Page 136 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 137
0= c 0 , 0= c 2 − c 1 ,
1
1
4 4
0= c 0 − c 1 + c 3 , 5
0= −c 2 − c 1 +5c 3 + c 4
2
Note that this step would in general yield linear constraints over a field extension, e.g.
over Q(γ 1 ) or Q(γ 2 ). They can always be written as a bigger number of constraints
over the ground field, here Q. This is the algorithmic side of the statement that the
divisor K [1,1] is Q-rational. Solving the system with respect to c 3 and setting c 3 =1,

(K [7,8] ):
2 2 3 2 2
we get g 1 = xy + x y + xy − x y . We do the same again for g 2 ∈L C
2
g 2 := c 0 y + c 1 xy + c 2 x y + c 3 y + c 4 xy + c 5 x y
2 2
2
2
2
2 2
g 2,1,[7,8] = c 0 + c 1 u 1 + c 2 u + c 3 v 1 + c 4 u 1 v 1 + c 5 u v 1
1 1
yields the following con-
and ν 2 (g 2,1,[7,8] ) ≥ α ν 2
Requiring ν 1 (g 2,1,[7,8] ) ≥ α ν 1
straints:
0= c 0 − c 1 + c 2 , 0= c 4 − 2c 2 − c 5 − c 3 + c 1 ,
1
1
1
4 4 4
0= c 0 − c 3 , 5
0= −c 4 − c 3 + c 1
2
Solving the system with respect to c 3 and c 4 and setting c 3 =2 and c 4 = −1 (which
is a matter of choice), we get g 2 =2y +4xy +2x y +2y − xy − 3x y .
2 2
2
2
2
The system for computing g 1 had got an 1-dimensional solution. This implies that
the divisor (g 1 ) [1,1] is equal to A. Hence its support is exactly the preimage of the

singular locus. On the other hand we were left with essentially one degree of freedom
when computing g 2 and therefore (g 2 ) [7,8] > A. Indeed we find that the support of

g 2 has an additional point corresponding to Q 3 ∈ U 1 , namely (u 1 ,v 1 )=(−1, −16).
Therefore we have to consider one more valuation ν 3 centered at Q 3 (which one
could get again from a Puiseux series solution at Q 3 ). All this is reflected in the
twisted orders of g 1 and g 2 :
= ν 1,[1,1] (g 1 )=2, β 1,ν 2 = ν 2,[1,1] (g 1 )=2, β 1,ν 3 = ν 3,[1,1] (g 1 )=0,
β 1,ν 1
= ν 1,[7,8] (g 2 )=2, β 2,ν 1 = ν 2,[7,8] (g 2 )=2, β 2,ν 3 = ν 3,[7,8] (g 2 )=1
β 2,ν 1
=0.
Since Q 3 ∈ C is a smooth point, the adjoint order is α ν 3
7.7.4 Linear system of an anticanonical divisor
(K [7,1] ):

Now we make an Ansatz for an element h ∈L C
h := c 1 + c 2 x + c 3 y + c 4 xy + c 5 x y + c 6 y + c 7 xy + c 8 x y
2 2
2
2
2
h 1,[7,1] := c 1 + c 2 u 1 + c 3 v 1 + c 4 u 1 v 1 + c 5 u v 1 + c 6 v + c 7 u 1 v + c 8 u v
2
2 2
2
2
1 1 1 1 1
−
+ β 2,ν 2
− α ν 1
+ β 2,ν 1
Requiring ν 1 (h 1,[7,1] ) ≥ β 1,ν 1
=2, ν 2 (h 1,[7,1] ) ≥ β 1,ν 2
=1 (i.e. h 1,[7,1] also has to vanish
α ν 2
=2 and ν 3 (h 1,[7,1] ) ≥ β 1,ν 3
− α ν 3
+ β 2,ν 3
on Q 3 ) yields the following constraints: