Lectures on Invariant Theory

232 Pages·2003·0.88 MB·English

by  Dolgachev I.

LECTURES ON INVARIANT
THEORY
IGOR V. DOLGACHEV
February 12, 2003
ii
Preface
This book is based on one-semester graduate courses I gave at Michigan in 1994
and 1998, and at Harvard in 1999. A part of the book is borrowed from an earlier
version of my lecture notes which were published by the Seoul National Univer-
sity [22]. The main changes consist of including several chapters on algebraic
invariant theory, simplifying and correcting proofs, and adding more examples
from classical algebraic geometry. The last Lecture of [22] which contains some
applications to construction of moduli spaces has been omitted. The book is lit-
erally intended to be a first course in the subject to motivate a beginner to study
more. AneweditionofD.Mumford’sbookGeometricInvariantTheorywithap-
pendices by J. Fogarty and F. Kirwan [75] as well as a survey article of V. Popov
and E. Vinberg [91] will help the reader to navigate in this broad and old subject
ofmathematics. Mostoftheresultsandtheirproofsdiscussedinthepresentbook
can be found in the literature. We include some of the extensive bibliography of
the subject (with no claim for completeness). The main purpose of this book is
to give a short and self-contained exposition of the main ideas of the theory. The
sole novelty is including many examples illustrating the dependence of the quo-
tientonalinearizationoftheactionaswellasincludingsomebasicconstructions
intoricgeometryasexamplesoftorusactionsonaffinespace. Wealsogivemany
examples related to classical algebraic geometry. Each chapter ends with a set of
exercises and bibliographical notes. We assume only minimal prerequisites for
students: a basic knowledge of algebraic geometry covered in the first two chap-
tersofShafarevich’sbook[104]and/orHartshorne’sbook[46],agoodknowledge
of multilinear algebra and some rudiments of the theory of linear representations
of groups. Although we often use some of the theory of affine algebraic groups,
theknowledgeofthegroupGL isenoughforourpurpose.
n
I am grateful to some of my students and colleagues for critical remarks and
catching numerous mistakesin my lecture notes. Special thanks go toAna-Maria
Castravet,MihneaPopa,JanisStipinsandIvanArzhantsev.
iii

Contents
Preface iii
Introduction ix
1 Thesymbolicmethod 1
1.1 Firstexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Polarizationandrestitution . . . . . . . . . . . . . . . . . . . . . 4
1.3 Bracketfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 TheFirstFundamentalTheorem 17
2.1 Theomega-operator . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Theproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Grassmannvarieties . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Thestraighteningalgorithm . . . . . . . . . . . . . . . . . . . . . 22
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Reductivealgebraicgroups 29
3.1 TheGordan–HilbertTheorem . . . . . . . . . . . . . . . . . . . 29
3.2 Theunitarytrick . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Affinealgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Nagata’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
v
vi
4 Hilbert’sFourteenthProblem 47
4.1 Theproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 TheWeitzenbo¨ckTheorem . . . . . . . . . . . . . . . . . . . . . 49
4.3 Nagata’scounterexample . . . . . . . . . . . . . . . . . . . . . . 52
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Algebraofcovariants 65
5.1 Examplesofcovariants . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Covariantsofanaction . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Linearrepresentationsofreductivegroups . . . . . . . . . . . . . 72
5.4 Dominantweights . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 TheCayley–Sylvesterformula . . . . . . . . . . . . . . . . . . . 79
5.6 Standardtableauxagain . . . . . . . . . . . . . . . . . . . . . . . 84
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Quotients 91
6.1 Categoricalandgeometricquotients . . . . . . . . . . . . . . . . 91
6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3 Rationalquotients . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Linearizationofactions 103
7.1 Linearizedlinebundles . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Theexistenceoflinearization . . . . . . . . . . . . . . . . . . . . 107
7.3 Linearizationofanaction . . . . . . . . . . . . . . . . . . . . . . 110
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8 Stability 115
8.1 Stablepoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.2 Theexistenceofaquotient . . . . . . . . . . . . . . . . . . . . . 117
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
vii
9 Numericalcriterionofstability 129
9.1 Thefunctionµ(x,λ) . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Thenumericalcriterion . . . . . . . . . . . . . . . . . . . . . . . 132
9.3 Theproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4 Theweightpolytope . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5 Kempf-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10 Projectivehypersurfaces 145
10.1 Nonsingularhypersurfaces . . . . . . . . . . . . . . . . . . . . . 145
10.2 Binaryforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.3 Planecubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.4 Cubicsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11 Configurationsoflinearsubspaces 165
11.1 Stableconfigurations . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 PointsinPn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11.3 LinesinP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
12 Toricvarieties 187
12.1 Actionsofatorusonanaffinespace . . . . . . . . . . . . . . . . 187
12.2 Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Bibliographicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography 205
IndexofNotation 215
Index 217

