Algebraic geometry II (Q499826)

This volume is a sequel to \textit{David Mumford}'s classical textbook [Algebraic Geometry. I: Complex projective varieties. Berlin: Springer-Verlag (1976; Zbl 0356.14002); see also Reprint of the corr. 2nd print. (1995; Zbl 0821.14001)], which first appeared almost 40 years ago. In fact, the genesis of this considerably delayed second volume is rather amazing, and can be briefly summarized as follows. It is well-known that Fields medalist David Mumford (born 1937) worked in algebraic geometry and related fields from 1959 until 1982, before his research interests shifted to areas in applied mathematics such as pattern theory, shape theory, vision theory, neuronal architectures, and stochastic models. During the first period of his academic career, D. Mumford gave various introductory courses in algebraic geometry at Harvard University, and he once aimed to publish the continuously evolving course notes as a three-volume textbook. However, what was really published is a preliminary version of the first three chapters, which appeared in 1967 as mimeographed notes bound with a red cover, the famous ``Red book'', and the above-mentioned first volume ``Algebraic geometry I'' [Zbl 0821.14001] of the originally planned textbook series. The mimeographed ``Red book'' from 1967 is now available as Springer Lecture Notes under the title ``The red book of varieties and schemes'' [Zbl 0658.14001; Zbl 0945.14001], and as such it still is one of the great, most popular classics among the textbooks in modern algebraic geometry. The intended volume ``Algebraic geometry II'' of the planned book series would have been based on the material of the ``Red book'', on the one hand, and on existing but unpublished notes of D. Mumford on the other. However, due to the appearance of \textit{R. Hartshorne}'s famous textbook [Algebraic geometry. York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001)] in the meantime, which already covered a good part of the material he had planned to include, and in view of his changed mathematical interests in the early 1980s, D. Mumford never completed this undertaking in the following nearly 40 years. Now, thanks to the effort of T. Oda, the old draft notes for that intended volume ``Algebraic geometry II'' by D. Mumford have been prepared for publication in book form, the result of which is exactly the book under review. With the assistance of several other colleagues, a number of new sections were added to improve the text, and a so-called ``penultimate draft'' for the book is posted on D. Mumford's web site since early 2015. While Volume I is devoted to classical projective algebraic varieties, the present, finally published second volume provides an introduction to schemes, their morphisms, and their sheaf cohomology. Roughly speaking, this volume covers the material in the ``Red book'' in more depth with several more topics added, as the authors pointed out in the preface. As to the precise contents, the book comprises nine chapters, each of which consists of several sections. Chapter 1 introduces the basics on schemes and module sheaves, including the spectrum of a ring, the sheaf \(\widetilde{M}\) associated to a module \(M\), the notion of scheme, products of schemes, quasi-coherent sheaves, the functor of points, relative schemes, and the definition of schemes as functors. Chapter 2 is titled ``Exploring the world of schemes'' and discusses classical varieties as schemes, fundamental properties of schemes, closed subschemes, separated schemes, projective schemes, and proper morphisms. Chapter 3 turns to the elementary global study of the projective scheme \(\operatorname{Proj}(R)\) of a ring \(R\), with focus on invertible sheaves and twists, the functor of \(\operatorname{Proj}(R)\), the technique of blowing-up, quasi-coherent sheaves on \(\operatorname{Proj}(R)\), ample invertible sheaves, and the description of invertible sheaves via cocycles, divisors, and line bundles. Chapter 4 addresses general questions concerning ground fields and base rings in scheme theory, thereby touching upon the Galois theory of schemes, the Frobenius morphism, the concepts of flatness and specialization, the dimension of fibres of a morphism, and Hensel's lemma. Chapter 5 treats singular and non-singular points, with particular emphasis on the concept of regularity, Kähler differentials, smooth morphisms, criteria for smoothness, normal schemes, Zariski's main theorem, and the notion of multiplicity à la A. Weil. Chapter 6 introduces group schemes, culminating in S. Lang's theorems on group schemes over finite fields. Chapter 7 explains the Čech cohomology of coherent sheaves over algebraic schemes, along with various methods of concrete computation, higher direct image sheaves and Leray's spectral sequence, Serre's ampleness criterion and functorial properties of ampleness for invertible sheaves, intersection numbers of invertible sheaves, the Nakai-Moishezon criterion for ampleness, and a section (by V. Balaji) on Seshadri constants. Section 8 gives some applications of cohomology theory, mainly with a view toward J. Tate's elegant proof of the Riemann-Roch theorem for curves via residues of differentials, the comparison of algebraic with analytic cohomology groups, the idea of de Rham cohomology, special phenomena occurring in characteristic \(p>0\), and the elements of algebraic deformation theory. The concluding Chapter 9, added in publication, presents two more recent applications of the ideas and methods developed so far in the book. The first topic concerns S. Mori's existence theorem of rational curves on smooth projective varieties, while the second application is G. V. Belyi's famous theorem on the geometric interpretation of the absolute Galois group \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), which is closely related to A. Grothendieck's theory of ``dessins d'enfants''. Also this chapter is due to V. Balaji. Each of the original Chapters 1--8 ends with a set of related exercises, consisting of those found originally in D. Mumford's manuscript plus further ones provided by other colleagues. The book has been supplied with a rich, up-dated bibliography as well as with a carefully compiled index, and the printing is modern and utmost reader-friendly. Certainly, in the course of the last four decades, numerous textbooks on modern algebraic geometry have been published, but the final edition of D. Mumford's draft for his once intended volume ``Algebraic geometry II'' is nevertheless something highly special. Apart from being of invaluable historical significance in the textbook literature, this book reflects D. Mumford's unique, matchless mastery in expository writing in the best possible way. Also, D. Mumford's approach to teaching algebraic geometry is still highly inspiring, even after so many decades, and this book is a very worthy contribution to the steadily growing textbook literature in the field. And, last but not least, Volume I from 1976 has finally found its once foretold sequel, after 40 years, which now really justifies the original numbering of volumes.