Deformations of Algebraic SchemesIn 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. |
Contents
1 | |
Formal deformation theory | 37 |
Examples of deformation functors 103 | 102 |
Hilbert and Quot schemes | 187 |
A Flatness | 269 |
B Differentials | 279 |
Smoothness | 293 |
Complete intersections 305 | 304 |
E Functorial language | 313 |
References 321 | 320 |
List of symbols | 329 |
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Common terms and phrases
affine open algebraic schemes Artin rings Assume automorphisms bijective called closed embedding closed subscheme codimension coherent sheaf commutative diagram complete intersection condition conormal sequence Consider Corollary cotangent deduce defined definition Defx denote element etale exact sequence Example fibres finite type first-order deformation flat family follows formal couple formal deformation formally smooth functor of Artin H¹(X H²(X Hilb Hilbert functor Hilbert polynomial Hilbert scheme homomorphism ideal sheaf induced infinitesimal deformation injective invertible sheaf IP¹ isomorphism isotrivial k-algebra k-rational k-rational point Kodaira-Spencer map Lemma Let f locally free locally trivial Math moduli morphism ƒ morphism of functors noetherian nonsingular curve Nx/y ob(A obstruction space parametrized projective nonsingular projective scheme Proof Proposition prorepresentable prove quotient regular embedding resp semiuniversal sheaves singular smooth morphism Spec(A subfunctor surjective Theorem trivial deformation unobstructed vector versal