Deformation theory / Robin Hartshorne.

Format:

Book

Author/Creator:

Hartshorne, Robin.

Series:

Graduate texts in mathematics ; 257.

Graduate texts in mathematics, 0072-5285 ; 257

Language:

English

Subjects (All):

Deformations of singularities.

Geometry, Algebraic.

Physical Description:

vi, 234 pages : illustrations ; 25 cm.

Place of Publication:

New York ; London : Springer, [2010]

Summary:

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.

This 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 exercise, 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.

Contents:

1 First-Order Deformations 5

1 The Hilbert Scheme 5

2 Structure over the Dual Numbers 9

3 The T i Functors 18

4 The Infinitesimal Lifting Property 26

5 Deformations of Rings 35

2 Higher-Order Deformations 45

6 Subschemes and Invertible Sheaves 45

7 Vector Bundles and Coherent Sheaves 53

8 Cohen-Macaulay in Codimension Three 58

9 Complete Intersections and Gorenstein in Codimension Three 73

10 Obstructions to Deformations of Schemes 78

11 Obstruction Theory for a Local Ring 85

12 Dimensions of Families of Space Curves 88

13 A Nonreduced Component of the Hilbert Scheme 91

3 Formal Moduli 99

14 Plane Curve Singularities 100

15 Functors of Artin Rings 106

16 Schlessinger's Criterion 111

17 Hilb and Pic are Pro-representable 118

18 Miniversal and Universal Deformations of Schemes 120

19 Versal Families of Sheaves 128

20 Comparison of Embedded and Abstract Deformations 131

21 Algebraization of Formal Moduli 138

22 Lifting from Characteristic p to Characteristic 0 144

4 Global Questions 149

23 Introduction to Moduli Questions 150

24 Some Representable Functors 156

25 Curves of Genus Zero 164

26 Moduli of Elliptic Curves 167

27 Moduli of Curves 177

28 Moduli of Vector Bundles 188

29 Smoothing Singularities 199.

Notes:

Includes bibliographical references (pages [217]-224) and index.

ISBN:

9781441915955

1441915958

1441915966

9781441915962

OCLC:

496229710