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Algebraic Geometry II

David Mumford and Tadao Oda
Hindustan Book Agency
Publication Date: 
Number of Pages: 
Texts and Reading in Mathematics 73
[Reviewed by
Fernando Q. Gouvêa
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Here is a book with an interesting history. David Mumford published Algebraic Geometry I: Complex Projective Varieties in 1976. In the introduction to that book, he explained that he had

… found it impractical to teach classical geometry and schemes at the same time. Therefore, the present volume, which is the first of several, introduces only complex projective varieties […] The next volume will deal with schemes, including cohomology of coherent sheaves on them and applications…

Soon after, however, Mumford’s mathematical interests changed, and there seemed to be no chance that those “several” volumes would ever appear. Springer later published The Red Book of Varieties and Schemes, based on older course notes that had circulated for years among students at Harvard.

In graduate school in the 1980s, I took “Algebraic Geometry I” with Joseph Harris. His approach followed Mumford’s lead in the sense that he too started with complex varieties. Harris later published Algebraic Geometry: A First Course (Springer GTM, 1992), giving his own take on this material. It turned out, however, that no one was scheduled to teach “Algebraic Geometry II” the following semester, so a group of graduate students persuaded David Mumford to oversee a reading course based on his existing notes for the course he had taught at Harvard before. This is how I ended up in possession of an early version of the book under review. The notes (I still have them!) cover about two-thirds of the material in the final book.

Many years later, Tadao Oda asked me and several others for copies of those lecture notes. He then worked with Mumford to finish the book. That is what we have here.

What is it like? One should first recall that Algebraic Geometry I is very much a graduate text, a terse book that covers a lot of classical algebraic geometry in less than 200 pages. (Harris’s book is 330 pages, and Shafarevich’s Basic Algebraic Geometry I, in the same spirit, is 310.) That has not changed here: we get a detailed but terse account of scheme-theoretic algebraic geometry.

The book is well written (as one would expect from Mumford) and insightful. I applaud the inclusion of a (short) chapter on group schemes, something not often found in other introductory texts. I also really like the chapter on the cohomology of coherent sheaves, which includes four sections on how one actually computes cohomology. There is also a rather useful discussion of the Leray spectral sequence.

To complete the book, Oda has added sections and has also included many notes and remarks. In particular, there is a new proof of the Riemann-Toch theorem due to John Tate, an account of Belyi’s three-point theorem written by Carlos Simpson and a section by Shigefumi Mori on his existence theorem for rational curves. These additions enrich and improve the book. Mori has marked them all as “added in publication”, a curious signal intended to tell us that Mumford is not responsible for those portions of the book.

There are some minor problems, however, when it comes to integrating the new material into the old. The first example appears on page 27. At the top of the page, Mumford gives the definition of a coherent sheaf. There follows:

Remark. (Added in publication) Although the notion of “locally of finite presentation” coincides with that of “coherent” for X locally Noetherian, the standard definition of the latter is slightly different for general X…

The (now?) standard definition follows. Two problems arise. The first issue (certainly minor) is that the reader gets no signal of when the “remark” ends. More seriously, however, how is the reader to know which definition is in use? There is a footnote that perhaps would help, telling us that coherent sheaves “will be used only on Noetherian X’s except in §4.4.” (The note, but not the exception clause, is from Mumford’s original notes.) This tells us that we need not worry except in that section… but a quick look at section 4.4 does not immediately reveal where non-Noetherian schemes are used. The effect is to leave the non-expert reader confused.

Something similar happens in section 1.8, also “added in publication.” The issue here is to discuss Grothendieck’s “functorial” approach, in which we want to understand schemes via their point functors. Mumford’s notes mention this in section 1.6, so updating the discussion is certainly relevant. It is cast at a high level, however, with references to all sorts of ideas and constructions that have not yet appeared in the book. Again this is fine for the expert, but it makes the book less useful as a textbook and especially for self-study.

Overall, however, the book’s brilliant exposition and savvy choice of topics far outweighs these inconveniences. Algebraic Geometry II is not an easy book, but working through it will provide a good base from which to proceed into research in algebraic geometry.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.


1. Schemes and sheaves: definitions

1.1. \(\mathrm{Spec}(R)\)

1.2. \(\tilde{M}\)

1.3. Schemes

1.4. Products

1.5. Quasi-coherent sheaves

1.6. The functor of points

1.7. Relativization

1.8. Defining schemes as functors

Appendix: Theory of sheaves

2. Exploring the world of schemes

2.1. Classical varieties as schemes

2.2. The properties: reduced, irreducible and finite type

2.3. Closed subschemes and primary decompositions

2.4. Separated schemes

2.5. \(\mathrm{Proj} R\)

2.6. Proper morphisms

3. Elementary global study of \(\mathrm{Proj} R\)

3.1. Invertible sheaves and twists

3.2. The functor of \(\mathrm{Proj} R\)

3.3. Blow ups

3.4. Quasi-coherent sheaves on \(\mathrm{Proj} R\)

3.5. Ample invertible sheaves

3.6. Invertible sheaves via cocycles, divisors, line bundles

4. Ground fields and base rings

4.1. Kronecker’s big picture

4.2. Galois theory and schemes

4.3. The Frobenius morphism

4.4. Flatness and specialization

4.5. Dimension of fibres of a morphism

4.6. Hensel’s lemma

5. Singular vs. non-singular

5.1. Regularity

5.2. Kähler differential

5.3. Smooth morphisms

5.4. Criteria for smoothness

5.5. Normality

5.6. Zariski’s Main Theorem

5.7. Multiplicities following Weil

6. Group schemes and applications

6.1. Group schemes

6.2. Lang’s theorems over finite fields

7. The cohomology of coherent sheaves

7.1. Basic Čech cohomology

7.2. The case of schemes: Serre’s theorem

7.3. Higher direct images and Leray’s spectral sequence

7.4. Computing cohomology (1): Push \(\mathcal{F}\) into a huge acyclic sheaf

7.5. Computing cohomology (2): Directly via the Čech complex

7.6. Computing cohomology (3): Generate \(\mathcal{F}\) by “known” sheaves

7.7. Computing cohomology (4): Push \(\mathcal{F}\) into a coherent acyclic sheaf

7.8. Serre’s criterion for ampleness

7.9. Functorial properties of ampleness

7.10. The Euler characteristic

7.11. Intersection numbers

7.12. The criterion of Nakai-Moishezon

7.13. Seshadri constants

8. Applications of cohomology

8.1. The Riemann-Roch theorem

Appendix: Residues of differentials on curves

8.2. Comparison of algebraic with analytic cohomology

8.3. De Rham cohomology

8.4. Characteristic p phenomena

8.5. Deformation theory

9. Two deeper results

9.1. Mori’s existence theorem of rational curves

9.2. Belyi’s three point theorem