Book Review

David Mumford—Selected Papers,Volume IIReviewed by Frans Oort

David Mumford—Selected Papers, Volume II: OnAlgebraic Geometry, including Correspondencewith GrothendieckEdited by Ching-Li Chai, Amnon Neeman, andTakahiro ShiotaSpringer, July 2010Price: US$99.00, 767 pagesISBN: 978-0-387-72491-1

The volume under review reproduces all papers byDavid Mumford in algebraic geometry not alreadyincluded in Volume I (see [1]) and mathematicalcorrespondence between Grothendieck and Mum-ford, plus several letters from Grothendieck toother mathematicians. Hence all papers in algebraicgeometry by David Mumford are now collected andavailable in these two volumes [1], [2]. Let me firstsay a few words about the mathematics of Mumford.

The Style of Mumford

From the Autobiography of David Mumford : “At Har-vard a classmate said ‘Come with me to hear Pro-fessor Zariski’s first lecture, even though we won’tunderstand a word’ and Oscar Zariski bewitched me.When he spoke the words ‘algebraic variety’, therewas a certain resonance in his voice that said dis-tinctly that he was looking into a secret garden. Iimmediately wanted to be able to do this too. It ledme to 25 years of struggling to make this world tan-gible and visible. Especially, I became obsessed witha kind of passion flower in this garden, the modulispace of Riemann. I was always trying to find newangles from which I could see them better.” (1996;see [5], p. 225.)

Frans Oort is emeritus professor in pure mathematics at theUniversity of Utrecht, the Netherlands. His email address isf.oort@uu.nl.

DOI: http://dx.doi.org/10.1090/noti951

During the period starting around 1960 and end-ing in the 1980s, David Mumford amazed us withnew approaches to old problems in algebraic geome-try. He rekindled interest in classical geometric ideasusing modern methods.

Meeting Mumford and attending lectures by himmeant an encounter with sparkling new ideas. Thesame holds true for reading the twelve books andmore than sixty papers Mumford wrote in the years1959–1982, when he was active in algebraic geom-etry. These carry fundamentally new insights, suchas:

• revival of old ideas and techniques, long for-gotten and reactivated by Mumford in a newspirit;

• unexpected views, questions, and directionsin mathematics; and

• a deep understanding of the material studied,developing a sure and precise grip on theessence of topics.

His publications give us back that beautiful geomet-ric feeling that was getting more and more lost inthe algebraization and the functorialization of geom-etry.

His style is unique and fascinating. In a periodwhen writing mathematics was increasingly done ina way where every symbol had many indices, wheretrees of definitions and concepts were difficult toclimb, Mumford found a mathematical language thatis clear and leads you straight to the central ideawithout losing precision. In reading Mumford, you’dbetter have a piece of paper and pencil at hand, be-cause many arguments have to be worked out ingreater detail by the reader himself, only to discoverat the end that the author is correct, that he musthave thought through all details of the situation be-ing considered. This alone makes reading Mumford’spapers fascinating and stimulating.

In many cases Mumford does not aim at thegreatest generality. The basic idea, the immediate

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intuitive approach to the problem studied is central.Often he brings new insight and a fresh approachto classical questions. His work opens windows andgives rise to new developments.

As John Tate describes this: “Mumford has car-ried forward, after Zariski, the project of makingalgebraic and rigorous the work of the Italian school... Mumford’s main interest [is] the theory of varietiesof moduli. This is a central topic in algebraic geom-etry having its origins in the theory of elliptic inte-grals. The development of the algebraic and globalaspects of this subject in recent years is due mainlyto Mumford, who attacked it with a brilliant com-bination of classical, almost computational, meth-ods and Grothendieck’s new scheme-theoretic tech-niques.” (1974, see [4]).

It would be wonderful to document developmentsarising from and stimulated by his pioneering work.He is generous to many of us, in contact and in writ-ing, producing beautiful ideas and results and leav-ing open roads to new thoughts. We can hardly un-derestimate the influence this has had on all of us.

From his rich source of ideas, he often awardedinspiration to other mathematicians, who thenfinished the details. I have myself experiencedthis twice, and I am still very grateful for thoseopportunities.

About Selected Papers, Volume I

For a review of the first volume, see [7] and [8]. Inthat book, just about half of the papers by Mum-ford in algebraic geometry were published. As thosetwo reviews point out, there were several flaws. Itwas not clear why some papers were not reproduced,and the papers did not appear in chronological or-der. That volume contains five different lists of ref-erences. No two agree. There are painful mistakes,such as page numbers that are omitted or wrong;names that are misspelled; references that are unsys-tematically abbreviated, even within the same list;papers from the volume itself that are referred toby different titles. We missed important papers byMumford, such as the paper with Deligne, “The ir-reducibility of the space of curves of given genus”,[69e], appearing now in Volume II (with 325 citations,one of the most influential papers in modern alge-braic geometry). But we also missed papers that werehard to find.

About Selected Papers, Volume II: Papers andNotes

The present volume corrects these flaws. The bookcontains a precise bibliography of Mumford. Thisis very useful; we can really trust this list. Further-more, this volume contains all papers not appearingin Volume I.

The editors have done a great job of writing notesabout the papers. In addition to correcting misprints,the notes indicate new developments and comment

on information not available at the time a particularpaper was written. This additional material makesreading these articles even more interesting.

Just sit down with this volume and be over-whelmed by ideas from, say, fifty years ago, stillnew and inspiring now. Read through the paper[61a] “The topology of normal singularities ...” or[65d] “Picard groups of moduli problems”, just tomention two of the papers featured here, and yousee the fresh look, the powerful approach, and theinspiration communicated to the reader. But alsoread [78d], “Fields Medals (IV): An instinct for thekey idea”, where Mumford and Tate describe workby the 1978 Fields Medalist Pierre Deligne; we seethe unique quality of work by Deligne, its place inthe history of algebraic geometry, and interactionwith mathematics by Grothendieck in just two pages.Such papers collected in this volume give insightinto this field in the years 1960–1980.

The Correspondence

Mumford wrote in 2008: “For me, personally,Grothendieck’s letters were priceless and en-abled me to understand many of his ideas intheir raw form before they were generalized toofar and embedded in the daunting machinery of his‘Élements’.” See [2], p. 5.

In this volume we see the fruit of a fascinatinglabor: reproducing a correspondence. It is a uniquesource of information and inspiration. We can com-pare a paper with the comments by Grothendieckon a preliminary version. Moreover, the editors haveprovided more than 160 footnotes explaining ideasin those letters and providing recent developmentsand references. Reviewing, typesetting, and editingthis material was a big task, and we can be very grate-ful to the editors for this valuable work. Togetherwith the Grothendieck-Serre correspondence [6], thisbook gives a beautiful picture and stimulating ideasaround the development of algebraic geometry inthe years 1960–1985. We know that Grothendieck de-stroyed many personal papers; as a consequence wemainly have the letters of Grothendieck to Mumford,while the greater part of the other half of the cor-respondence is lacking. This makes guessing evenmore interesting, though we would have appreciatedseeing the other half.

Let me say some words about the interaction be-tween Grothendieck and Mumford, the deep respecton both sides, and the difference in their style ofresearch, thinking, and writing.

First Interaction between Mumford andGrothendieck

I remember meeting Grothendieck in 1961 in aParis street; both of us were going to the samelecture. Grothendieck mentioned a constructionmade by a young American mathematician. In a1961 letter to Grothendieck, Mumford described

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his proof of “the key theorem in a constructionof the arithmetic scheme of moduli of curves ofany genus.” Grothendieck was excited about thisidea, apparently completely new to him. Later Mum-ford pinned down the notion of a “coarse modulischeme”, necessary in case the obvious modulifunctor is not representable by a variety (or by ascheme). (See [2], pp. 635–638, where we see thisexcitement of Grothendieck reflected in several let-ters to Mumford.) Grothendieck explained that for“higher levels” he could represent moduli functors,but for all levels he could not perform the necessaryconstruction. Mumford announced this methodand explained it in his paper [61c]. Grothendieck,although positive about the new ideas of the youngMumford, wrote in 1961: “It seems to me that, be-cause of your lack of some technical backgroundon schemata, some proofs are rather awkward andunnatural” ([2], p. 636).

