An Introduction to Invariants and Moduli

, by Shigeru Mukai , Translated by W. M. Oxbury

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ISBN: 9780521809061 | 0521809061
Cover: Hardcover
Copyright: 9/8/2003

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Incorporated in this volume are the first two books in Mukai's series on moduli theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Amongst other things this volume includes an improved presentation of the classical foundations of invarant theory that, in addition to geometers, would be useful to those studying representation theory. This translation gives an accurate account of Mukai's influential Japanese texts.

Preface

Acknowledgements

xiii

Introduction

(a) What is a moduli space?

(b) Algebraic varieties and quotients of algebraic varieties

xvi

xix

1 Invariants and moduli

(50)

1.1 A parameter space for plane conics

(8)

1.2 Invariants of groups

(10)

(a) Hilbert series

(4)

(b) Molien's formula

(2)

(4)

1.3 Classical binary invariants

(13)

(a) Resultants and discriminants

(7)

(b) Binary quartics

(6)

1.4 Plane curves

(9)

(a) Affine plane curves

(3)

(b) Projective plane curves

(6)

1.5 Period parallelograms and cubic curves

(10)

(a) Invariants of a lattice

(3)

(b) The Weierstrass function

(3)

(4)

2 Rings and polynomials

(26)

2.1 Hilbert's Basis Theorem

(4)

2.2 Unique factorisation rings

(3)

2.3 Finitely generated rings

(3)

2.4 Valuation rings

(7)

(a) Power series rings

(2)

(b) Valuation rings

(5)

2.5 A diversion: rings of invariants which are not finitely generated

(9)

(a) Graded rings

(1)

(b) Nagata's trick

(3)

(4)

3 Algebraic varieties

(39)

3.1 Affine varieties

(13)

(a) Affine space

(3)

(b) The spectrum

(5)

Morphims

(1)

Products

(1)

General spectra and nilpotents

(1)

Dominant morphisms

(1)

Open immersions

(1)

Local properties

(1)

3.2 Algebraic varieties

(7)

(a) Gluing affine varieties

(4)

(b) Projective varieties

(3)

3.3 Functors and algebraic groups

(5)

(a) A variety as a functor from algebras to sets

(2)

(b) Algebraic groups

100

(3)

3.4 Completeness and toric varieties

103

(13)

(a) Complete varieties

103

(4)

(b) Toric varieties

107

(4)

111

(5)

4 Algebraic groups and rings of invariants

116

(42)

4.1 Representations of algebraic groups

117

(5)

4.2 Algebraic groups and their Lie spaces

122

(8)

(a) Local distributions

122

(2)

(b) The distribution algebra

124

(4)

128

(2)

4.3 Hilbert's Theorem

130

(7)

(a) Linear reductivity

130

(5)

(b) Finite generation

135

(2)

4.4 The Cayley-Sylvester Counting Theorem

137

(15)

(a) SL(2)

137

(3)

(b) The dimension formula for SL(2)

140

(2)

142

(1)

(d) The Cayley-Sylvester Formula

143

(5)

(e) Some computational examples

148

(4)

4.5 Geometric reductivity of SL(2)

152

(6)

5 The construction of quotient varieties

158

(23)

5.1 Affine quotients

159

(8)

(a) Separation of orbits

159

(4)

(b) Surjectivity of the affine quotient map

163

(2)

165

(2)

5.2 Classical invariants and the moduli of smooth hypersurfaces in Pn

167

(14)

(a) Classical invariants and discriminants

167

(4)

(b) Stability of smooth hypersurfaces

171

(3)

174

(1)

(d) Nullforms and the projective quotient map

175

(6)

6 The projective quotient

181

(30)

6.1 Extending the idea of a quotient: from values to ratios

182

(15)

(a) The projective spectrum

186

(3)

(b) The Proj quotient

189

(6)

195

(2)

6.2 Linearisation and Proj quotients

197

(4)

6.3 Moving quotients

201

(10)

(a) Flops

201

(4)

(b) Toric varieties as quotient varieties

205

(3)

208

(3)

7 The numerical criterion and some applications

211

(23)

7.1 The numerical criterion

212

(7)

(a) 1-parameter subgroups

212

(1)

(b) The proof

213

(6)

7.2 Examples and applications

219

(15)

(a) Stability of projective hypersurfaces

219

(5)

(b) Cubic surfaces

224

(6)

230

(4)

8 Grassmannians and vector bundles

234

(53)

8.1 Grassmannians as quotient varieties

235

(16)

(a) Hilbert series

237

(2)

(b) Standard monomials and the ring of invariants

239

(2)

241

(4)

(d) Grassmannians as projective varieties

245

(2)

