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3 Real Line Arrangements and Surfaces with Many Real Nodes 51
These numbers are the same as the numbers of complex nodes of Chmutov’s
surfaces Chm A 2 (x, y, z). To our knowledge, the result gives new lower bounds for
d
the maximum number µ (d) of real singularities on a surface of degree d in P ( )
3
for d =9, 11 and d ≥ 13, see table 3.1. Notice that all best known lower bounds for
µ (d) are attained by surfaces with only conical nodes which is not astonishing in
view of the upper bounds for solitary points mentioned in the introduction.
3.3 On Two-Colorings of Real Simple Line Arrangements
The real folding polynomials F A 2 (x, y) used in the previous section are in fact real
,d
simple (straight) line arrangements in 2 , i.e., lines no three of which meet in a
point. Such arrangements can be 2-colored in a natural way (see fig. 3.2): We label
in black those regions (cells)of 2 \{F A 2 (x, y)=0} in which F A 2 (x, y) takes
,d ,d
negative values, the others in white. The bounded black regions in fig. 3.2 contain
exactly one critical point with critical value −1 each.
Harborth has shown in [11] that the maximum number M b (d) of black cells in
such real simple line arrangements of d lines satisfies:
d + d, d odd,
1
1 2
M b (d) ≤ 3 3 (3.6)
d + d, d even.
1
1 2
3 6
d of these cells are unbounded. This is a purely combinatorial result which is strongly
related to the problem of determining the maximum number of triangles in such
arrangements which has a long and rich history (see [10]). Notice that this bound is
better than the one obtained by Kharlamov using Hodge theory [13]. It is known that
the bound (3.6) is exact for infinitely many values of d. The real folding polynomials
F A 2 (x, y) almost achieve this bound. Moreover, our arrangements have the very
,d
special property that all critical points with a negative (resp. positive) critical value
have the same critical value −1 (resp. +8).
To translate the upper bound on the number of black cells into an upper bound
on critical points we use the following lemma:
Lemma 3 (see Lemme 10, 11 in [18]). Let f be a real simple line arrangement
consisting of d ≥ 3 lines. Then f has exactly d−1 bounded open cells each of
2
which contains exactly one critical point. Moreover, all the critical points of f are
non-degenerate. No unbounded open cell contains a critical point.
It is easy to prove the lemma, e.g. by counting the number of bounded cells and by
observing that each such cell contains at least one critical point. Comparing this with
d
the number (d−1) − = d−1 of all critical points with non-zero critical values
2
2 2
gives the result. Now we can show that our real line arrangements are asymptotically
the best possible ones for constructing surfaces with many singularities:
Theorem 4. The maximum number of critical points with the same non-zero real
critical value 0 = v ∈ of a real simple line arrangement is bounded by M b (d)−d,
where d is the number of lines. In particular, the maximum number of critical points
d
on two levels of such an arrangement does not exceed + M b (d) − d ≈ d .
5 2
2 6