Book Review David Mumford— Selected Papers, Volume II Reviewed by Frans Oort During the period starting around 1960 and end- David Mumford—Selected Papers, Volume II: On ing in the 1980s, David Mumford amazed us with Algebraic Geometry, including Correspondence new approaches to old problems in algebraic geome- with Grothendieck try. He rekindled interest in classical geometric ideas Edited by Ching-Li Chai, Amnon Neeman, and using modern methods. Takahiro Shiota Meeting Mumford and attending lectures by him Springer, July 2010 meant an encounter with sparkling new ideas. The Price: US$99.00, 767 pages same holds true for reading the twelve books and ISBN: 978-0-387-72491-1 more than sixty papers Mumford wrote in the years 1959–1982, when he was active in algebraic geom- The volume under review reproduces all papers by etry. These carry fundamentally new insights, such David Mumford in algebraic geometry not already as: included in Volume I (see [1]) and mathematical • revival of old ideas and techniques, long for- correspondence between Grothendieck and Mum- gotten and reactivated by Mumford in a new ford, plus several letters from Grothendieck to spirit; other mathematicians. Hence all papers in algebraic • unexpected views, questions, and directions geometry by David Mumford are now collected and in mathematics; and available in these two volumes [1], [2]. Let me first • a deep understanding of the material studied, say a few words about the mathematics of Mumford. developing a sure and precise grip on the essence of topics. The Style of Mumford His publications give us back that beautiful geomet- From the Autobiography of David Mumford: “At Har- ric feeling that was getting more and more lost in vard a classmate said ‘Come with me to hear Pro- the algebraization and the functorialization of geom- fessor Zariski’s first lecture, even though we won’t etry. understand a word’ and Oscar Zariski bewitched me. His style is unique and fascinating. In a period When he spoke the words ‘algebraic variety’, there when writing mathematics was increasingly done in was a certain resonance in his voice that said dis- a way where every symbol had many indices, where tinctly that he was looking into a secret garden. I trees of definitions and concepts were difficult to immediately wanted to be able to do this too. It led climb, Mumford found a mathematical language that me to 25 years of struggling to make this world tan- is clear and leads you straight to the central idea gible and visible. Especially, I became obsessed with without losing precision. In reading Mumford, you’d a kind of passion flower in this garden, the moduli better have a piece of paper and pencil at hand, be- space of Riemann. I was always trying to find new cause many arguments have to be worked out in angles from which I could see them better.” (1996; greater detail by the reader himself, only to discover see [5], p. 225.) at the end that the author is correct, that he must have thought through all details of the situation be- Frans Oort is emeritus professor in pure mathematics at the ing considered. This alone makes reading Mumford’s University of Utrecht, the Netherlands. His email address is papers fascinating and stimulating. [email protected]. In many cases Mumford does not aim at the DOI: http://dx.doi.org/10.1090/noti951 greatest generality. The basic idea, the immediate 214 Notices of the AMS Volume 60, Number 2 intuitive approach to the problem studied is central. on information not available at the time a particular Often he brings new insight and a fresh approach paper was written. This additional material makes to classical questions. His work opens windows and reading these articles even more interesting. gives rise to new developments. Just sit down with this volume and be over- As John Tate describes this: “Mumford has car- whelmed by ideas from, say, fifty years ago, still ried forward, after Zariski, the project of making new and inspiring now. Read through the paper algebraic and rigorous the work of the Italian school [61a] “The topology of normal singularities ...” or ... Mumford’s main interest [is] the theory of varieties [65d] “Picard groups of moduli problems”, just to of moduli. This is a central topic in algebraic geom- mention two of the papers featured here, and you etry having its origins in the theory of elliptic inte- see the fresh look, the powerful approach, and the grals. The development of the algebraic and global inspiration communicated to the reader. But also aspects of this subject in recent years is due mainly read [78d], “Fields Medals (IV): An instinct for the to Mumford, who attacked it with a brilliant com- key idea”, where Mumford and Tate describe work bination of classical, almost computational, meth- by the 1978 Fields Medalist Pierre Deligne; we see ods and Grothendieck’s new scheme-theoretic tech- the unique quality of work by Deligne, its place in niques.” (1974, see [4]). the history of algebraic geometry, and interaction It would be wonderful to document developments with mathematics by Grothendieck in just two pages. arising from and stimulated by his pioneering work. Such papers collected in this volume give insight He is generous to many of us, in contact and in writ- into this field in the years 1960–1980. ing, producing beautiful ideas and results and leav- ing open roads to new thoughts. We can hardly un- The Correspondence derestimate the influence this has had on all of us. Mumford wrote in 2008: “For me, personally, From his rich source of ideas, he often awarded Grothendieck’s letters were priceless and en- inspiration to other mathematicians, who then abled me to understand many of his ideas in finished the details. I have myself experienced their raw form before they were generalized too this twice, and I am still very grateful for those far and embedded in the daunting machinery of his opportunities. ‘Élements’.” See [2], p. 5. In this volume we see the fruit of a fascinating About Selected Papers, Volume I labor: reproducing a correspondence. It is a unique For a review of the first volume, see [7] and [8]. In source of information and inspiration. We can com- that book, just about half of the papers by Mum- pare a paper with the comments by Grothendieck ford in algebraic geometry were published. As those on a preliminary version. Moreover, the editors have two reviews point out, there were several flaws. It provided more than 160 footnotes explaining ideas was not clear why some papers were not reproduced, in those letters and providing recent developments and the papers did not appear in chronological or- and references. Reviewing, typesetting, and editing der. That volume contains five different lists of ref- this material was a big task, and we can be very grate- erences. No two agree. There are painful mistakes, ful to the editors for this valuable work. Together such as page numbers that are omitted or wrong; with the Grothendieck-Serre correspondence [6], this names that are misspelled; references that are unsys- book gives a beautiful picture and stimulating ideas tematically abbreviated, even within the same list; around the development of algebraic geometry in papers from the volume itself that are referred to the years 1960–1985. We know that Grothendieck de- by different titles. We missed important papers by stroyed many personal papers; as a consequence we Mumford, such as the paper with Deligne, “The ir- mainly have the letters of Grothendieck to Mumford, reducibility of the space of curves of given genus”, while the greater part of the other half of the cor- [69e], appearing now in Volume II (with 325 citations, respondence is lacking. This makes guessing even one of the most influential papers in modern alge- more interesting, though we would have appreciated braic geometry). But we also missed papers that were seeing the other half. hard to find. Let me say some words about the interaction be- tween Grothendieck and Mumford, the deep respect About Selected Papers, Volume II: Papers and on both sides, and the difference in their style of Notes research, thinking, and writing. The present volume corrects these flaws. The book contains a precise bibliography of Mumford. This First Interaction between Mumford and is very useful; we can really trust this list. Further- Grothendieck more, this volume contains all papers not appearing I remember meeting Grothendieck in 1961 in a in Volume I. Paris street; both of us were going to the same The editors have done a great job of writing notes lecture. Grothendieck mentioned a construction about the papers. In addition to correcting misprints, made by a young American mathematician. In a the notes indicate new developments and comment 1961 letter to Grothendieck, Mumford described February 2013 Notices of the AMS 215 his proof of “the key theorem in a construction Some Remarks about Grothendieck of the arithmetic scheme of moduli of curves of Alexandre (Alexander) Grothendieck was active in al- any genus.” Grothendieck was excited about this gebraic geometry in the period 1958–1970. His style idea, apparently completely new to him. Later Mum- was fundamental. Anything he considered would be ford pinned down the notion of a “coarse moduli done in the most general situation. If there is a gen- scheme”, necessary in case the obvious moduli eral solution, we can be sure Grothendieck puts us functor is not representable by a variety (or by a on the right track. The revolution he started (in al- scheme). (See [2], pp. 635–638, where we see this gebraic geometry) has been fruitful, although not excitement of Grothendieck reflected in several let- many of us can perform on the same heights and ters to Mumford.) Grothendieck explained that for at the same level of abstraction.