Deformation theory / Robin Hartshorne.

Includes bibliographical references and index.

First-Order Deformations -- The Hilbert Scheme -- Structure over the Dual Numbers -- The Ti Functors -- The Infinitesimal Lifting Property -- Deformations of Rings -- Higher-Order Deformations -- Subschemes and Invertible Sheaves -- Vector Bundles and Coherent Sheaves -- Cohen-Macaulay in Codimension Three -- Complete Intersections and Gorenstein in Codimension Three -- Obstructions to Deformations of Schemes -- Obstruction Theory for a Local Ring -- Dimensions of Families of Space Curves -- A Nonreduced Component of the Hilbert Scheme -- Formal Moduli -- Plane Curve Singularities -- Functors of Artin Rings -- Schlessinger's Criterion -- Hilb and Pic are Pro-representable -- Miniversal and Universal Deformations of Schemes -- Versal Families of Sheaves -- Comparison of Embedded and Abstract Deformations -- Algebraization of Formal Moduli -- Lifting from Characteristic p to Characteristic -- Global Questions -- Introduction to Moduli Questions -- Some Representable Functors -- Curves of Genus Zero -- Moduli of Elliptic Curves -- Moduli of Curves -- Moduli of Vector Bundles -- Smoothing Singularities.

The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck.Topics include:* deformations over the dual numbers;* smoothness and the infinitesimal lifting property;* Zariski tangent space and obstructions to deformation problems;* pro-representable functors of Schlessinger;* infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles.The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.

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