rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Deformations of Algebraic Schemes
  name:
    name part: Sernesi, Edoardo.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2006
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XI, 342 p. online resource.
  abstract: 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, e.g. the study of the local properties of moduli spaces.Today deformation theory is highly formalized and has ramified widely within mathematics. 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 proofs that are relatively little known, yet of everyday relevance to algebraic geometers. Based on Grothendieck's functorial approach it covers formal deformation theory, algebraization, isotriviality, Hilbert schemes, Quot schemes and flag Hilbert schemes. It 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. Many examples are provided. Most of the algebraic results needed are proved. The style of exposition is kept at a level amenable to graduate students with an average background in algebraic geometry.
  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.
  note: by Edoardo Sernesi.
  subject:
    topic: Mathematics
  subject:
    topic: Algebraic geometry
  subject:
    topic: Commutative algebra
  subject:
    topic: Commutative rings
  subject:
    topic: Geometry
  subject:
    topic: Mathematics
  subject:
    topic: Geometry
  subject:
    topic: Algebraic Geometry
  subject:
    topic: Commutative Rings and Algebras
  classification: QA440-699
  classification: 516
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: A Series of Comprehensive Studies in Mathematics, 334
  identifier: 9783540306153
  identifier: http://dx.doi.org/10.1007/978-3-540-30615-3
  record info:
    record creation date: 100301
    record change date: 20151204164219.0
    record identifier: 978-3-540-30615-3