##
**Deformation theory.**
*(English)*
Zbl 1186.14004

Graduate Texts in Mathematics 257. Berlin: Springer (ISBN 978-1-4419-1595-5/hbk; 978-1-4419-1596-2/ebook). vi, 234 p. (2010).

Robin Hartshorne’s popular textbook “Algebraic geometry” [Graduate Texts in Mathematics, 52. New York - Heidelberg - Berlin: Springer-Verlag (1977; Zbl 0367.14001)], is well known to everyone who ever tried to get acquainted with the basic modern aspects of the subject during the last three decades. Due to its comprehensiveness, versatility, and expository mastery, Hartshorne’s “Algebraic Geometry” is by far the most widely used introductory text (and reference book) in this discipline of contemporary mathematics. Generations of algebraic geometers have acquired a profound fundamental knowledge of the subject from R. Hartshorne’s book ever since its first appearance, and the book itself has never been out of print so far.

Now, more than thirty years after the publishing of his standard text on modern algebraic geometry, R. Hartshorne obliges with another book in the field. Being more specialized, the present volume is to provide an introduction to the main ideas of deformation theory in algebraic geometry and to illustrate their use in a number of typical concrete situations. Actually, as the author points out in the preface, this book has an amazingly long history that began exactly thirty years ago. Namely, in the fall semester of 1979, R. Hartshorne taught a course on algebraic deformation theory at Berkeley, mainly with the goal to understand completely Grothendieck’s approach to the local study of the Hilbert scheme using cohomological methods. The handwritten notes of this course circulated quietly for many years until D. Eisenbud urged the author to complete and publish them. Then, five years ago, R. Hartshorne expanded the old notes into a rough draft, which he used to teach a course in the spring of 2005. Finally, he rewrote those notes once more and, with the addition of numerous exercises, turned them into the book under review.

As the present text is intended to be of introductory nature, no effort has been made to develop the theory of deformations in full generality. Instead, the author has preferred to elaborate the basic ideas underlying the theory, without letting them get buried in too many technical details. Also, the approach has been kept as elementary as possible, thereby assuming only a basic familiarity with the concepts and methods of algebraic geometry as developed in the author’s above-mentioned standard text.

In this vein, and very much to the benefit of the rather unexperienced reader, the author has not striven for stating results in their most general form, nor has he attempted to use the more recent state-of-the-art framework of Grothendieck topologies and algebraic stacks to its full extent. Overall, the purpose of this book is to explain the basic concepts and methods of deformation theory, to bring forth clearly their fundamental essence, to show how they work in various standard situations, and to provide some instructive examples and applications from the literature.

As for the contents, the book is divided into four chapters which, altogether comprise twenty-nine sections. The author’s guiding principle is to focus on four standard situations:

(A) Deformations of subschemes of a fixed ambient scheme \(X\).

(B) Deformations of line bundles on a fixed scheme \(X\).

(C) Deformations of coherent sheaves on a fixed scheme \(X\).

(D) Deformations of abstract schemes,including the local study of deformations of singularities as well as the global study of deformations of non-singular varieties (global moduli).

For each of these particular situations, a number of typical problems is discussed, with the ultimate goal to establish a global parameter space classifying the isomorphism classes of the objects in question and, moreover, to describe its geometric properties.

However, in this introductory text, the technically involved proofs of the existence of these global classifying spaces are not provided, as the author’s primary goal is rather to lay the foundations of the respective deformation theory that allow to describe the local structure of the (assumed) global parameter space.

Chapter 1 is titled “First-Order Deformations” and deals with algebraic deformations over the ring \(D:=k[t]/(t^2)\) of dual numbers associated to an algebraically closed field \(k\). Starting with the concept of Hilbert scheme as a deformation space of a closed subscheme of the projective space \(\mathbb{P}^n_k\), which serves as a model in the sequel, deformations over \(D\) are discussed for the particular situations (A), (B) and (C). Then, after an introduction to the cotangent complex and the \(T^i\) functors of Lichtenbaum and Schlessinger, deformations of abstract schemes (as in situation (D)) are explained, thereby using the infinitesimal lifting property with regard to non-singular varieties.

Chapter 2 turns to the more general case of higher-order deformations, that is, to deformations over arbitrary Artin rings, together with the according obstruction theories for the respective situations (A), (B), (C), and (D). Along the way, three special cases are treated in greater detail, namely Cohen-Macaulay subschemes of codimension 2, locally complete intersection schemes, and Gorenstein subschemes in codimension 3. Additional illustrating material concerns the obstruction theory for local rings, the classical bound on the dimension of the Hilbert scheme of projective space curves, and Mumford’s “pathological” example of a family of non-singular projective space curves whose Hilbert scheme is generically non-reduced.

