Deformation Theory

  • Robin Hartshorne
Part of the Graduate Texts in Mathematics book series (GTM, volume 257)

Table of contents

  1. Front Matter
    Pages i-vi
  2. Robin Hartshorne
    Pages 1-4
  3. Robin Hartshorne
    Pages 5-44
  4. Robin Hartshorne
    Pages 45-98
  5. Robin Hartshorne
    Pages 99-148
  6. Robin Hartshorne
    Pages 149-216
  7. Back Matter
    Pages 1-18

About this book

Introduction

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.

Keywords

Hilbert scheme deformation theory first-order deformations formal moduli higher-order deformations infinitesimal lifting property invertible sheaves vector bundles

Authors and affiliations

  • Robin Hartshorne
    • 1
  1. 1.University of California at BerkeleyDepartment of MathematicsBerkeleyU.S.A.

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-1596-2
  • Copyright Information Robin Hartshorne 2010
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-1595-5
  • Online ISBN 978-1-4419-1596-2
  • Series Print ISSN 0072-5285
  • About this book