Deformations of Algebraic Schemes

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Springer Science & Business Media, 20.04.2007 - 342 Seiten
In one sense, deformation theory is as old as algebraic geometry itself: this is because all algebro-geometric objects can be “deformed” by suitably varying the coef?cients of their de?ning equations, and this has of course always been known by the classical geometers. Nevertheless, a correct understanding of what “deforming” means leads into the technically most dif?cult parts of our discipline. It is fair to say that such technical obstacles have had a vast impact on the crisis of the classical language and on the development of the modern one, based on the theory of schemes and on cohomological methods. The modern point of view originates from the seminal work of Kodaira and Spencer on small deformations of complex analytic manifolds and from its for- lization and translation into the language of schemes given by Grothendieck. I will not recount the history of the subject here since good surveys already exist (e. g. [27], [138], [145], [168]). Today, while this area is rapidly developing, a self-contained text covering the basic results of what we can call “classical deformation theory” seems to be missing. Moreover, a number of technicalities and “well-known” facts are scattered in a vast literature as folklore, sometimes with proofs available only in the complex analytic category. This book is an attempt to ?ll such a gap, at least p- tially.
 

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Inhalt

343 Morphisms from a nonsingular curve with fixed target
172
The Hilbert schemes and the Quot schemes
187
42 Flatness in the projective case
194
422 Stratifications
198
423 Flattening stratifications
200
43 Hilbert schemes
206
432 Linear systems
207
433 Grassmannians
209

Formal deformation theory
37
22 Functors of Artin rings
44
23 The theorem of Schlessinger
54
24 The local moduli functors
64
242 Obstruction spaces
69
243 Algebraic surfaces
72
25 Formal versus algebraic deformations
75
26 Automorphisms and prorepresentability
89
262 Isotriviality
96
Examples of deformation functors
102
312 The second cotangent module and obstructions
109
313 Comparison with deformations of the nonsingular locus
117
314 Quotient singularities
120
32 Closed subschemes
122
32 Obstructions
129
323 The forgetful morphism
132
324 The local relative Hilbert functor
136
33 Invertible sheaves
137
332 Deformations of sections I
141
333 Deformations of pairs X L
145
334 Deformations of sections II
152
34 Morphisms
156
341 Deformations of a morphism leaving domain and target fixed
157
342 Deformations of a morphism leaving the target fixed
161
434 Existence
213
442 Local properties
223
45 Flag Hilbert schemes
227
452 Local properties
230
46 Examples and applications
235
462 An obstructed nonsingular curve in IP³
237
463 An obstructed nonreduced scheme
238
464 Relative grassmannians and projective bundles
240
465 Hilbert schemes of points
247
466 Schemes of morphisms
249
467 Focal loci
251
47 Plane curves
254
472 The Severi varieties
256
473 Nonemptiness of Severi varieties
262
Flatness
269
Differentials
279
Smoothness
293
Complete intersections
304
D2 Relative complete intersection morphisms
307
Functorial language
313
References
320
List of Symbols
329
Index
333
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Über den Autor (2007)

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.

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