Introduction
Geometric invariant theory arises in an attempt to construct a quotient of an al-
gebraic variety X by an algebraic action of a linear algebraic group G. In many
applications X is the parametrizing space of certain geometric objects (algebraic
curves,vectorbundles,etc.) andtheequivalencerelationontheobjectsisdefined
by a group action. The main problem here is that the quotient space X/G may
not exist in the category of algebraic varieties. The reason is rather simple. Since
one expects that the canonical projection f : X → X/G is a regular map of al-
gebraic varieties and so has closed fibres, all orbits must be closed subsets in the
Zariski topology of X. This rarely happens when G is not a finite group. A pos-
sible solution to this problem is to restrict the action to an invariant open Zariski
subset U, as large as possible, so that U → U/G exists. The geometric invariant
theory (GIT) suggests a method for choosing such a set so that the quotient is a
quasi-projective algebraic variety. The idea goes back to David Hilbert. Suppose
X = V is a linear space and G is a linear algebraic group acting on V via its
linearrepresentation. ThesetofpolynomialfunctionsonV invariantwithrespect
tothisactionisacommutativealgebraAoverthegroundfield. Hilbertprovesthat
A is finitely generated if G = SL or GL and any set of generators f ,...,f
n n 1 N
of A defines an invariant regular map from X to some affine algebraic variety Y
contained in affine space AN whose ring of polynomial functions is isomorphic
to A. By a theorem of Nagata the same is true for any reductive linear algebraic
group. The map f : X → Y has a universal property for G-invariant maps of
X and is called the categorical quotient. The inverse image of the origin is the
closed subvariety defined by all invariant homogeneous polynomials of positive
degree. It is called the null-cone. Its points cannot be distinguished by invariant
functions; they are called unstable points. The remaining points are called semi-
stable points. When we pass to the projective space P(V) associated to V, the
images of semi-stable points form an invariant open subset P(V)ss and the map
f induces a regular map f¯ : P(V)ss → Y¯, where Y¯ (denoted by P(V)ss//G) is
ix
x INTRODUCTION
a projective algebraic variety with the projective coordinate algebra isomorphic
to A. In applications considered by Hilbert, P(V) parametrizes projective hyper-
¯
surfaces of certain degree and dimension, and the projective algebraic variety Y
is the “moduli space” of these hypersurfaces. The hypersurfaces represented by
unstable points are left out from the moduli space; they are “too degenerate”. A
¯
nonsingularhypersurfaceisalwaysrepresentedbyasemi-stablepoint. SinceY is
a projective variety, it is considered as a “compactification” of the moduli space
of nonsingular hypersurfaces. The fibres of the map P(V)ss → P(V)ss//G are
not orbits in general; however, each fibre contains a unique closed orbit so that
P(V)ss//Gparametrizesclosedorbitsinthesetofsemi-stablepoints.
Sincetheequationsofthenull-conearehardtofindwithoutcomputingexpli-
citly the ring of invariant polynomials, one uses another approach. This approach
is to describe the set of semi-stable points by using the Hilbert–Mumford numer-
ical criterion of stability. In many cases it allows one to determine the set P(V)ss
veryexplicitlyandtodistinguishstablepointsamongsemi-stableones. Theseare
the points whose orbits are closed in P(V)ss and whose stabilizer subgroups are
finite. The restriction of the map P(V)ss → P(V)ss//G to the set of stable points
P(V)s isanorbitmapP(V)s → P(V)s/G. Itiscalledageometricquotient.
More generally, if G is a reductive algebraic group acting on a projective al-
gebraic variety X, the GIT approach to constructing the quotient consists of the
following steps. First one chooses a linearization of the action, a G-equivariant
embedding of X into a projective space P(V) with a linear action of G as above.
The choice of a linearization is a parameter of the construction; it is defined by
a G-linearized ample line bundle on X. Then one sets Xss = X ∩ P(V)ss and
definesthecategoricalquotientXss → Xss//Gastherestrictionofthecategorical
quotient P(V)ss → P(V)ss//G. The image variety Xss//G is a closed subvariety
ofP(V)ss//G.
Letusgiveabriefcommentonthecontentofthebook.
In Chapters 1 and 2 we consider the classical example of invariant theory in
which the general linear group GL(V) of a vector space V of dimension n over
a field k acts naturally on the space of homogeneneous polynomials Pol (V) of
d
some degree d. We explain the classical symbolic method which allows one to
identify an invariant polynomial function of degree m on this space with an ele-
ment of the projective coordinate algebra k[Gr(n,m)] on the Grassmann variety
Gr(n,m)ofn-dimensionallinearsubspacesinkm initsPlu¨ckerembedding. This
interpretationisbasedontheFirstFundamentalTheoremofInvariantTheory. The
proof of this theorem uses a rather technical algebraic tool, the so-called Clebsch
omega-operator. Wechoosethislessconceptualapproachtoshowtheflavorofthe