Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris.

*(English)*Zbl 1235.14002
Grundlehren der Mathematischen Wissenschaften 268. Berlin: Springer (ISBN 978-3-540-42688-2/hbk; 978-3-540-69392-5/ebook). xxii, 928 p. (2011).

This voluminous book is the long-awaited second part of the authors’ comprehensive monograph on the geometry of algebraic curves. The first volume [Geometry of algebraic curves. Volume I. Grundlehren der mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)] was published more than 25 years ago, at that time with the main goal to provide the first unified and systematic presentation of the basic old and new ideas, methods, and results in the theory of algebraic curves. In particular, the central topic of the first volume was the study of linear series on a fixed curve, together with the then very recent developments in the theory of special divisors, varieties of special linear series on a curve, and Brill-Noether theory.

Also, the authors announced the second volume as a forthcoming book, which was to contain an exposition of the fundamentals of deformation theory and of the main properties of the moduli spaces of curves, some of the further results of Brill-Noether theory, a presentation of the basic properties of the varieties of special linear series on a moving curve, and a proof of the (then new) theorem that the moduli space of curves of sufficiently high genus is of general type.

However, since the 1980s, the study of the moduli spaces of curves has undergone a rapid, nearly explosive development and still continues to do so, especially in view of the increasingly growing interrelation with mathematical physics. As the authors point out, the subject has developed in ways that are incredibly richer than could have been imagined at the time of writing Volume I in 1985. Taking these developments into account and trying to keep up with them, the authors have radically changed the disposition of the second volume, apparently revised the contents continuously, and finally presented the somewhat different, now realy existing second volume of their monograph “Geometry of Algebraic Curves”.

The main purpose of this book is to provide comprehensive and detailed foundations for the theory of moduli of complex algebraic curves, and that from multiple perspectives and various points of view. In fact, this book treats for the first time the different aspects of moduli theory in a coherent manner, thereby instructively combining algebro-geometric, complex-analytic, topological, and combinatorial methods. The originally envisioned centerpiece of the second volume, namely the further study of linear series on a general or variable curve, culminating in a proof of the Petri conjecture, is still an important part of the book, but it is not the central aspect anymore.

As for the contents of the current text, recall that the first volume contained the first eight chapters, while the present book comprises the following thirteen chapters, starting with Chapter IX and ending with Chapter XXI.

Chapter IX provides a self-contained introduction to the Hilbert scheme, thereby stressing the significance of the concept of flatness and the special case of curves. Chapter X presents the fundamental results for the construction of the moduli space \(M_{g,n}\) of stable \(n\)-pointed curves of genus \(g\), including the so-called stable reduction theorem and such basic constructions as lutching, projection, and stabilization. The central objects of study in this chapter are nodal curves. Chapter XI deals with the deformation theory of (stable) nodal curves, including the Kodaira-Spencer approach, Kuranishi families, the Hilbert scheme of \(n\)-canonical curves, the period map and the local Torelli theorem, Hodge bundles and their curvature, deformations of symmetric products, and other related topics. Chapter XII is then devoted to the construction of the moduli space \(\overline M_{g,n}\) in different ways.

This space is first exhibited as an analytic space, then as an algebraic space, and finally as an orbifold and as a Deligne-Mumford stack. Along the way, the first properties of the boundary strata of these moduli spaces are studied, and an essentially self-contained introduction to the theory of stacks is given as well.

Chapter XIII discusses the theory of line bundles on moduli stacks of stable curves. The reader gets acquainted with the necessary theory of descent, the Hodge bundle, the tangent bundle to the stack, the canonical bundle, and the normal bundles to the various boundary strata. Furthermore, the determinant of cohomology, the Deligne pairing, Mumford’s class, and various notions of Picard groups of moduli stacks are analyzed in great detail.

Chapter XIV gives a proof of the projectivity of the moduli space of stable curves. The authors’ proof uses an interesting combination of two techniques, namely geometric invariant theory and the Hilbert-Mumford criterion of stability, on the one hand, and certain numerical inequalities among cycles in moduli spaces and positivity results, on the other hand. Chapter XV provides a self-contained introduction to the Teichmüller point of view in the theory of complex curves. Actually, Teichmüller theory is needed in the authors’ subsequent discussion of smooth Galois covers of moduli spaces, and therefore this chapter explains the Teichmüller space and the mapping class group, quadratic differentials and Teichmüller deformations, the proof of Teichmüller’s uniqueness theorem, the boundary of Teichmüller space, and the simple connectedness of the moduli stack of stable curves. These techniques are used in Chapter XVI, where smooth Galois covers of moduli spaces are described. More precisely, the authors construct moduli spaces of stable pointed curves as quotients of smooth varieties by finite groups, thereby presenting a variation of E. Looijenga’s construction due to Abramovich, Corti, and Vistoli. The central subject of Chapter XVII is the theory of cycles in the moduli space \(\overline M_{g,n}\), including the necessaryintersection theory of certain stacks, tautological classes on moduli spaces of curves, tautological relations and the tautological ring, Mumford’s relations for the Hodge classes, and a description of the Chow ring of the moduli space \(\overline M_{0,P}\).

Chapter XVIII turns to the combinatorial study of the moduli space \(M_{g,n}\). To this end, the authors introduce a number of simplicial complexes associated to a pointed oriented surface, and these complexes are used to define certain cell decompositions of the Teichmüller space and of its bordification. These decompositions descend to orbicell-decompositions of the moduli space \(M_{g,P}\) and of suitable compactifications of it.

The basic tools for the combinatorial description of the associated Teichmüller spaces are certain graphs, the so-called ribbon graphs, Jenkins-Strebel differentials, uniformization theory, and the hyperbolic geometry of Riemann surfaces.

Chapter XIX discusses first consequences of the cellular decomposition of moduli spaces as studied in the foregoing chapter. The authors use the cellular decomposition to compute the rational cohomology of the space \(\overline M_{g,n}\) in degrees one and two, touch upon Harer’s stability theorem, the Madsen-Weiss theorem and the Tillmann theorem on the stable rational cohomology of \(M_{g,n}\) and give then a proof of Harer’s theorem on the second homology of \(M_{g,n}\) by applying Deligne’s spectral sequence for the complement of a divisor with normal crossings. At the end of this chapter, further uses of the cellular decomposition are presented by deriving some combinatorial formulae of M. Kontsevich in this context.

Chapter XX presents a nearly self-contained version of Kontsevich’s proof of Witten’s conjecture on the intersection numbers of the cohomological (so-called) \(\psi\)-classes.This chapter is titled “Intersection Theory of Tautological Classes” and explains Witten’s generating series, Virasoro operators and the KdV hierarchy, Feynman diagrams and matrix models, equivariant cohomology and the virtual Euler-Poincaré characteristic of \(M_{g,n}\), Gromov-Witten invariants, and other related material.

Chapter XXI returns to one of the central themes of the first volume: Brill-Noether theory. Here the authors study the Brill-Noether theory for smooth curves “moving with modul”. As the authors point out, this chapter is based on a draft version from the 1980s, when it was planned as a major part of the second volume. Thus it is meant here to give a snapshot of what Brill-Noether theory looked some twenty-five ago, both in content and style, and does not completely reflect the present state of the theory. Among the topics included in this final chapter are the following: the relative Picard variety, Brill-Noether varieties on moving curves, Looijenga’s vanishing theorem for the tautological ring of \(M_g\), Lazarsfeld’s proof of Petri’s conjecture, Horikawa theory, the Hurwitz scheme and its irreducibility, the unirationality of \(M_g\) for genus \(g\leq 10\) and other meanwhile classical results.

Each chapter comes with two extra sections at the end. One of them provides bibliographical notes and hints for further reading concerning the respective chapter, and the other gives a wealth of related exercises, many of which were contributed by J. Harris.

As in the first volume, the exercises are arranged in well-structured series, most of which usually cover an additional topic in the context of the respective section.

The bibliography at the end of the book is extremely rich and very up-to-date. There are 695 references listed in this bibliography, which must be seen as a highly valuable service to the reader, too.

Altogether, the present second volume of “Geometry of Algebraic Curves” has appeared with some delay, but it was worth waiting for it. The authors have presented a wealth of topical material from the moduli theory of curves, and that in a unique blend of different aspects, old and new viewpoints, relations to mathematical physics, and in great versatility. The presentation is utmost lucid, detailed, inspiring and instructive, written in a very motivating and user-friendly style.

