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.

Autorentext

Edoardo Sernesi - vita

Present position:
Professore ordinario di Geometria, Facoltà di Scienze MFN, Università Roma Tre

Education:

• Laurea in Matematica- Università di Roma, 1969

• Ph.D. in Mathematics - Brandeis University, 1976

  Professional experience:

• Assistente ordinario di Geometria, Università di Ferrara, 1974-1980.
• Professore straordinario di Geometria Università di Roma ``La Sapienza", 1980-1983.
• Professore ordinario di Geometria Università di Roma ``La Sapienza", 1983-1992.

• Professore ordinario di Geometria Università Roma Tre, from 1992.

  Zusammenfassung
  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.

  Inhalt
  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.