##
**Classical algebraic geometry. A modern view.**
*(English)*
Zbl 1252.14001

Cambridge: Cambridge University Press (ISBN 978-1-107-01765-8/hbk; 978-1-139-08443-7/ebook). xii, 639 p. (2012).

Algebraic geometry has a long and eventful history, in the course of which the subject has undergone several re-formations of its foundations, both conceptually as well as methodologically. If modern algebraic geometry is understood as the period that began in the second half of the twentieth century,when its blending with commutative algebra, homological algebra, category theory, sheaf theory, and arithmetic formed its new shape in the hands of A. Grothendieck, K. Kodaira, D. Mumford, J.-P. Serre, and many others, classical algebraic geometry should be seen as the rich theory developed in the nineteenth and early twentieth century, especially by the famous schools of algebraic geometry in Italy, Germany, France, Britain, and North America. However, many of the great achievements of past generations of algebraic geometers have been generalized in the framework of modern algebraic geometry, and seemingly have become old -fashioned (or even obsolete) in our times. As a result, much of the research in classical algebraic geometry only exists in its original form, that is, in a mathematical language incomprehensible to most contemporary scholars in the field.

With a view towards this very fact, the main purpose of the monograph under review is to present some of the most beautiful topics in classical algebraic geometry in a modern context, thereby saving them from sinking into oblivion, on the one hand, and making them accessible to younger generations of algebraic geometers on the other.

Actually, the present book grew out of the author’s lecture notes on the subject, which were (and still are) available from his website at http://www.math.Isa.umich.edu/idolga/CAG.pdf, along with various other interesting notes.

According to its purpose, the book is definitely not a textbook, as it is not meant to provide an introduction to algebraic geometry. Instead, the author’s idea was to reconstruct various classical results by using modern techniques, without necessarily explaining their (often ingenious but obscure) original, purely geometrical proofs. Also, as the author pointed out in the preface, his intention was to present a compendium of material that either cannot be found, is to dispersed to be found easily, or is simply not treated adequately in the contemporary research literature. Generally, a profound background knowledge of basic modern algebraic geometry is required and assumed, perhaps as based on the standard texts by I. R. Shafarevich, R. Hartshorne, and others.

As for the contents, the book consists of ten chapters, each of which is divided into several sections, including a list of related exercises and some relevant historical notes.

Chapter 1 deals with the classical topic of polarity for projective hypersurfaces with particular emphasis on dual hypersurfaces, polar \(s\)-hedra, and dual homogeneous forms. The classical works of J. Steiner, J. Plücker, J. Sylvester, G. Salmon, J. Rosanes, and other geometers of the 19th century are carefully analyzed in this chapter.

Chapter 2 studies conics and quadric surfaces in further detail, especially with a view towards their polarity properties and their allied duality theory, Chapter 3 turns to the classical theory of plane cubic curves, with focus on their Hessians, their polars, their projective generation, and their invariant theory. Determinantal equations are the topic discussed in Chapter 4, where a modern approach to determinantal plane curves and their moduli, a treatment of determinantal hypersurfaces, and linear determinantal representations of surfaces via arithmetically Cohen-Macaulay sheaves play a central role. Chapter 5 is devoted to theta characteristics of algebraic curves, including related topics such as theta functions, quadratic forms and their Arf invariants hyperelliptic curves. Jacobians, Steiner complexes, the Scorza correspondence for curves, and contact hyperplanes of canonical curves.

Chapter 6 analyzes plane quartics, their bitangents, their determinant equations, their even theta characteristics, their particular invariant theory, and their automorphisms. A wealth of classic results from the 19th century is illuminated in this chapter, together with many references to modern developments in the research on plane quartics.

Cremona transformations are the subject of study in Chapter 7, where also the related theory of birational maps between algebraic surfaces is introduced, and that in its modern, sheaf-theoretic and cohomological setting. Chapter 8 takes up another classical topic, namely the study of del Pezzo surfaces (or Fano surfaces). The account given here is particularly comprehensive and detailed, with many subtle and refined results concerning del Pezzo surfaces of low degree.

The rich theory of cubic surfaces is reviewed in Chapter 9, with the central topics being the lines on a nonsingular cubic in \(\mathbb{P}^3\), Schur’s quadrics, singularities of cubic surfaces, determinantal equations of normal cubic surfaces, moduli of cubic surfaces, automorphisms of cubic surfaces, and various special cubic surfaces. Finally, Chapter 10 is titled “Geometry of lines”. In this concluding part of the book, the author explains Grassmannians and Schubert varieties, secant varieties of Grassmannians of lines, linear line complexes in Grassmannians of lines and their applications, linear systems of linear line complexes, quadratic line complexes, the geometry of the intersection of two quadrics, Kummer surfaces, harmonic line complexes, the geometry of ruled surfaces, the Cayley-Zeuthen formulas, and numerous concrete examples. All this classical material is instructively presented in a modern context, thereby effectively combining beautiful classical ideas with the powerful contemporary techniques in algebraic geometry.

As mentioned before, each chapter comes with its own list of related exercises. Most of these problems are rather challenging and of further-leading theoretical nature. Nevertheless, they provide a wealth of additional results and examples in quite a unique manner, and an invitation to active research likewise. Just as useful are the historical notes at the end of each chapter, which also provide invaluable hints to the respective classical research literature. The bibliography contains more than 600 precise references, which surely represents another great feature of this masterful treatise.

All together, the author has rendered a great, truly invaluable service to the community of algebraic geometers worldwide. No doubt, this great book is a product of ultimate enthusiasm, ethical principles, and expertise, which will help preserve the precious legacy of classical algebraic geometry for further generations of researchers, teachers, and students in the field.

With a view towards this very fact, the main purpose of the monograph under review is to present some of the most beautiful topics in classical algebraic geometry in a modern context, thereby saving them from sinking into oblivion, on the one hand, and making them accessible to younger generations of algebraic geometers on the other.

Actually, the present book grew out of the author’s lecture notes on the subject, which were (and still are) available from his website at http://www.math.Isa.umich.edu/idolga/CAG.pdf, along with various other interesting notes.

According to its purpose, the book is definitely not a textbook, as it is not meant to provide an introduction to algebraic geometry. Instead, the author’s idea was to reconstruct various classical results by using modern techniques, without necessarily explaining their (often ingenious but obscure) original, purely geometrical proofs. Also, as the author pointed out in the preface, his intention was to present a compendium of material that either cannot be found, is to dispersed to be found easily, or is simply not treated adequately in the contemporary research literature. Generally, a profound background knowledge of basic modern algebraic geometry is required and assumed, perhaps as based on the standard texts by I. R. Shafarevich, R. Hartshorne, and others.

As for the contents, the book consists of ten chapters, each of which is divided into several sections, including a list of related exercises and some relevant historical notes.

Chapter 1 deals with the classical topic of polarity for projective hypersurfaces with particular emphasis on dual hypersurfaces, polar \(s\)-hedra, and dual homogeneous forms. The classical works of J. Steiner, J. Plücker, J. Sylvester, G. Salmon, J. Rosanes, and other geometers of the 19th century are carefully analyzed in this chapter.

Chapter 2 studies conics and quadric surfaces in further detail, especially with a view towards their polarity properties and their allied duality theory, Chapter 3 turns to the classical theory of plane cubic curves, with focus on their Hessians, their polars, their projective generation, and their invariant theory. Determinantal equations are the topic discussed in Chapter 4, where a modern approach to determinantal plane curves and their moduli, a treatment of determinantal hypersurfaces, and linear determinantal representations of surfaces via arithmetically Cohen-Macaulay sheaves play a central role. Chapter 5 is devoted to theta characteristics of algebraic curves, including related topics such as theta functions, quadratic forms and their Arf invariants hyperelliptic curves. Jacobians, Steiner complexes, the Scorza correspondence for curves, and contact hyperplanes of canonical curves.

Chapter 6 analyzes plane quartics, their bitangents, their determinant equations, their even theta characteristics, their particular invariant theory, and their automorphisms. A wealth of classic results from the 19th century is illuminated in this chapter, together with many references to modern developments in the research on plane quartics.

Cremona transformations are the subject of study in Chapter 7, where also the related theory of birational maps between algebraic surfaces is introduced, and that in its modern, sheaf-theoretic and cohomological setting. Chapter 8 takes up another classical topic, namely the study of del Pezzo surfaces (or Fano surfaces). The account given here is particularly comprehensive and detailed, with many subtle and refined results concerning del Pezzo surfaces of low degree.

The rich theory of cubic surfaces is reviewed in Chapter 9, with the central topics being the lines on a nonsingular cubic in \(\mathbb{P}^3\), Schur’s quadrics, singularities of cubic surfaces, determinantal equations of normal cubic surfaces, moduli of cubic surfaces, automorphisms of cubic surfaces, and various special cubic surfaces. Finally, Chapter 10 is titled “Geometry of lines”. In this concluding part of the book, the author explains Grassmannians and Schubert varieties, secant varieties of Grassmannians of lines, linear line complexes in Grassmannians of lines and their applications, linear systems of linear line complexes, quadratic line complexes, the geometry of the intersection of two quadrics, Kummer surfaces, harmonic line complexes, the geometry of ruled surfaces, the Cayley-Zeuthen formulas, and numerous concrete examples. All this classical material is instructively presented in a modern context, thereby effectively combining beautiful classical ideas with the powerful contemporary techniques in algebraic geometry.

As mentioned before, each chapter comes with its own list of related exercises. Most of these problems are rather challenging and of further-leading theoretical nature. Nevertheless, they provide a wealth of additional results and examples in quite a unique manner, and an invitation to active research likewise. Just as useful are the historical notes at the end of each chapter, which also provide invaluable hints to the respective classical research literature. The bibliography contains more than 600 precise references, which surely represents another great feature of this masterful treatise.

All together, the author has rendered a great, truly invaluable service to the community of algebraic geometers worldwide. No doubt, this great book is a product of ultimate enthusiasm, ethical principles, and expertise, which will help preserve the precious legacy of classical algebraic geometry for further generations of researchers, teachers, and students in the field.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

14H50 | Plane and space curves |

14J25 | Special surfaces |

14E07 | Birational automorphisms, Cremona group and generalizations |

14H42 | Theta functions and curves; Schottky problem |

14Mxx | Special varieties |

14N15 | Classical problems, Schubert calculus |

14N25 | Varieties of low degree |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14Nxx | Projective and enumerative algebraic geometry |