Math 274Lectures onDeformation TheoryRobin Hartshornec2004PrefaceMy goal in these notes is to give an introduction to deformationtheory by doing some basic constructions in careful detail in their sim-plest cases, by explaining why people do things the way they do, withexamples, and then giving some typical interesting applications. Theearly sections of these notes are based on a course I gave in the Fall of1979.Warning: The present state of these notes is rough. The notation andnumbering systems are not consistent (though I hope they are consis-tent within each separate section). The cross-references and referencesto the literature are largely missing. Assumptions may vary from onesection to another. The safest way to read these notes would be asa loosely connected series of short essays on deformation theory. Theorder of the sections is somewhat arbitrary, because the material doesnot naturally fall into any linear order.I will appreciate comments, suggestions, with particular referenceto where I may have fallen into error, or where the text is confusing ormisleading.Berkeley, September 6, 2004iiiCONTENTS iiiContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iChapter 1. Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Structures over the dual numbers . . . . . . . . . . . . . . . . . . . . 43. The Tifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114. The infinitesimal lifting property . . . . . . . . . . . . . . . . . . . . 185. Deformation of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Chapter 2. Higher Order Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 336. Higher order deformations and obstruction theory . . . 337. Obstruction theory for a lo cal ring . . . . . . . . . . . . . . . . . . 398. Cohen–Macaulay in codimension two . . . . . . . . . . . . . . . . 439. Complete intersections and Gorenstein incodimension three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410. Obstructions to deformations of schemes . . . . . . . . . . . 5811. Dimensions of families of space curves . . . . . . . . . . . . . 6412. A non-reduced component of the Hilbert scheme . . . 68Chapter 3. Formal Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513. Plane curve singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514. Functors of Artin rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215. Schlessinger’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8616. Fibred products and flatness . . . . . . . . . . . . . . . . . . . . . . . 9117. Hilb and Pic are pro-representable . . . . . . . . . . . . . . . . . 9418. Miniversal and universal deformations of schemes . . 9619. Deformations of sheaves and the Quot functor . . . . 10420. Versal families of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 10821. Comparison of embedded and abstractdeformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Chapter 4. Globe Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117CONTENTS iv22. Introduction to moduli questions . . . . . . . . . . . . . . . . . 11723. Curves of genus zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12224. Deformations of a morphism . . . . . . . . . . . . . . . . . . . . . . 12625. Lifting from characteristic p to characteristic 0 . . . . 13126. Moduli of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 13827. Moduli of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157CHAPTER 1: GETTING STARTED 1CHAPTER 1Getting Started1 IntroductionDeformation theory is the loc al study of deformations. Or, seen fromanother point of view, it is the infinitesimal study of a family in theneighborhood of a given element. A typical situation would be a flatmorphism of schemes f : X → …