Real Line Arrangements and Surfaces with Many Real Nodes

Abstract

A long standing question is if maximum number $\mu(d)$ of nodes on a surface of degree $d$ in $\dP^3(\dC)$ can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, $\mu(d) \gtrapprox {5/12}d^3$, is provided by Chmutov's construction from 1992 which gives surfaces whose nodes have non-real coordinates. Using explicit constructions of certain real line arrangements we show that Chmutov's construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov's lower bound (i.e., for $d=3,4,...,8,10,12$) can also be realized with only real singularities. Thus, our result shows that, up to now, all known lower bounds can be attained with only real singularities. We conclude with an application of the theory of real line arrangements which shows that our arrangements are aymptotically the best possible ones. This proves a special case of a conjecture of Chmutov.

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arXiv:math/0507234v2 [math.AG] 23 Nov 2005
REAL LINE ARRANGEMENTS AND SURFACES
WITH MANY REAL NODES
SONJA BRESKE, OLIVER LABS, AND DUCO VAN STRATEN
Abstract. A long standing question is if maximum number µ(d) of nodes on
a surface of degree d in
reals which has only real singularities. The currently best known asymptotic
lower bound, µ(d) ?
which gives surfaces whose nodes have non-real coordinates.
Using explicit constructions of certain real line arrangements we show that
Chmutov’s construction can be adapted to give only real singularities. All
currently best known constructions which exceed Chmutov’s lower bound (i.e.,
for d = 3,4,··· ,8,10,12) can also be realized with only real singularities.
Thus, our result shows that, up to now, all known lower bounds for µ(d) can
be attained with only real singularities.
We conclude with an application of the theory of real line arrangements
which shows that our arrangements are aymptotically the best possible ones.
This proves a special case of a conjecture of Chmutov.
?3( ?) can be achieved by a surface defined over the
5
12d3, is provided by Chmutov’s construction from 1992
Introduction
A node (or A1singularity) in
form x2+y2+z2= 0 in some local coordinates. We denote by µ(d) the maximum
possible number of nodes on a surface in
has a long and rich history. Currently, µ(d) is only known for d = 1,2,...,6 (see
[1, 11] for sextics and [15] for a recent improvement for septics).
In this paper, we consider the relationship between µ(d) and the maximum
possible number of real nodes on a surface in
Obviously, µ
the maximum number of nodes be achieved with real surfaces with real singularities?
The previous question arises naturally because all results in low degree d ≤ 12
suggest that it could be true (see [1, 15, 8, 7, 19] and table 1). But the best known
asymptotic lower bound, µ(d) ?
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 1 on the following page). In the real case we can distinguish between two
types of nodes, conical nodes (x2+y2−z2= 0) and solitary points (x2+y2+z2= 0):
Our construction produces only conical nodes.
Notice that in general there are no better real upper bounds for µ
than the well-known complex ones of Miyaoka [17] and Varchenko [20]. But for
solitary points there exist better bounds via the relation to the zerothBetti number
?3is a singular point which can be written in the
?3( ?). The question of determining µ(d)
?3( ?) which we denote by µ
?(d).
?(d) ≤ µ(d), but do we even have µ
?(d) = µ(d)? In other words: Can
5
12d3, follows from Chmutov’s construction [5]
?(d) known
Date: February 2, 2008.
2000 Mathematics Subject Classification. Primary 14J17, 14J70; Secondary 14P25.
Key words and phrases. algebraic geometry, real algebraic geometry, many real singularities.
1

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2SONJA BRESKE, OLIVER LABS, AND DUCO VAN STRATEN
(see e.g., [12]). E.g., Rohn showed in 1913 that a real quartic surface in
cannot have more than 10 solitary points although it can have 16 conical nodes.
We show a real upper bound of ≈5
on two levels of real simple line arrangements consisting of d lines. In [6], Chmutov
conjectured this to be the maximum number for all complex plane curves of degree
d. He also 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 asymptotically the best possible real line arrangements for this purpose.
?3( ?)
6d2for the maximum number of critical points
d 1 2 3 456789 101112 13
d
µ(d),µ
?(d) ≤ 0 1 4 16 31 65 104 174 246 360 480 645 832
4
9d(d − 1)2
≈
µ(d),µ
?(d) ≥ 0 1 4 16 31 65 99 168 216 345 425 600 732
5
12d3
Table 1. The currently known bounds for the maximum number
µ(d) (resp. µ
?(d)) of nodes on a surface of degree d in
?3( ?) (resp.
?3( ?)) are equal. The bold numbers indicate in which cases our
result improves the previously known lower bound for µ
?(d).
1. Variants of Chmutov’s Surfaces with Many Real Nodes
Let Td(z) ∈
ues −1 and +1 (see fig. 1 on page 4), i.e.: T0(z) := 1,
2·z·Td−1(z)−Td−2(z) for d ≥ 2. Chmutov [5] uses them together with the so-called
folding polynomials FA2
d(x,y) ∈
struct surfaces ChmA2
folding polynomials are defined as follows:
?[z] be the Tchebychev polynomial of degree d with critical val-
T1(z) := z,
Td(z) :=
?[x,y] associated to the root-system A2 to con-
d(x,y,z) := FA2
d(x,y)+1
2(Td(z)+1) with many nodes. These
(1) FA2
d (x,y) := 2 + det












x 1
2y x
3 y
0 1
...
...
0 ··· ··· 0
0 ··· ··· ··· 0
......
.........
............
............... 0
............ 1
...
...
...
1
y x












+ det












y
10 ··· ··· ··· 0
......
.........
...............
............... 0
............ 1
0 ··· ··· 0
2x y
3 x
0 1
...
...
...
...
1
x y












.
The FA2
−1, and 8. Thus, the surface ChmA2
which the critical values of FA2
both are 0 or the first is −1 and the second is +1).
Notice that the plane curve defined by FA2
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
FA2
d(x,y) has only few different critical values. In [3], the first author computed
the critical points of the other folding polynomials. Among these, there are the
d(x,y) have critical points with only three different critical values: 0,
d(x,y,z) is singular exactly at those points at
d(x,y) and1
2(Td(z) + 1) sum up to zero (i.e., either
d(x,y) consists in fact of d lines. But

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REAL LINE ARRANGEMENTS AND SURFACES WITH MANY REAL NODES3
following examples which are the real line arrangements we have been looking for
(see [3, p. 87–89]):
We define the real folding polynomial FA2
system A2as (see also fig. 1 on the following page)
?,d(x,y) ∈
?[x,y] associated to the root
(2)
FA2
?,d(x,y) := FA2
d(x + iy, x − iy),
where i is the imaginary number. It is easy to see that the FA2
real coefficients. The numbers of critical points are the same as those of FA2
but now they have real coordinates as the following lemma shows:
?,d(x,y) have indeed
d(x,y);
Lemma 1. The real folding polynomial FA2
has?d
1
3d2− d
?,d(x,y) associated to the root system A2
2
?real critical points with critical value 0 and
if d ≡ 0mod 3,
1
3d2− d +2
3
otherwise(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 FA2
h1:
?,d, we use the map
?2→
?2, defined by
(u,v) ?→?cos(2π(u+v))+cos(2πu)+cos(2πv), sin(2π(u+v))−sin(2πu)−sin(2πv)?.
This is in fact just the real and imaginary part of the first component of the gener-
alized cosine h considered by Withers [21] and Chmutov [5]. It is easy to see that
h1is a coordinate change if u − v > 0, u + 2v > 0, and 2u + v < 1. It transforms
the polynomial FA2
d
?,dinto the function GA2
:
?2→
?2, defined by
GA2
d(u,v) := FA2
?,d(h1(u,v)) = 2cos(2πdu) + 2cos(2πdv) + 2cos(2πd(u + v)) + 2.
The calculation of the critical points of GA2
in [5]. As the function GA2
defined by u−v > 0, u+2v > 0, and 2u+v < 1, the images of these points under
the map h1are all the critical points of the real folding polynomial FA2
d. In contrast to [5], we get real critical points because h1is a map from
itself.
d
is exactly the same as the one performed
has (d − 1)2distinct real critical points in the region
d
?,dof degree
?2into
?
None of the other root systems yield more critical points on two levels. But as
mentioned in [16], the real folding polynomials associated to the root system B2
give hypersurfaces in
for the maximum number of nodes in higher dimensions slightly (see [16]; [3] gives
a detailed discussion of all these folding polynomials and their critical points).
The lemma immediately gives the following variant of Chmutov’s nodal surfaces:
?n, n ≥ 5, which improve the previously known lower bounds
Theorem 1. Let d ∈
?. The real projective surface of degree d defined by
(4) ChmA2
?,d(x,y,z) := FA2
?,d(x,y) +1
2(Td(z) + 1) ∈
?[x,y,z]

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4SONJA BRESKE, OLIVER LABS, AND DUCO VAN STRATEN
Figure 1. For degree d = 9 we show the Tchebychev polynomial
T9(z), the real folding polynomial FA2
system A2, and the surface ChmA2
in which FA2
?,9(x,y) associated to the root
?,9(x,y,z). The bounded regions
?,9(x,y) takes negative values are marked in black.
has the following number of real nodes:
(5)
1
12
1
12
1
12
1
12
?5d3− 13d2+ 12d?
?5d3− 13d2+ 16d − 8?
?5d3− 14d2+ 13d − 4?
?5d3− 14d2+ 9d?
d(x,y,z). To our knowledge, the result gives new lower bounds for
the maximum number µ
for d = 9,11 and d ≥ 13, see table 1 on page 2. Notice that all best known
lower bounds for µ
not astonishing in view of the upper bounds for solitary points mentioned in the
introduction.
if d ≡ 0 mod 6,
if d ≡ 2,4mod 6,
if d ≡ 1,5mod 6,
if d ≡ 3mod 6.
These numbers are the same as the numbers of complex nodes of Chmutov’s
surfaces ChmA2
?(d) of real singularities on a surface of degree d in
?3( ?)
?(d) are attained by surfaces with only conical nodes which is
2. On Two-Colorings of Real Simple Line Arrangements
The real folding polynomials FA2
real simple (straight) line arrangements in
a point. Such arrangements can be 2-colored in a natural way (see fig. 1): We label
in black those regions (cells) of
negative values, the others in white. The bounded black regions in fig. 1 contain
exactly one critical point with critical value −1 each.
Harborth has shown in [10] that the maximum number Mb(d) of black cells in
such real simple line arrangements of d lines satisfies:
?,d(x,y) used in the previous section are in fact
?2, i.e., lines no three of which meet in
?2\ {FA2
?,d(x,y) = 0} in which FA2
?,d(x,y) takes
(6)
Mb(d) ≤
?
1
3d2+1
1
3d2+1
3d,
6d,
d odd,
d even.
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 [9]). Notice that this
bound is better than the one obtained by Kharlamov using Hodge theory [13]. It is
known that the bound (6) is exact for infinitely many values of d. The real folding

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REAL LINE ARRANGEMENTS AND SURFACES WITH MANY REAL NODES5
polynomials FA2
have the very 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:
?,d(x,y) almost achieve this bound. Moreover, our arrangements
Lemma 2 (see Lemme 10, 11 in [18]). Let f be a real simple line arrangement
consisting of d ≥ 3 lines. f has exactly
contains exactly one critical point. All the critical points of f are non-degenerate.
?d−1
2
?
bounded open cells each of which
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 the number (d−1)2−?d
best possible ones for constructing surfaces with many singularities:
2
?=?d−1
2
?of all non-zero critical points gives the
result. Now we can show that our real line arrangements are asymptotically the
Theorem 2. The maximum number of critical points with the same non-zero crit-
ical value 0 ?= v ∈
where d is the number of lines. In particular, the maximum number of critical
points on two levels of such an arrangement does not exceed?d
Proof. In view of the upper bound (6) for the maximum number Mb(d) of black
cells of a real simple line arrangement we only have to verify that any bounded cell
contains only one critical point. But this follows from the preceding lemma.
? of a real simple line arrangement is bounded by Mb(d) − d,
2
?+Mb(d)−d ≈5
6d2.
?
Chmutov showed a much more general result ([4], see [6] for the case of non-
degenerate critical points): For a plane curve of degree d the maximum number
of critical points on two levels does not exceed ≈
≈5
ments FA2
d(x,y) he used for his construction (and also by the real line arrangement
FA2
particular case of real simple line arrangements. As Chmutov remarked in [5], such
an upper bound immediately implies an upper bound on the maximum number of
nodes on a surface in separated variables:
7
8d2.In [6], he conjectured
6d2to be the actual maximum which is attained by the complex line arrange-
?,d(x,y)). Thus, our theorem 2 is the verification of Chmutov’s conjecture in the
Corollary 3. A surface of the form p(x,y) + q(z) = 0 cannot have more than
≈1
2d =
number is attained by the surfaces ChmA2
2d2·1
2d +1
3d2·15
12d3nodes if p(x,y) is a real simple line arrangement. This
?,d(x,y,z) defined in theorem 1.
Comparing this number to the upper bound ≈
(see e.g., [12, p. 533]) one is tempted to ask if it is possible to deform our singular
surfaces to get examples with many real connected components. But our surfaces
ChmA2
y2−z2= 0). When removing the singularities from the zero-set of the surface every
connected component contains at least three of the singularities. Thus, the zeroth
Betti number of a small deformation of our surfaces are not larger than ≈
which is far below the number ≈13
Conversely, we may ask if it is always possible to move the lines of a simple real
line arrangement in such a way that all critical points which have a critical value of
the same sign can be chosen to have the same critical value. If this were true then
it would be possible to improve our lower bound for the maximum number µ
5
12d3on the zerothBetti number
?,d(x,y,z) only contain A−
1singularities which locally look like a cone (x2+
5
3·12d3
36d3resulting from Bihan’s construction [2].
?(d)