Page 51 - Geometric Modeling and Algebraic Geometry

48 S. Breske et al.
which yields only singularities with non-real coordinates. In this paper, we show that
his construction can be adapted to give surfaces with only real singularities (see table
3.1). In the real case we can distinguish between two types of nodes, conical nodes
(x +y −z =0) and solitary points (x +y +z =0): Our construction produces
2
2
2
2
2
2
only conical nodes.
Notice that in general there are no better real upper bounds for µ (d) known
than the well-known complex ones of Miyaoka [17] and Varchenko [20]. But in
some cases, for solitary points there exist better bounds via the relation to the zero th
Betti number. E.g., it has been shown by Nikulin that a K3 surface cannot have more
than 10 solitary points (although it can have 16 conical nodes). For quartic surfaces
in P this result is probably due to R.W.H.T. Hudson (see [7] for an overview on
3
related results).
d for the maximum number of real critical
We show an upper bound of ≈ 5 2
6
points on two levels of real simple line arrangements consisting of d lines; here, sim-
ple means that no three lines meet in a common point. In [6], Chmutov conjectured
this to be the maximum number for all complex plane curves of degree d.Healso
noticed [5] that such a bound directly implies an upper bound for the number of real
nodes of certain surfaces. Our upper bound shows that our examples are asymptoti-
cally the best possible real line arrangements for this purpose.
d 1 2 3 4 5 6 7 8 9 10 11 12 13 d
µ(d),µ (d) ≤ 0 1 4 16 31 65 104 174 246 360 480 645 832 4 d(d − 1) 2
9
µ(d),µ (d) ≥ 0 1 4 16 31 65 99 168 216 345 425 600 732 ≈ 5 d 3
12
Table 3.1. The currently known bounds for the maximum number µ(d) (resp. µ (d)) of nodes
on a surface of degree d in P ( ) (resp. P ( )) are equal. The bold numbers indicate in which
3
3
cases our result improves the previously known lower bound for µ (d).

3.2 Variants of Chmutov’s Surfaces with Many Real Nodes

Let T d (z) ∈ [z] be the Tchebychev polynomial of degree d with critical values −1
and +1 (see fig. 3.2). This can either be defined recursively by T 0 (z):=1, T 1 (z):=
z, T d (z):=2·z·T d−1 (z) − T d−2 (z) for d ≥ 2, or implicitly by T d (cos(z)) =
cos(dz). Chmutov [5] uses them together with the so-called folding polynomials
F A 2 (x, y) ∈ [x, y] associated to the root-system A 2 to construct surfaces
d
1
Chm A 2 (x, y, z):= F A 2 (x, y)+ (T d (z)+1)
d d
2
with many nodes. These folding polynomials are defined as follows: