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Selected Papers, Volume II: On Algebraic Geometry, including Correspondence with Grothendieck

David Mumford
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Darren Glass
, on

Several years ago I wrote a review of the first volume of David Mumford’s Selected Papers. At that time I wrote that “There is little doubt that David Mumford is one of the most influential mathematicians of the second half of the twentieth century” and how much I appreciated having many of his works in algebraic geometry collected in one place. Five years have passed, and my feelings about the book, as well as Mumford’s work, have stayed the same. I pull the volume off of my shelves at least once a month to look something up or just to peruse the papers due to their combination of deep mathematics and quality exposition. There are two recent developments, however, that make me revisit this subject: Mumford has been announced as a winner of the National Medal of Science, the highest honor the US government bestows on scientists, and Springer has published a second volume of Mumford’s writings, subtitled On Algebraic Geometry, including correspondence with Grothendieck.

Selected Papers Volume II collects twenty-nine articles by Mumford, along with four previously unpublished pieces and dozens of letters between Mumford and Grothendieck. The articles begin with the first article Mumford published, “The topology of normal singularities of an algebraic surface and a criterion for simplicity” which appeared in 1961, and continue through to one of Mumford’s final papers in algebraic geometry, “What can be computed in Algebraic Geometry?” which was written jointly with Dave Bayer in 1993. These papers, along with those collected in Volume I make up Mumford’s body of work in pure mathematics before he switched research areas and began working in computer vision, an area he is still active in to this day.

In addition to all of the technical papers on topics such as polarized varieties, algebraic theta functions, and deformations of commutative group schemes, this book also collects biographical sketches and descriptions of the work of both Oscar Zariski and Pierre Deligne, as well as a piece written in memoriam of George Kempf.

The last 130 pages of the volume are dedicated to correspondence. Most of these are letters from Alexandre Grothendieck to Mumford, although there are also letters that Mumford wrote in response, and several letters from Grothendieck to others such as Zariski, Tate, and Hartshorne. The first of these letters was written by Grothendieck in August of 1958 and opens with a story about lost visa papers in the first paragraph before jumping into some results on the topology of schemes. The letters span the next thirty years, and anyone who knows Grothendieck’s story (and, if you don’t, you should certainly check out the vast collection of materials online at The Grothendieck Circle) will not be surprised to learn that the letters are fascinating both for their mathematics and for their commentaries on other matters. They give interesting insights into the algebraic geometry that was done in the second half of the twentieth century as well as into the people who did it, and I had a hard time putting this part of the book down. The final letter collected was written from Mumford to Grothendieck in February of 1986 and opens with

Thank you for your long and moving letter which I have thought about quite a bit… after all this, I really must say I don’t agree with you that there has been a general degradation in the manners and customs of the mathematical community. By moving into different fields, what I have found is that on the contrary pure mathematics has better manners and is much more gentle than any of the other fields I have touched… I feel that there are lapses in the mathematical world, but they are rare, the people involved are usually guilty more due to oversight than to intent, and almost everyone tries to rectify the errors. On the other hand there is more of a general tendency in mathematics to forget whatever the previous generation did!

Hopefully, these collections will make it harder to forget what Mumford’s generation accomplished, and the editors of this volume, Ching-Li Chai, Amnon Neeman, and Takahiro Shiota, deserve a lot of credit in assembling this volume. It seems to me that a collection of Mumford’s Selected Papers is as close to a “review-proof” math book as I can imagine — those of you who were interested in the book as soon as you saw it listed probably could not be dissuaded by anything negative I would say (not that I have anything negative to say!) and those of you who were not interested by the title probably could not be convinced, which is just as well as this is not meant as an introduction to the field. But I was asked to review the book anyways, so I will simply say that I feel about this book the same way I felt about the first volume: this is a book that most algebraic geometers — and all libraries — will not want to do without.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His research interests include algebraic geometry, number theory, and cryptography. He can be reached at

Topology of normal singularities and a criterion for simplicity.- The canononical ring of an algebraic surface.- Some aspects of the problem of moduli.- Two fundamental theorems on deformations of polarized varieties.- A remark on Mordell's conjecture.- Picard groups of moduli problems.- Abelian quotients of the Teichmuller modular group.- Deformations and liftings of finite, commutative group schemes.- Bi-extentions of formal groups.- The irreducibility of the space of curves of given genus.- Varieties defined by quadratric equations, with an appendix by G. Kempf.- A remark on Mahler's compactness theorem.- Introduction to the theory of moduli.- An example of a unirational 3-fold which is not rational.- A remark on the paper of M. Schlessinger.- Matsusaka's big theorem.- The self-intersection formula and the "forumle-clef".- Hilbert's fourteenth problem-the finite generation of subrings such as rings of invariants.- The projectivity of the moduli space of stable curves. I. Preliminaries on "det" and "Div".- An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de Vries equation and related nonlinear equation.- The work of C.P. Ramanujam in algebraic geometry.- Some footnotes to the work of C.P. Ramanujam.- Fields medals. IV. An instinct for the key idea.- The spectrum of difference operators and algebraic curves.- Proof of the convexity theorem.- Oscar Zariski: 1899-1986.- Foreward for non-mathematicians.- What can be computed in algebraic geometry.- In memoriam: George R. Kempf 1944-2002.- Boundary points on modular varieties.- Further comments on boundary points.- Abstract theta functions.- Abstract theta functions over local fields.