Deformation TheorySpringer Science & Business Media, 10 déc. 2009 - 234 pages In the fall semester of 1979 I gave a course on deformation theory at Berkeley. My goal was to understand completely Grothendieck’s local study of the Hilbert scheme using the cohomology of the normal bundle to characterize the Zariski tangent space and the obstructions to deformations. At the same timeIstartedwritinglecturenotesforthecourse.However,thewritingproject soon foundered as the subject became more intricate, and the result was no more than ?ve of a projected thirteen sections, corresponding roughly to s- tions 1, 2, 3, 5, 6 of the present book. These handwritten notes circulated quietly for many years until David Eisenbud urged me to complete them and at the same time (without consu- ing me) mentioned to an editor at Springer, “You know Robin has these notes on deformation theory, which could easily become a book.” When asked by Springer if I would write such a book, I immediately refused, since I was then planning another book on space curves. But on second thought, I decided this was,afterall,aworthyproject,andthatbywritingImight?nallyunderstand the subject myself. So during 2004 I expanded the old notes into a rough draft, which I used to teach a course during the spring semester of 2005. Those notes, rewritten once more, with the addition of exercises, form the book you are now reading. Mygoalinthisbookistointroducethemainideasofdeformationtheoryin algebraicgeometryandtoillustratetheiruseinanumberoftypicalsituations. |
Table des matières
Introduction | 1 |
1 FirstOrder Deformations | 5 |
2 HigherOrder Deformations | 45 |
3 Formal Moduli | 99 |
4 Global Questions | 149 |
References | 217 |
225 | |
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Expressions et termes fréquents
abstract deformations ACM curves algebraically closed Artin ring automorphisms B-module base extension closed point closed subscheme coarse moduli space codimension Cohen-Macaulay coherent sheaf cohomology complete intersection consider corresponding curves of degree curves of genus defined deformation theory deformations of Xo dimension divisor dual numbers element elliptic curves étale exact sequence example exists families of curves family X/S finite type flat family formal family functor F genus g given gives global hence Hilbert polynomial Hilbert scheme homomorphism ideal induced invertible sheaf irreducible component isomorphism classes k-algebra lemma let F Let Xo locally free Math miniversal family modular family nonsingular curve obstruction theory obstructions to deforming obtain open affine open subset pro-representable projective scheme Proof quartic surface quotient regular local ring residue field sheaves smooth Spec surjective tangent space theorem torsor trivial vector bundles Zariski zero