Page 123 - Geometric Modeling and Algebraic Geometry

124 T. Beck and J. Schicho U 5

U 6

R 6 L 6 R 5 L 5 U 4
U 7
V 6
L 7
T ϕ 1,4 R 4 L 4 U 3 V 7 E 7 E 6 E 5 V 5 E 3 V 4
ψ 4
E 4
R 7
R 3
V 3
E 2
ϕ 1,7
E 1
ψ 1
L 3
V 2
V 1
U 1
R 2
L 1
R 1
L 2
ϕ 1,2
U 2
Fig. 7.3. Construction of the toric surface S
ϕ i,j := ψ j ◦ ψ −1 : U i,j → U j,i .
i
The above morphisms are compatible in the sense that ϕ j,i = ϕ −1 and for each k
i,j
we have ϕ i,j (U i,j ∩ U i,k )= U j,i ∩ U j,k and ϕ i,k = ϕ j,k ◦ ϕ i,j .
Now we can define an abstract surface S as a finite quotient S/ ≡ where S :=
˙ U i is the disjoint union of affine planes. We identify points along the morphisms

i
ϕ i,j , see figure 7.3. That is for i = j a point (a, b) ∈ U i and a point (c, d) ∈ U j are
equivalent (a, b) ≡ (c, d) if (c, d)= ϕ i,j (a, b) whenever this expression is defined.
This corresponds to the general gluing construction for schemes, see e.g. [9,
exercise II.2.12]. By abuse of notation we will from now on identify U i and its image
in S. Then {U i } 1≤i≤n is an open cover.
It is not hard to see that E i := R i ∪L i+1 is an irreducible curve on S and isomor-
phic to P .Wecallitan edge curve. The curves E i−1 and E i intersect transversally
1
K
in a point V i ∈ U i , corresponding to the origin (u i ,v i )=(0, 0) of the corresponding
chart. For non-neighboring indices i and j the edge curves E i and E j are disjoint.
The complement of the union of all edge curves is the torus T, which is also the
intersection of all open sets U i .
Comparing figures 7.2 and 7.3 suggests that there are certain correspondences
between a smooth polygon, here Π 2 , and the toric surface constructed from it. For
example there is a line P corresponding to each edge of the polygon and they inter-
1
K
sect accordingly. On the other hand the constructed surface is invariant with respect
to the scaling of the polygon and even the actual length of its edges. The only impor-
tant data is the set of normal vectors.
We briefly summarize important properties of the toric surface S: