Page 50 - Geometric Modeling and Algebraic Geometry

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Real Line Arrangements
and Surfaces with Many Real Nodes

Sonja Breske , Oliver Labs , and Duco van Straten 1
1
2
Institut f¨ur Mathematik, Johannes Gutenberg Universit¨ at, 55099 Mainz, Germany
1
Breske@Mathematik.Uni-Mainz.de,
Straten@Mathematik.Uni-Mainz.de
Institut f¨ur Mathematik, Universit¨ at des Saarlandes, Geb. E2.4, 66123 Saarbr¨ucken,
2
Germany
Labs@Math.Uni-Sb.de, Mail@OliverLabs.net

Summary. A long standing question is if the maximum number µ(d) of nodes on a surface
of degree d in P ( ) can be achieved by a surface defined over the reals which has only
3
real singularities. The currently best known asymptotic lower bound, µ(d) 5 d ,ispro-
3
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vided 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 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 for the purpose of constructing
surfaces with many nodes. This proves a special case of a conjecture of Chmutov.

3.1 Introduction

A node (or A 1 singularity) in 3 is a singular point which can be written in the
form x + y + z =0 in some local coordinates. We denote by µ(d) the maximum
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2
2
possible number of nodes on a surface in P ( ). The question of determining µ(d)
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has a long and rich history. Currently, µ(d) is only known for d =1, 2,..., 6 (see
[1, 12] for sextics and [15] for a recent improvement for septics).
In this paper, we consider the relationship between µ(d) and the maximum pos-
sible number of real nodes on a surface in P ( ) which we denote by µ (d).Obvi-
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ously, µ (d) ≤ µ(d), but do we even have µ (d)= µ(d)? In other words: Can 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, 8, 9, 15, 19] and table 3.1). But the best known
asymptotic lower bound, µ(d) 5 d , follows from Chmutov’s construction [5]
3
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