(e) A digression: the degree of the Grassmannian

247

(4)

8.2 Modules over a ring

251

(11)

(a) Localisation

251

(3)

(b) Local versus global

254

(3)

257

(2)

(d) Tensor products and flat modules

259

(3)

8.3 Locally free modules and flatness

262

(6)

(a) Locally free modules

262

(2)

(b) Exact sequences and flatness

264

(4)

8.4 The Picard group

268

(8)

(a) Algebraic number fields

268

(3)

(b) Two quadratic examples

271

(5)

8.5 Vector bundles

276

(11)

(a) Elementary sheaves of modules

277

(2)

(b) Line bundles and vector bundles

279

(3)

282

(2)

(d) The tangent space of the functor

284

(3)

9 Curves and their Jacobians

287

(61)

9.1 Riemann's inequality for an algebraic curve

288

(9)

(a) Prologue: gap values and the genus

290

(2)

(b) Divisors and the genus

292

(2)

294

(3)

9.2 Cohomology spaces and the genus

297

(12)

(a) Cousin's problem

297

(4)

(b) Finiteness of the genus

301

(3)

304

(3)

(d) Generation by global sections

307

(2)

9.3 Nonsingularity of quotient spaces

309

(7)

(a) Differentials and differential modules

309

(2)

(b) Nonsingularity

311

(2)

313

(3)

9.4 An algebraic variety with the Picard group as its set of points

316

(11)

(a) Some preliminaries

316

(3)

(b) The construction

319

(4)

323

(4)

9.5 Duality

327

(7)

(a) Dualising line bundles

327

(3)

(b) The canonical line bundle

330

(2)

332

(2)

9.6 The Jacobian as a complex manifold

334

(14)

(a) Compact Riemann surfaces

335

(2)

(b) The comparison theorem and the Jacobian

337

(5)

342

(6)

10 Stable vector bundles on curves

348

(50)

10.1 Some general theory

349

(16)

(a) Subbundles and quotient bundles

350

(2)

(b) The Riemann-Roch formula

352

(3)

355

(4)

(d) Grothendieck's Theorem

359

(2)

(e) Extensions of vector bundles

361

(4)

10.2 Rank 2 vector bundles

365

(6)

(a) Maximal line subbundles

365

(1)

(b) Nonstable vector bundles

366

(3)

369

(2)

10.3 Stable bundles and Pfaffian semiinvariants

371

(15)

(a) Skew-symmetric matrices and Pfaffians

371

(5)

Even skew-symmetric matrices

372

(2)

Odd skew-symmetric matrices

374

(1)

Skew-symmetric matrices of rank 2

375

(1)

(b) Gieseker points

376

(3)

379

(7)

10.4 An algebraic variety with SUc (2,L) as its set of points

386

(12)

(a) Tangent vectors and smoothness

387

(4)

(b) Proof of Theorem 10.1

391

(3)

394

(4)

11 Moduli functors

398

(39)

11.1 The Picard functor

400

(13)

(a) Fine moduli and coarse moduli

400

(3)

(b) Cohomology modules and direct images

403

(4)

407

(3)

(d) Poincaré line bundles

410

(3)

11.2 The moduli functor for vector bundles

413

(9)

(a) Rank 2 vector bundles of odd degree

416

(2)

(b) Irreducibility and rationality

418

(1)

419

(3)

11.3 Examples

422

(15)

(a) The Jacobian of a plane quantic

422

(2)

(b) The affine Jacobian of a spectral curve

424

(1)

425

(6)

(d) Vector bundles on a spectral curve

431

(2)

(e) Vector bundles on a curve of genus 2

433

(4)

12 Intersection numbers and the Veninde formula

437

(50)

12.1 Sums of inverse powers of trigonometric functions

439

(8)

(a) Sine sums

439

(2)

(b) Variations

441

(3)

444

(3)

12.2 Riemann-Roch theorems

447

(10)

(a) Some preliminaries

448

(2)

(b) Hirzebruch-Riemann-Roch

450

(3)

453

(2)

(d) Riemann-Roch with involution

455

(2)

12.3 The standard line bundle and the Mumford relations

457

(8)

(a) The standard line bundle

457

(4)

(b) The Newstead classes

461

(2)

463

(2)

12.4 From the Mumford relations to the Verlinde formula

465

(11)

(a) Warming lip: secant rings

466

(4)

(b) The proof of formulae (12.2) and (12.4)

470

(6)

12.5 An excursion: the Veninde formula for quasiparabolic bundles

476

(11)

(a) Quasiparabolic vector bundles

476

(4)

(b) A proof of (12.6) using Riemann-Roch and the Mumford relations

480

(3)

483

(4)

Bibliography

487

(8)

Index

495

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