Ruled Varieties: An Introduction to Algebraic Differential Geometry

152 Pages·2001·6.687 MB·English

by  Prof. Dr. Gerd Fischer, Dr. Jens Piontkowski (auth.)

Gerd Fischer
Jens Piontkowski
Ruled Varieties
Advanced Lectures
in Mathematics
Editorial board:
Prof. Dr. Martin Aigner, Freie Universitat Berlin, Germany
Prof. Dr. Michael Griiter, Universitat des Saarlandes, Saarbriicken, Germany
Prof. Dr. Rudolf Scharlau, Universitat Dortmund, Germany
Prof. Dr. Gisbert Wiistholz, ETR Ziirich, Switzerland
Introduction to Markov Chains
Erhard Behrends
Einfiihrung in die Symplektische Geometrie
Rolf Berndt
Wavelets - Eine Einfiihrung
Christian Blatter
Local Analytic Geometry
Theo de Jong, Gerhard Pfister
Ruled Varieties
Gerd Fischer, Jens Piontkowski
Dirac-Operatoren in der Riemannschen Geometrie
Thomas Friedrich
Hypergeometric Summation
Wolfram Koepf
The Steiner Tree Problem
Rans-Jiirgen Promel, Angelika Steger
The Basic Theory of Power Series
Jesus M. Ruiz
vieweg ________________- --'
Gerd Fischer
Jens Piontkowski
Ruled Varieties
An Introduction to
Algebraic Differential Geometry
aI
vleweg
Prof. Dr. Gerd Fischer
Dr. Jens Piontkowski
Heinrich-Heine-Universitat Dusseldorf
Mathematisches lnstitut
UniversitatsstraBe 1
40255 Dusseldorf, Germany
[email protected]
[email protected]
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First edition, May 2001
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Printed on acid-free paper
ISBN-13: 978-3-528-03138-1 e-ISBN-13: 978-3-322-80217-0
DOl: 10.1007/978-3-322-80217-0
Er redet nur von Regelflachen,
ich werd'mich an dem Flegel rachen!
Dedicated to the memory of Karl Stein (1913 -2000)
Preface
The simplest surfaces, aside from planes, are the traces of a line moving in ambient space or,
more precisely, the unions of one-parameter families of lines. The fact that these lines can
be produced using a ruler explains their name, "ruled surfaces." The mechanical production
of ruled surfaces is relatively easy, and they can be visualized by means of wire models.
These surfaces are not only of practical use, but also provide artistic inspiration.
Mathematically, ruled surfaces are the subject of several branches of geometry, especially
differential geometry and algebraic geometry. In classical geometry, we know that surfaces
of vanishing Gaussian curvature have a ruling that is even developable. Analytically, de
velopable means that the tangent plane is the same for all points of the ruling line, which is
equivalent to saying that the surface can be covered by pieces of paper. A classical result
from algebraic geometry states that rulings are very rare for complex algebraic surfaces
in three-space: Quadrics have two rulings, smooth cubics contain precisely twenty-seven
lines, and in general, a surface of degree at least four contains no line at all. There are ex
ceptions, such as cones or tangent surfaces of curves. It is also well-known that these two
kinds of surfaces are the only developable ruled algebraic surfaces in projective three-space.
The natural generalization of a ruled surface is a ruled variety, i.e., a variety of arbitrary
dimension that is "swept out" by a moving linear subspace of ambient space. It should
be noted that a ruling is not an intrinsic but an extrinsic property of a variety, which only
makes sense relative to an ambient affine or projective space. In this book, we consider
ruled varieties mainly from the point of view of complex projective algebraic geometry,
where the strongest tools are available. Some local techniques could be generalized to
complex analytic varieties, but in the real analytic or even differentiable case there is little
hope for generalization: the reason being that rulings, and especially developable rulings,
have the tendency to produce severe singularities.