The current book is an excellent research monograph and reference book in the theory of complex algebraic curves and their moduli, which is very likely to become an indispensable source for researchers and graduate students in both complex geometry and mathematical physics.

Also, the authors announced the second volume as a forthcoming book, which was to contain an exposition of the fundamentals of deformation theory and of the main properties of the moduli spaces of curves, some of the further results of Brill-Noether theory, a presentation of the basic properties of the varieties of special linear series on a moving curve, and a proof of the (then new) theorem that the moduli space of curves of sufficiently high genus is of general type.

However, since the 1980s, the study of the moduli spaces of curves has undergone a rapid, nearly explosive development and still continues to do so, especially in view of the increasingly growing interrelation with mathematical physics. As the authors point out, the subject has developed in ways that are incredibly richer than could have been imagined at the time of writing Volume I in 1985. Taking these developments into account and trying to keep up with them, the authors have radically changed the disposition of the second volume, apparently revised the contents continuously, and finally presented the somewhat different, now realy existing second volume of their monograph “Geometry of Algebraic Curves”.

The main purpose of this book is to provide comprehensive and detailed foundations for the theory of moduli of complex algebraic curves, and that from multiple perspectives and various points of view. In fact, this book treats for the first time the different aspects of moduli theory in a coherent manner, thereby instructively combining algebro-geometric, complex-analytic, topological, and combinatorial methods. The originally envisioned centerpiece of the second volume, namely the further study of linear series on a general or variable curve, culminating in a proof of the Petri conjecture, is still an important part of the book, but it is not the central aspect anymore.

As for the contents of the current text, recall that the first volume contained the first eight chapters, while the present book comprises the following thirteen chapters, starting with Chapter IX and ending with Chapter XXI.

Chapter IX provides a self-contained introduction to the Hilbert scheme, thereby stressing the significance of the concept of flatness and the special case of curves. Chapter X presents the fundamental results for the construction of the moduli space \(M_{g,n}\) of stable \(n\)-pointed curves of genus \(g\), including the so-called stable reduction theorem and such basic constructions as lutching, projection, and stabilization. The central objects of study in this chapter are nodal curves. Chapter XI deals with the deformation theory of (stable) nodal curves, including the Kodaira-Spencer approach, Kuranishi families, the Hilbert scheme of \(n\)-canonical curves, the period map and the local Torelli theorem, Hodge bundles and their curvature, deformations of symmetric products, and other related topics. Chapter XII is then devoted to the construction of the moduli space \(\overline M_{g,n}\) in different ways.

This space is first exhibited as an analytic space, then as an algebraic space, and finally as an orbifold and as a Deligne-Mumford stack. Along the way, the first properties of the boundary strata of these moduli spaces are studied, and an essentially self-contained introduction to the theory of stacks is given as well.

Chapter XIII discusses the theory of line bundles on moduli stacks of stable curves. The reader gets acquainted with the necessary theory of descent, the Hodge bundle, the tangent bundle to the stack, the canonical bundle, and the normal bundles to the various boundary strata. Furthermore, the determinant of cohomology, the Deligne pairing, Mumford’s class, and various notions of Picard groups of moduli stacks are analyzed in great detail.

Chapter XIV gives a proof of the projectivity of the moduli space of stable curves. The authors’ proof uses an interesting combination of two techniques, namely geometric invariant theory and the Hilbert-Mumford criterion of stability, on the one hand, and certain numerical inequalities among cycles in moduli spaces and positivity results, on the other hand. Chapter XV provides a self-contained introduction to the Teichmüller point of view in the theory of complex curves. Actually, Teichmüller theory is needed in the authors’ subsequent discussion of smooth Galois covers of moduli spaces, and therefore this chapter explains the Teichmüller space and the mapping class group, quadratic differentials and Teichmüller deformations, the proof of Teichmüller’s uniqueness theorem, the boundary of Teichmüller space, and the simple connectedness of the moduli stack of stable curves. These techniques are used in Chapter XVI, where smooth Galois covers of moduli spaces are described. More precisely, the authors construct moduli spaces of stable pointed curves as quotients of smooth varieties by finite groups, thereby presenting a variation of E. Looijenga’s construction due to Abramovich, Corti, and Vistoli. The central subject of Chapter XVII is the theory of cycles in the moduli space \(\overline M_{g,n}\), including the necessaryintersection theory of certain stacks, tautological classes on moduli spaces of curves, tautological relations and the tautological ring, Mumford’s relations for the Hodge classes, and a description of the Chow ring of the moduli space \(\overline M_{0,P}\).

Chapter XVIII turns to the combinatorial study of the moduli space \(M_{g,n}\). To this end, the authors introduce a number of simplicial complexes associated to a pointed oriented surface, and these complexes are used to define certain cell decompositions of the Teichmüller space and of its bordification. These decompositions descend to orbicell-decompositions of the moduli space \(M_{g,P}\) and of suitable compactifications of it.

The basic tools for the combinatorial description of the associated Teichmüller spaces are certain graphs, the so-called ribbon graphs, Jenkins-Strebel differentials, uniformization theory, and the hyperbolic geometry of Riemann surfaces.

Chapter XIX discusses first consequences of the cellular decomposition of moduli spaces as studied in the foregoing chapter. The authors use the cellular decomposition to compute the rational cohomology of the space \(\overline M_{g,n}\) in degrees one and two, touch upon Harer’s stability theorem, the Madsen-Weiss theorem and the Tillmann theorem on the stable rational cohomology of \(M_{g,n}\) and give then a proof of Harer’s theorem on the second homology of \(M_{g,n}\) by applying Deligne’s spectral sequence for the complement of a divisor with normal crossings. At the end of this chapter, further uses of the cellular decomposition are presented by deriving some combinatorial formulae of M. Kontsevich in this context.

Chapter XX presents a nearly self-contained version of Kontsevich’s proof of Witten’s conjecture on the intersection numbers of the cohomological (so-called) \(\psi\)-classes.This chapter is titled “Intersection Theory of Tautological Classes” and explains Witten’s generating series, Virasoro operators and the KdV hierarchy, Feynman diagrams and matrix models, equivariant cohomology and the virtual Euler-Poincaré characteristic of \(M_{g,n}\), Gromov-Witten invariants, and other related material.

Chapter XXI returns to one of the central themes of the first volume: Brill-Noether theory. Here the authors study the Brill-Noether theory for smooth curves “moving with modul”. As the authors point out, this chapter is based on a draft version from the 1980s, when it was planned as a major part of the second volume. Thus it is meant here to give a snapshot of what Brill-Noether theory looked some twenty-five ago, both in content and style, and does not completely reflect the present state of the theory. Among the topics included in this final chapter are the following: the relative Picard variety, Brill-Noether varieties on moving curves, Looijenga’s vanishing theorem for the tautological ring of \(M_g\), Lazarsfeld’s proof of Petri’s conjecture, Horikawa theory, the Hurwitz scheme and its irreducibility, the unirationality of \(M_g\) for genus \(g\leq 10\) and other meanwhile classical results.

Each chapter comes with two extra sections at the end. One of them provides bibliographical notes and hints for further reading concerning the respective chapter, and the other gives a wealth of related exercises, many of which were contributed by J. Harris.

As in the first volume, the exercises are arranged in well-structured series, most of which usually cover an additional topic in the context of the respective section.

The bibliography at the end of the book is extremely rich and very up-to-date. There are 695 references listed in this bibliography, which must be seen as a highly valuable service to the reader, too.

Altogether, the present second volume of “Geometry of Algebraic Curves” has appeared with some delay, but it was worth waiting for it. The authors have presented a wealth of topical material from the moduli theory of curves, and that in a unique blend of different aspects, old and new viewpoints, relations to mathematical physics, and in great versatility. The presentation is utmost lucid, detailed, inspiring and instructive, written in a very motivating and user-friendly style.

The current book is an excellent research monograph and reference book in the theory of complex algebraic curves and their moduli, which is very likely to become an indispensable source for researchers and graduate students in both complex geometry and mathematical physics.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

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

14H10 | Families, moduli of curves (algebraic) |

14H15 | Families, moduli of curves (analytic) |

14H42 | Theta functions and curves; Schottky problem |

30F60 | Teichmüller theory for Riemann surfaces |

32G13 | Complex-analytic moduli problems |

14H51 | Special divisors on curves (gonality, Brill-Noether theory) |

14A20 | Generalizations (algebraic spaces, stacks) |

14H81 | Relationships between algebraic curves and physics |