AN INVITATION TO WEB GEOMETRY From Abel's addition Theorem to the algebraization of ...

254 Pages·2009·1.79 MB·English

“coloquio”
i i
2009/6/2
page 1
i i
AN INVITATION TO WEB GEOMETRY
From Abel’s addition Theorem to the
algebraization of codimension one webs
Jorge Vit´orio Pereira and Luc Pirio
June 2, 2009
i i
i i
“coloquio”
i i
2009/6/2
page v
i i
Para Dayse e Jorge,
os meus Pastores.
J.V.P.
Pour Min’
L.P.
i i
i i
“coloquio”
i i
2009/6/2
page vi
i i
i i
i i
“coloquio”
i i
2009/6/2
page vii
i i
Preface
Thefirstpurposeofthisbookistoserveassupportingmaterialfor
a mini-course on web geometry to be delivered at the 27th Brazilian
MathematicalColloquiumwhichwilltakeplaceatIMPAinlastweek
ofJuly2009. But,initsalmost250pagesthereismuchmorethanone
can possibly cover in five lectures of one hour each. The abundance
of material is due to the second purpose of this text: convey some
of the beauty of web geometry and to provide an account, as self-
contained as possible, of some of the exciting advancements the field
has witnessed in the last few years.
We have tried to write a book which is not very demanding in
termsofpre-requisites. Itistruethatatsomepointsfamiliaritywith
the basic language of algebraic/complex geometry is welcome but,
except at very few passages,not much more is needed. An effort has
been made to explain, even if sometimes superficially, every single
unusual concept appearing in the text.
At an early stage of this project we decided to use the third in-
stead of the first person. In retrospect, it is hard to understand why
two authors,none of them particularly comfortable with the English
language,tookthisdecision. Today,the onlyexplanationthatcomes
to mindisasubconsciousattemptto expirethe sinsoftwobadwrit-
ers. We apologize for the awkwardness of the prose and hope that
those more familiar with English than us will find some amusement
with the clumsiness of it.
Thistextwouldtakemuchlongertocometolightwithoutthein-
vitation of M´arcio Gomes Soares to submit a mini-course proposal
to the 27th Brazilian Mathematical Colloquium. Besides Soares,
i i
i i
“coloquio”
i i
2009/6/2
page viii
i i
viii
we would like to thank Hernan Falla Luza and Paulo Sad, whom
catched a number of misprints and mistakes appearing in prelimi-
nary versions. We are also indebted to Annie Bruter for her help in
translatingtoEnglishadraftoftheintroductionoriginallywrittenin
French. Evenmoreimportantly,JorgewantstothankDayseandLuc
wants to thank Mina for all the patience and unconditional support
gracefully given during the writing of this book.
Rio de Janeiro and Rennes, May 2009.
Jorge Vit´orio Pereira and Luc Pirio
Jorge Vit´orio Pereira Luc Pirio
IMPA IRMAR – UMR 6625 du CNRS
[email protected] [email protected]
i i
i i
“coloquio”
i i
2009/6/2
page ix
i i
Contents
Preface vii
Introduction 1
Conventions 15
1 Local and global webs 17
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 Germs of smooth webs . . . . . . . . . . . . . . 18
1.1.2 Equivalence and first examples . . . . . . . . . 19
1.1.3 Algebraic webs . . . . . . . . . . . . . . . . . . 21
1.2 Planar 3-webs . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.1 Holonomy and hexagonal webs . . . . . . . . . 23
1.2.2 Curvature for planar 3-webs . . . . . . . . . . . 26
1.2.3 Germs of hexagonal webs on the plane . . . . . 31
1.2.4 Algebraic planar 3-webs are hexagonal . . . . . 32
1.3 Singular and global webs . . . . . . . . . . . . . . . . . 36
1.3.1 Germs of singular webs I . . . . . . . . . . . . 36
1.3.2 Germs of singular webs II . . . . . . . . . . . . 38
1.3.3 Global webs . . . . . . . . . . . . . . . . . . . . 40
1.3.4 Discriminant . . . . . . . . . . . . . . . . . . . 42
1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.4.1 Global webs on projective spaces . . . . . . . . 48
1.4.2 Algebraic webs revisited . . . . . . . . . . . . . 49
1.4.3 Projective duality . . . . . . . . . . . . . . . . 54
1.4.4 Webs attached to projective surfaces . . . . . . 56
i i
i i
“coloquio”
i i
2009/6/2
page x
i i
x CONTENTS
2 Abelian relations 61
2.1 Determining the abelian relations . . . . . . . . . . . . 63
2.1.1 Abel’s method . . . . . . . . . . . . . . . . . . 63
2.1.2 Webs with infinitesimal automorphisms . . . . 68
2.2 Bounds for the rank . . . . . . . . . . . . . . . . . . . 72
2.2.1 Bounds for ℓj( ) . . . . . . . . . . . . . . . . 73
W
2.2.2 Bounds for the rank . . . . . . . . . . . . . . . 75
2.2.3 Webs of maximal rank . . . . . . . . . . . . . . 78
2.3 Conormals of webs of maximal rank . . . . . . . . . . 78
2.3.1 Rational normal curves . . . . . . . . . . . . . 80
2.3.2 Proof of Castelnuovo Lemma . . . . . . . . . . 83
2.3.3 Normal forms for webs of maximal rank . . . . 85
2.3.4 A generalizationof Castelnuovo Lemma . . . . 87
3 Abel’s addition Theorem 91
3.1 Abel’s Theorem I: smooth curves . . . . . . . . . . . . 92
3.1.1 Trace under ramified coverings . . . . . . . . . 92
3.1.2 Trace relative to the family of hyperplanes. . . 94
3.1.3 Abel’s Theorem for smooth curves . . . . . . . 94
3.1.4 Abelian relations for algebraic webs . . . . . . 96
3.1.5 Castelnuovo’s bound . . . . . . . . . . . . . . . 97
3.2 Abel’s Theorem II: arbitrary curves. . . . . . . . . . . 98
3.2.1 Residues and traces . . . . . . . . . . . . . . . 99
3.2.2 Abelian differentials . . . . . . . . . . . . . . . 101
3.2.3 Abel’s addition Theorem . . . . . . . . . . . . 105
3.2.4 Abelian differentials for curves on surfaces . . . 106
3.3 Algebraic webs of maximal rank . . . . . . . . . . . . 110
3.3.1 Curves on the Veronese surface . . . . . . . . . 111
3.3.2 Curves on rational normal scrolls . . . . . . . 111
3.3.3 Webs of maximal rank . . . . . . . . . . . . . . 113
3.4 Webs and families of hypersurfaces . . . . . . . . . . . 115
3.4.1 Dual webs with respect to a family . . . . . . . 115
3.4.2 Trace relative to families of hypersurfaces . . . 116
3.4.3 Bounds for rank and genus . . . . . . . . . . . 117
3.4.4 Families of theta translates . . . . . . . . . . . 118
i i
i i
“coloquio”
i i
2009/6/2
page xi
i i
CONTENTS xi
4 The ubiquitous algebraization tool 121
4.1 The converse of Abel’s Theorem . . . . . . . . . . . . 122
4.1.1 Statement . . . . . . . . . . . . . . . . . . . . . 122
4.1.2 Dual formulation . . . . . . . . . . . . . . . . . 123
4.2 Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 Reduction to dimension two . . . . . . . . . . . 126
4.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . 127
4.2.3 Lifting to the incidence variety . . . . . . . . . 129
4.2.4 Back to the projective plane. . . . . . . . . . . 130
4.2.5 Globalizing to conclude . . . . . . . . . . . . . 133
4.3 Algebraization of smooth 2n-webs . . . . . . . . . . . 134
4.3.1 The Poincar´emap . . . . . . . . . . . . . . . . 135
4.3.2 Linearization . . . . . . . . . . . . . . . . . . . 135
4.3.3 Algebraization . . . . . . . . . . . . . . . . . . 137
4.3.4 Poincar´e map for planar webs . . . . . . . . . . 138
4.3.5 Canonical maps. . . . . . . . . . . . . . . . . . 140
4.4 Double-translation hypersurfaces . . . . . . . . . . . . 141
4.4.1 Examples . . . . . . . . . . . . . . . . . . . . . 143
4.4.2 Abel-Jacobi map . . . . . . . . . . . . . . . . . 144
4.4.3 Classification . . . . . . . . . . . . . . . . . . . 145
5 Algebraization 147
5.1 Tr´epreau’s Theorem . . . . . . . . . . . . . . . . . . . 148
5.1.1 Statement . . . . . . . . . . . . . . . . . . . . . 148
5.1.2 The geometry of Castelnuovo curves . . . . . . 149
5.1.3 On the geometry of maximal rank webs . . . . 152
5.2 Maps naturally attached to . . . . . . . . . . . . . 153
W
5.2.1 Canonical maps. . . . . . . . . . . . . . . . . . 154
5.2.2 Poincar´e’smap . . . . . . . . . . . . . . . . . . 154
5.2.3 Properties . . . . . . . . . . . . . . . . . . . . . 155
5.3 Poincar´e-Blaschke’smap . . . . . . . . . . . . . . . . . 158
5.3.1 Interpolation of the canonical maps . . . . . . 160
5.3.2 Definition of Poincar´e-Blaschke’smap . . . . . 163
5.3.3 The rank of Poincar´e-Blaschke’smap. . . . . . 164
5.4 Poincar´e-Blaschke’ssurface . . . . . . . . . . . . . . . 174
5.4.1 Rational normal curves everywhere . . . . . . . 175
5.4.2 Algebraization I : X is algebraic . . . . . . . 175
W
5.4.3 Algebraization II and conclusion . . . . . . . . 177
i i
i i
“coloquio”
i i
2009/6/2
page xii
i i
xii CONTENTS
6 Exceptional Webs 181
6.1 Criterium for linearization . . . . . . . . . . . . . . . . 182
6.1.1 Characterizationof linear webs . . . . . . . . . 182
6.1.2 Affine and projective connections . . . . . . . . 184
6.1.3 Linearization of planar webs. . . . . . . . . . . 187
6.1.4 First examples of exceptional webs . . . . . . . 190
6.1.5 Algebraization of planar webs . . . . . . . . . 191
6.2 Infinitesimal automorphisms . . . . . . . . . . . . . . . 193
6.2.1 Basics on differential algebra . . . . . . . . . . 193
6.2.2 Variation of the rank . . . . . . . . . . . . . . . 195
6.2.3 Infinitely many families of exceptional webs . . 199
6.3 Pantazi-H´enautcriterium . . . . . . . . . . . . . . . . 200
6.3.1 Linear differential systems . . . . . . . . . . . . 200
6.3.2 The differential system . . . . . . . . . . . 207
W
S
6.3.3 Pantazi-H´enautcriterium . . . . . . . . . . . . 210
6.3.4 Mih˘aileanu criterium . . . . . . . . . . . . . . . 211
6.4 Classification of CDQL webs . . . . . . . . . . . . . . 211
6.4.1 Definition . . . . . . . . . . . . . . . . . . . . . 212
6.4.2 On the projective plane . . . . . . . . . . . . . 212
6.4.3 On Hopf surfaces . . . . . . . . . . . . . . . . . 216
6.4.4 On abelian surfaces . . . . . . . . . . . . . . . 216
6.4.5 Outline of the proof . . . . . . . . . . . . . . . 218
6.5 Further examples . . . . . . . . . . . . . . . . . . . . . 221
6.5.1 Polylogarithmic webs. . . . . . . . . . . . . . . 221
6.5.2 A very simple series of exceptional webs . . . . 225
6.5.3 An exceptional 11-web . . . . . . . . . . . . . . 226
6.5.4 Terracini and Buzano’s webs . . . . . . . . . . 227
Bibliography 231
List of Figures 241
Index 243
i i
i i
“coloquio”
i i
2009/6/2
page 1
i i
Introduction
It seems impossible to grasp the ins and outs of a mathematical
field without setting it back in its historical context. An attempt,
certainly incomplete and biased, is made in the next few pages. At
the end of the Introduction, one finds a description of the contents
of this book, and suggestions on how to use it.
Historical Notes
Ifthebirthofwebgeometrycanbeascribedtothemiddleofthe1930s
in Hambourg (see below), some precedents can be found as early as
the middle of the XIXth century. The concepts and problems of
web geometry springs from two different fields of the XIXth century
mathematics : projective differential geometry and nomography.
Itismainlyfromthefirstthatwebgeometrycomesfrom. Atthat
time, projective differential geometry mainly consisted of the study
of projective properties of curves and surfaces in R3, that is of their
differential properties that are invariant up to homographies.
Gaussiangeometry,whichhadappearedbefore,studiedtheprop-
erties of (curves and) surfaces in ordinary euclidian space that are
invariant up to isometric transformations. Gauss and other mathe-
maticians pointed out how useful the first and second fundamental
forms are for the study of surfaces. They also brought to light the
relevance of derived concepts, such as the principal, asymptotic and
conjugated directions. When considering the integralcurves of these
tangent direction fields, the mathematicians of the time were con-
i i
i i