As in the classical case of surfaces, there is a strong relationship between the subject of this
book, ruled varieties, and differential geometry. For our purpose, however, the Hermitian
Fubini-Study metric and the related concepts of curvature are not necessary. In order to
detect developable rulings, it suffices to consider a bilinear second fundamental form that
is the differential of the GauS map. This method does not give curvature as a number, but
rather measures the degree of vanishing of curvature; this point of view has been used in
a fundamental paper of GRIFFITHS and HARRIS [GH21. The purposes of this book are
to make parts of this paper more accessible, to give detailed and more elementary proofs,
and to report on recent progress in this area. This text can also serve as an introduction to
"bilinear" complex differential geometry, a useful method in algebraic geometry.
In Chapter 0 we recall some classical facts about developable surfaces in real affine and
complex projective three-space. The basic results are a) the linearity of the fibers of the
GauS map and b) the classification of developable surfaces, locally in the real differentiable
case and globally in the complex projective case. We present very elementary proofs, which
can be adapted to more general situations.
VI
Families of linear spaces correspond to subsets of Grassmannians. Chapter 1 contains all
the facts we need about these varieties and more; for instance, there is a meticulous and
yet simple proof of the Pliicker relations, which is not easy to find in the existing literature.
Varieties that are unions of one-parameter families of linear spaces correspond to curves
in the Grassmannian. Following [GH2] , we derive a normal form representation of such
curves, which then implies the classification of developable varieties of GauB rank one in
Chapter 2.
Chapter 2 is the core of the book. We first introduce the different concepts of rulings of
a variety and methods to construct them. Then we concentrate on the additional condition
of developability; this class of ruled varieties is accessible to the methods of differential
geometry, especially the GauB map and the associated second fundamental form. The basic
theorem of this chapter is the linearity of the fibers of the GauB map, for which we give
proofs from various points of view. This result shows the GauS ruling to be the maximal
developable ruling and the GauB rank to be an invariant of the variety.
The central problem concerning developable varieties is their classification, and there are
two cases where it can be solved:
- Arbitrary dimension and GauB rank one (2.2.8).
- Codimension one and GauB rank two (2.5.6).
The key to the second case is duality or, more precisely, the dual variety of a given variety,
which is a subvariety of the dual projective space.
Chapter 3 is devoted to a very important class of ruled varieties: tangent and secant va
rieties. In general, the canonical ruling of the tangent variety is not developable, but it
contains a developable subruling by lines. An amazing result is the formation of a devel
opable ruling with larger fiber dimension if the tangent variety has a smaller than expected
dimension. The dimension of the tangent variety, like nearly all other invariants in this
book, can be computed from the second fundamental form. This is not the case, however,
for the dimension of the secant variety, where third order invariants become important. Un
til now, all attempts to find the right third or higher order invariant for the problem of this
dimension have failed, even for smooth varieties. Thus the book closes with one more open
problem.
The only prerequisites for this textbook are the basic concepts of complex affine and pro
jective algebraic geometry, the results of which are outlined in Chapter 1. The relatively
elementary methods that we use throughout are based on linear algebra with algebraic or
holomorphic parameters.
We wish to express our thanks to our collaborators Hung-Hsi Wu and Joseph Landsberg,
to Gabriele SuB for producing the camera-ready TEX-manuscript, and to our students for
their help in proofreading. We are also indebted to Vieweg for publishing this advanced
text.
Dusseldorf, March 2001 Gerd Fischer
Jens Piontkowski
Contents
o
Review from Classical Differential and Projective Geometry 1
0.1 Developable Rulings ... 1
0.2 Vanishing GauS Curvature 3
0.3 Hessian Matrices . . . . . 5
0.4 Classification of Developable Surfaces in]R3 . 7
0.5 Developable Surfaces in 1P3 (re) . . . . . . . . 9
1 Grassmannians 12
1.1 Preliminaries 12
1.1.1 Algebraic Varieties 12
1.1.2 Rational Maps .. 16
1.1.3 Holomorphic Linear Combinations 19
1.1.4 Limit Direction of a Holomorphic Path 20
1.1.5 Radial Paths 21
1.2 Plticker Coordinates . 22
1.2.1 Local Coordinates 22
1.2.2 The Plticker Embedding 23
1.2.3 Lines in 1P3 . . . . 24
1.2.4 The Plucker Image 25
1.2.5 Plucker Relations . 27
1.2.6 Systems of Vector Valued Functions 29
1.3 Incidences and Duality . . . . . . . . . . . 31
1.3.1 Equations and Generators in Terms of Plucker Coordinates. 31
1.3.2 Flag Varieties . . . . . . . 32
1.3.3 Duality of Grassmannians 33
1.3.4 Dual Projective Spaces .. 33
VIII
1.4 Tangents to Grassmannians . . . . . 34
1.4.1 Tangents to Projective Space 34
1.4.2 The Tangent Space of the Grassmannian . 35
1.5 Curves in Grassmannians 37
1.5.1 The Drill ... 37
1.5.2 Derived Curves 39
1.5.3 Sums and Intersections. 43
1.5.4 Associated Curves and Curves with Prescribed Drill 45
1.5.5 Normal Form 47
2 Ruled Varieties 49
2.1 Incidence Varieties and Duality . 49
2.1.1 Unions of Linear Varieties 49
2.1.2 Fano Varieties . 50
2.1.3 Joins ..... 51
2.1.4 Conormal Bundle and Dual Variety 52
2.1.5 Duality Theorem 55
2.1.6 The Contact Locus 56
2.1.7 The Dual Curve . 57
2.1.8 Rational Curves . 59
2.2 Developable Varieties . 61
2.2.1 Rulings.... 61
2.2.2 Adapted Parameterizations . 63
2.2.3 Germs of Rulings . . . . . . 64
2.2.4 Developable Rulings and Focal Points . 65
2.2.5 Developability of Joins . . . . . . . . . 69
2.2.6 Dual Varieties of Cones and Degenerate Varieties 71
2.2.7 Tangent and Osculating Scrolls ........ . 74
2.2.8 Classification of Developable One Parameter Rulings . 77
IX
2.2.9 Example of a ''Twisted Plane" . . . . 78
2.2.10 Characterization of Drill One Curves 82
2.3 The GauB Map .......... . 85
2.3.1 Definition of the GauB Map 85
2.3.2 Linearity of the Fibers . . . 86
2.3.3 GauB Map and Developability 88
2.3.4 GauB Image and Dual Variety 88
2.3.5 Existence of Varieties with Given GauB Rank 89
2.4 The Second Fundamental Form. . . . . . . . . . . . 93
2.4.1 Definition of the Second Fundamental Form . 93
2.4.2 The Degeneracy Space 96
2.4.3 The Degeneracy Map . 97
2.4.4 The Singular and Base Locus 98
2.4.5 The Codimension of a Uniruled Variety 99
2.4.6 Fibers of the GauB Map ..... 101
2.4.7 Characterization of GauS Images. 103
2.4.8 Singularities of the GauB Map 106
2.5 GauB Defect and Dual Defect. . . . . 109
2.5.1 Dual Defect of Segre Varieties 110
2.5.2 GauB Defect and Singular Locus . 111
2.5.3 Dual Defect and Singular Locus 112
2.5.4 Computation of the Dual Defect 113
2.5.5 The Surface Case ....... . 115
2.5.6 Classification of Developable Hypersurfaces 116
2.5.7 Dual Defect of Uniruled Varieties .... 117
2.5.8 Varieties with Very Small Dual Varieties. 118
x
3 Tangent and Secant Varieties 119
3.1 Zak's Theorems ..... 119
3.1.1 Tangent Spaces, Tangent Cones, and Tangent Stars 119
3.1.2 Zak's Theorem on Tangent and Secant Varieties. 121
3.1.3 Theorem on Tangencies .. , 124
3.2 Third and Higher Fundamental Forms 125
3.2.1 Definition .......... . 125
3.2.2 Vanishing of Fundamental Forms 128
3.3 Tangent Varieties . . . . . . . . . . . . . 129
3.3.1 The Dimension of the Tangent Variety. 129
3.3.2 Developability of the Tangent Variety 130
3.3.3 Singularities of the Tangent Variety 133
3.4 The Dimension of the Secant Variety . 135
Bibliography 137
Index 140
List of Symbols 142