Deformations of Algebraic Schemes (eBook)

Edoardo Sernesi - vita Present position:Professore ordinario di Geometria, Facoltà di Scienze MFN, Università Roma Tre Education:- Laurea in Matematica- Università di Roma, 1969 - Ph.D. in Mathematics - Brandeis University, 1976 Professional experience:- Assistente ordinario di Geometria, Università di Ferrara, 1974-1980.- Professore straordinario di Geometria Università di Roma ``La Sapienza", 1980-1983.- Professore ordinario di Geometria Università di Roma ``La Sapienza", 1983-1992.- Professore ordinario di Geometria Università Roma Tre, from 1992.

Preface 5
Contents 8
Terminology and notation 11
Introduction 13
1 Infinitesimal deformations 19
1.1 Extensions 19
1.2 Locally trivial deformations 30
2 Formal deformation theory 47
2.1 Obstructions 47
2.2 Functors of Artin rings 54
2.3 The theorem of Schlessinger 64
2.4 The local moduli functors 74
2.5 Formal versus algebraic deformations 85
2.6 Automorphisms and prorepresentability 99
3 Examples of deformation functors 113
3.1 Affine schemes 113
3.2 Closed subschemes 132
3.3 Invertible sheaves 147
3.4 Morphisms 166
4 The Hilbert schemes and the Quot schemes 197
4.1 Castelnuovo-Mumford regularity 197
4.2 Flatness in the projective case 204
4.3 Hilbert schemes 216
4.4 Quot schemes 229
4.5 Flag Hilbert schemes 237
4.6 Examples and applications 245
4.7 Plane curves 264
A Flatness 279
B Differentials 289
C Smoothness 303
D Complete intersections 315
D.1 Regular embeddings 315
D.2 Relative complete intersection morphisms 317
E Functorial language 323
References 331
List of Symbols 339
Index 343