Deformations of Algebraic Schemes

• Main Author: Sernesi, Edoardo (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2006.
• Series: A Series of Comprehensive Studies in Mathematics, 334
• Subjects: Mathematics.; Algebraic geometry.; Commutative algebra.; Commutative rings.; Geometry.; Algebraic Geometry.; Commutative Rings and Algebras.

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 pl...

Table of Contents:

• Introduction
• Infinitesimal Deformations: Extensions. Locally Trivial Deformations
• Formal Deformation Theory: Obstructions. Extensions of Schemes. Functors of Artin Rings. The Theorem of Schlessinger. The Local Moduli Functors
• Formal Versus Algebraic Deformations. Automorphisms and Prorepresentability
• Examples of Deformation Functors: Affine Schemes. Closed Subschemes. Invertible Sheaves. Morphisms
• Hilbert and Quot Schemes: Castelnuovo-Mumford Regularity. Flatness in the Projective Case. Hilbert Schemes. Quot Schemes. Flag Hilbert Schemes. Examples and Applications. Plane Curves
• Appendices: Flatness. Differentials. Smoothness. Complete Intersections. Functorial Language
• List of Symbols
• Bibliography.