| Deformations of Algebraic Schemes 2006 Edition Contributor(s): Sernesi, Edoardo (Author) |
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ISBN: 3540306080 ISBN-13: 9783540306085 Publisher: Springer OUR PRICE: $151.99 Product Type: Hardcover - Other Formats Published: July 2006 Annotation: The study of small and local deformations of algebraic varieties originates in the classical work of Kodaira and Spencer and its formalization by Grothendieck in the late 1950's. It has become increasingly important in algebraic geometry in every context where variational phenomena come into play, and in classification theory. Today deformation theory is highly formalized and has ramified widely. This self-contained account of deformation theory in classical algebraic geometry (over an algebraically closed field) brings together for the first time some results previously scattered in the literature, with relatively little known proofs, yet of everyday relevance to algebraic geometers. It also includes applications to the construction and properties of Severi varieties of families of plane nodal curves, space curves, deformations of quotient singularities, Hilbert schemes of points, local Picard functors, etc. The exposition, amenable at graduate student level, includes many examples. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Algebraic - Mathematics | Algebra - General - Mathematics | Algebra - Abstract |
| Dewey: 516.35 |
| LCCN: 2006924565 |
| Series: Grundlehren Der Mathematischen Wissenschaften (Springer Hardcover) |
| Physical Information: 0.96" H x 6.32" W x 9.52" (1.44 lbs) 342 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |