Page 52 - Geometric Modeling and Algebraic Geometry

3 Real Line Arrangements and Surfaces with Many Real Nodes 49
⎛ ⎞ ⎛ ⎞
x 1 0 ··· ··· ··· 0 y 1 0 ··· ··· ··· 0
⎜ . . . . . . ⎟ ⎜ . . . . . . ⎟
⎜ 2yx . . . ⎟ ⎜ 2xy . . . ⎟
⎜ . . . . ⎟ ⎜ . . . . ⎟
⎜ 3 y . ⎟ ⎜ 3 x . ⎟
⎜ . . . . . . . ⎟ ⎜ . . . . . . . ⎟
.
.
F A 2 (x, y):=2 + det ⎜ . . . . . . . . . ⎟ +det ⎜ . . . . . . . . . ⎟ .
d ⎜ 01 . . . . . ⎟ ⎜ 01 . . . . . ⎟
⎜ . . . . . . ⎟ ⎜ . . . . . . ⎟
⎜ . . . . . . . . . . . 0 ⎟ ⎜ . . . . . . . . . . . 0 ⎟
⎜ . . . . . . ⎟ ⎜ . . . . . . ⎟
⎝ . . . . . ⎠ ⎝ . . . . . ⎠
. . . . . 1 . . . . . 1
0 ··· ··· 0 1 yx 0 ··· ··· 0 1 xy
(3.1)
The F A 2 (x, y) have critical points with only three different critical values: 0, −1,
d
and 8. Thus, the surface Chm A 2 (x, y, z) is singular exactly at those points at which
d
the critical values of F A 2 (x, y) and (T d (z)+1) sum up to zero (i.e., either both
1
d
2
are 0 or the first is −1 and the second is +1).
Notice that the plane curve defined by F A 2 (x, y) consists in fact of d lines. But
d
these are not real lines and the critical points of this folding polynomial also have
non-real coordinates. It is natural to ask whether there is a real line arrangement
which leads to the same number of critical points. The term folding polynomials was
introduced in [21] (here we use a slightly different definition). In his article, Withers
also described many of their properties, but it was Chmutov [5] who noticed that
F A 2 (x, y) has only few different critical values. In [3], the first author computed the
d
critical points of the other folding polynomials. Among these, there are the following
examples which are the real line arrangements we have been looking for (see [3, p.
87–89]):
We define the real folding polynomial F A 2 (x, y) ∈ [x, y] associated to the
,d
root system A 2 as (see also fig. 3.2)
F A 2 (x, y):= F A 2 (x + iy, x − iy), (3.2)
,d d
where i is the imaginary number. It is easy to see that the F A 2 (x, y) have indeed
,d
real coefficients. The numbers of critical points are the same as those of F A 2 (x, y);
d
but now they have real coordinates as the following lemma shows:
Lemma 1. The real folding polynomial F A 2
,d (x, y) associated to the root system A 2

d
has real critical points with critical value 0 and
2
1 1 2
d − d if d ≡ 0mod 3, d − d + otherwise (3.3)
2
2
3 3 3
real critical points with critical value −1. The other critical points also have real
coordinates and have critical value 8.
Proof. We proceed similar to the case discussed in [5], see [3, p. 87–95] for details.
To calculate the critical points of the real folding polynomial F A 2 ,weuse themap
,d
h : 2 → 2 , defined by
1

cos(2π(u + v)) + cos(2πu) + cos(2πv)
(u, v) → .
sin(2π(u + v)) − sin(2πu) − sin(2πv)