i 0 −1 > 1.
We insert a new index, for simplicity say i 0 − ,set a 1 := 1 and determine
1
i 0 −
2 2
b 1 by integer division s.t. 0 ≤ a i 0 −1 b 1 − b i 0 −1 <a i 0 −1 . It follows
i 0 − i 0 −
2 2
a − a i 0 i 0 − 1 =1 · 1 − 0 · b =1 and
b
i 0 − 1 b i 0 i 0 − 1
2 2 2
a i 0 −1 b 1 − a 1 b i 0 −1 = a i 0 −1 b
i 0 − i 0 − i 0 − 1 − 1 · b i 0 −1 <a i 0 −1
2 2 2
b
− a i 0 i 0 −1 .
= a i 0 −1 · 1 − 0 · b i 0 −1
= a i 0 −1 b i 0
By inserting the additional support half plane with normal vector (a 1 ,b 1 )
i 0 − i 0 −
2 2
b
−a i 0 i 0 −1
, we “substitute” the value a i 0 −1 b i 0
and support line through the vertex v i 0
by the smaller value a i 0 −1 b 1 − a 1 b i 0 −1 and add a b 1 =1
i 0 − i 0 − i 0 − 1 b i 0 − a i 0 i 0 −
2 2 2 2
to the list. All other values stay fixed. Repeating this process statement (7.3) can be
achieved.
Constructing the surface
Now we construct a surface following the examples of P and P ×P .For 1 ≤ i ≤
2
1
1
K K K
n let U i := A be copies of the affine plane with coordinates u i and v i .Again we
2
K
identify U 0 and U n . We denote the coordinate axes by L i := {(u i ,v i ) ∈ U i | u i =
0} and R i := {(u i ,v i ) ∈ U i | v i =0} and define open embeddings of the algebraic
∗ ∗
torus T := (K ) where K = K \{0}:
2
ψ i : T → U i :(x, y) → (u i ,v i )=(x y ,x −b i−1 a i−1 )
y
b i −a i
The isomorphic image of ψ i is U i \ (L i ∪ R i ) and there it has the inverse
v ).
U i \ (L i ∪ R i ) → T :(u i ,v i ) → (x, y)=(u a i−1 a i b i−1 b i
v ,u
i i i i
For i, j ∈{1,...,n}, i = j we define open subsets
⎧
if i ≡ j − 1mod n,
⎨ U i \ L i
U i,j := U i \ R i if i ≡ j +1 mod n,
U i \ (L i ∪ R i )= ψ i (T) else.
⎩
For 1 ≤ i ≤ n the following maps are mutually inverse and therefore isomorphisms:
ϕ i−1,i : U i−1,i → U i,i−1 :(u i−1 ,v i−1 ) → (u i ,v i )=(u a i−2 b i −a i b i−2 v i−1 ,u −1 )
i−1 i−1
ϕ i,i−1 : U i,i−1 → U i−1,i :(u i ,v i ) → (u i−1 ,v i−1 )=(v −1 ,u i v a i−2 b i −a i b i−2 )
i i
If i and j are non-neighboring indices we set