Chapter 3 is devoted to the study of formal moduli spaces. Starting from the explicit case of plane curve singularities, the author discusses the general problem in terms of functors of Artin rings, including Schlessinger’s criterion for pro-representability as the crucial technical tool in this context. In the sequel, this formal apparatus is applied to each of the standard situations (A), (B), (C) and (D), along with numerous concrete examples and applications. Further material concerns a comparison of embedded and abstract deformations, with a special view toward general surfaces in \(\mathbb{P}^4_k\) of degree greater than 3 the problem of algebraization of formal moduli (à la M. Artin), and – as a further application – the question of lifting varieties from characteristic \(p> 0\) to characteristic \(0\).

Chapter 4 comes with the headline “Global Questions”. Here the methods of infinitesimal and formal deformations from the previous chapters are applied to study global moduli problems. After introducing the notions of fine moduli space and coarse moduli space, the Hilbert functor and the Picard functor are described as examples of representable moduli functors.

The rest of this concluding chapter is devoted to the discussion of a number of concrete classical moduli spaces and their geometric properties. More precisely, the author illuminates in detail the global moduli spaces of rational and elliptic curves, Mumford’s concept of modular families of curves of higher genus (together with the idea of stacks), the moduli space of stable vector bundles over a curve, and the notions of formally smoothable schemes and smoothable singularities. As an application of the general theory, the reader encounters here Mori’s theorem on the existence of rational curves in non-singular varieties in characteristic \(p>0\) whose canonical divisor is not numerically effective, on the one hand, and instructive examples of non-smoothable singularities on the other.

Much more material is covered by the huge number of further-leading exercises complementing each section of the book. These exercises provide much more of the respective theories as well as a wealth of additional examples. Most of the exercises are quite challenging, but they are also well-structured and equipped with guiding remarks or hints.

The author’s approach to deformation theory shows lucidly how far the subject can be treated with relatively elementary methods, without providing the technically complicated proofs of the existence of the main classifying schemes, and without using more advanced toolkits like geometric invariant theory, Artin’s approximation theorems, simplicial complexes, differential graded algebras, fibered categories, stacks, or derived categories.

No doubt, this masterly written book gives an excellent first introduction to algebraic deformation theory, and a perfect motivation for further, more advanced reading likewise. It is the author’s masterful style of expository writing that makes this text particularly valuable for seasoned graduate students and for future researchers in the field. The list of 177 references at the end of the book, which the author frequently refers to throughout the text, is another special feature of the volume under review.

As for complementary and parallel reading, the recent monograph “Deformations of Algebraic Schemes” by E. Sernesi [Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften 334. Berlin: Springer (2006; Zbl 1102.14001)] might be quite instructive and useful,especially in view of the technical prerequisites and subtleties as well as for working some of the exercises.

Now, more than thirty years after the publishing of his standard text on modern algebraic geometry, R. Hartshorne obliges with another book in the field. Being more specialized, the present volume is to provide an introduction to the main ideas of deformation theory in algebraic geometry and to illustrate their use in a number of typical concrete situations. Actually, as the author points out in the preface, this book has an amazingly long history that began exactly thirty years ago. Namely, in the fall semester of 1979, R. Hartshorne taught a course on algebraic deformation theory at Berkeley, mainly with the goal to understand completely Grothendieck’s approach to the local study of the Hilbert scheme using cohomological methods. The handwritten notes of this course circulated quietly for many years until D. Eisenbud urged the author to complete and publish them. Then, five years ago, R. Hartshorne expanded the old notes into a rough draft, which he used to teach a course in the spring of 2005. Finally, he rewrote those notes once more and, with the addition of numerous exercises, turned them into the book under review.

As the present text is intended to be of introductory nature, no effort has been made to develop the theory of deformations in full generality. Instead, the author has preferred to elaborate the basic ideas underlying the theory, without letting them get buried in too many technical details. Also, the approach has been kept as elementary as possible, thereby assuming only a basic familiarity with the concepts and methods of algebraic geometry as developed in the author’s above-mentioned standard text.

In this vein, and very much to the benefit of the rather unexperienced reader, the author has not striven for stating results in their most general form, nor has he attempted to use the more recent state-of-the-art framework of Grothendieck topologies and algebraic stacks to its full extent. Overall, the purpose of this book is to explain the basic concepts and methods of deformation theory, to bring forth clearly their fundamental essence, to show how they work in various standard situations, and to provide some instructive examples and applications from the literature.

As for the contents, the book is divided into four chapters which, altogether comprise twenty-nine sections. The author’s guiding principle is to focus on four standard situations:

(A) Deformations of subschemes of a fixed ambient scheme \(X\).

(B) Deformations of line bundles on a fixed scheme \(X\).

(C) Deformations of coherent sheaves on a fixed scheme \(X\).

(D) Deformations of abstract schemes,including the local study of deformations of singularities as well as the global study of deformations of non-singular varieties (global moduli).

For each of these particular situations, a number of typical problems is discussed, with the ultimate goal to establish a global parameter space classifying the isomorphism classes of the objects in question and, moreover, to describe its geometric properties.

However, in this introductory text, the technically involved proofs of the existence of these global classifying spaces are not provided, as the author’s primary goal is rather to lay the foundations of the respective deformation theory that allow to describe the local structure of the (assumed) global parameter space.

Chapter 1 is titled “First-Order Deformations” and deals with algebraic deformations over the ring \(D:=k[t]/(t^2)\) of dual numbers associated to an algebraically closed field \(k\). Starting with the concept of Hilbert scheme as a deformation space of a closed subscheme of the projective space \(\mathbb{P}^n_k\), which serves as a model in the sequel, deformations over \(D\) are discussed for the particular situations (A), (B) and (C). Then, after an introduction to the cotangent complex and the \(T^i\) functors of Lichtenbaum and Schlessinger, deformations of abstract schemes (as in situation (D)) are explained, thereby using the infinitesimal lifting property with regard to non-singular varieties.

Chapter 2 turns to the more general case of higher-order deformations, that is, to deformations over arbitrary Artin rings, together with the according obstruction theories for the respective situations (A), (B), (C), and (D). Along the way, three special cases are treated in greater detail, namely Cohen-Macaulay subschemes of codimension 2, locally complete intersection schemes, and Gorenstein subschemes in codimension 3. Additional illustrating material concerns the obstruction theory for local rings, the classical bound on the dimension of the Hilbert scheme of projective space curves, and Mumford’s “pathological” example of a family of non-singular projective space curves whose Hilbert scheme is generically non-reduced.

Chapter 3 is devoted to the study of formal moduli spaces. Starting from the explicit case of plane curve singularities, the author discusses the general problem in terms of functors of Artin rings, including Schlessinger’s criterion for pro-representability as the crucial technical tool in this context. In the sequel, this formal apparatus is applied to each of the standard situations (A), (B), (C) and (D), along with numerous concrete examples and applications. Further material concerns a comparison of embedded and abstract deformations, with a special view toward general surfaces in \(\mathbb{P}^4_k\) of degree greater than 3 the problem of algebraization of formal moduli (à la M. Artin), and – as a further application – the question of lifting varieties from characteristic \(p> 0\) to characteristic \(0\).

Chapter 4 comes with the headline “Global Questions”. Here the methods of infinitesimal and formal deformations from the previous chapters are applied to study global moduli problems. After introducing the notions of fine moduli space and coarse moduli space, the Hilbert functor and the Picard functor are described as examples of representable moduli functors.

The rest of this concluding chapter is devoted to the discussion of a number of concrete classical moduli spaces and their geometric properties. More precisely, the author illuminates in detail the global moduli spaces of rational and elliptic curves, Mumford’s concept of modular families of curves of higher genus (together with the idea of stacks), the moduli space of stable vector bundles over a curve, and the notions of formally smoothable schemes and smoothable singularities. As an application of the general theory, the reader encounters here Mori’s theorem on the existence of rational curves in non-singular varieties in characteristic \(p>0\) whose canonical divisor is not numerically effective, on the one hand, and instructive examples of non-smoothable singularities on the other.

Much more material is covered by the huge number of further-leading exercises complementing each section of the book. These exercises provide much more of the respective theories as well as a wealth of additional examples. Most of the exercises are quite challenging, but they are also well-structured and equipped with guiding remarks or hints.

The author’s approach to deformation theory shows lucidly how far the subject can be treated with relatively elementary methods, without providing the technically complicated proofs of the existence of the main classifying schemes, and without using more advanced toolkits like geometric invariant theory, Artin’s approximation theorems, simplicial complexes, differential graded algebras, fibered categories, stacks, or derived categories.

No doubt, this masterly written book gives an excellent first introduction to algebraic deformation theory, and a perfect motivation for further, more advanced reading likewise. It is the author’s masterful style of expository writing that makes this text particularly valuable for seasoned graduate students and for future researchers in the field. The list of 177 references at the end of the book, which the author frequently refers to throughout the text, is another special feature of the volume under review.

As for complementary and parallel reading, the recent monograph “Deformations of Algebraic Schemes” by E. Sernesi [Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften 334. Berlin: Springer (2006; Zbl 1102.14001)] might be quite instructive and useful,especially in view of the technical prerequisites and subtleties as well as for working some of the exercises.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14Dxx | Families, fibrations in algebraic geometry |

14B10 | Infinitesimal methods in algebraic geometry |

14D15 | Formal methods and deformations in algebraic geometry |

14B12 | Local deformation theory, Artin approximation, etc. |

14B07 | Deformations of singularities |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14C05 | Parametrization (Chow and Hilbert schemes) |