To French mathematicians of that period a con-struction in algebraic geometry should be done bysolving a “universal problem”, in their terminology,by “representing a functor”. Igusa constructed (Ann.Math. 72 (1960), 621–649, in terminology devel-oped by Mumford later) the coarse moduli schemeof curves of genus 2 over Z. To the “functorialthinking” of Paris mathematicians in 1961 this wasstrange, and we taste this atmosphere in the de-scription by Samuel of this construction by Igusa:“Signalons aussitôt que le travail d’Igusa ne résoudpas, pour les courbes de genre 2, le ‘problème desmodules’ tel qu’il a été posé par Grothendieck àdiverses reprises dans ce Séminaire. (We note thatthe work by Igusa for curves of genus 2 does notsolve the problem of ‘moduli of curves’ as proposedby Grothendieck several times in this seminar.)” Seethe first lines of [3]. This is the background of thedifference between Paris and Harvard mathematicsat that time. Grothendieck classified work by Igusaas “most discouraging to read”; see [2], p. 636. (Ilike that paper of Igusa; it was on my desk on manyoccasions.)

Mumford completed this construction of modulispaces over any base scheme (curves of any genus,polarized abelian varieties); it was a starting pointof new developments. A clear line led from clas-sical invariant theory to ideas by Igusa for g = 2to Mumford’s geometric invariant theory, therebycreating an aspect of research that complementedGrothendieck’s work on these topics.

The stimulating difference between these two gi-ants, Grothendieck and Mumford, their insight, andthe respect they had for each other are a source ofrich ideas. What a privilege for us to feel stimulat-ing aspects contained in Mumford’s papers repro-duced in these volumes and to be able to enjoy theexchange of their ideas, surviving in (part of) theircorrespondence, which is now available thanks tothis beautiful volume.

Some Remarks about Grothendieck

Alexandre (Alexander) Grothendieck was active in al-gebraic geometry in the period 1958–1970. His stylewas fundamental. Anything he considered would bedone in the most general situation. If there is a gen-eral solution, we can be sure Grothendieck puts uson the right track. The revolution he started (in al-gebraic geometry) has been fruitful, although notmany of us can perform on the same heights andat the same level of abstraction.

We now have (partial) access to a fascinating biog-raphy. Who Is Alexandre Grothendieck? is a projected3-volume biography of Alexandre Grothendieck.

Volume I.: Anarchy (by W. Scharlau) covers thestory of Grothendieck’s parents and his life1928–1948.

Volume II.: Mathematics (by L. Schneps) coversGrothendieck’s life and mathematics 1948–1970.

Volume III.: Spirituality (by W. Scharlau) coversthe years 1970–1991.

Volume I is complete and available in German andEnglish (2009). Volume III is complete and availablein German (2010). Volume II is in preparation.

A few years ago Grothendieck sought to blockpublication of his unpublished writings:

Declaration of intent of non-publication. I donot intend to publish or republish any workor text of which I am the author, in any formwhatsoever, printed or electronic, whetherin full or in excerpts, texts of personal na-ture, of scientific character, or otherwise,or letters addressed to anybody, and anytranslation of texts of which I am the author.Any edition or dissemination of such textswhich have been made in the past withoutmy consent, or which will be made in thefuture and as long as I live, is against my willexpressly specified here and is unlawful inmy eyes. As I learn of these, I will ask the per-son responsible for such pirated editions, orof any other publication containing withoutpermission texts from my hand (beyond pos-sible citations of a few lines each), to removefrom commerce these books; and librariansholding such books to remove these booksfrom those libraries.

If my intentions, clearly expressed here,should go unheeded, then the shame of itfalls on those responsible for the illegaleditions, and those responsible for the li-braries concerned (as soon as they have beeninformed of my intention).

Written at my home, January 3, 2010,Alexandre Grothendieck.

(Translated by Scott Morrison; see http://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter/. For the original French

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version of this letter, see http://tqft.net/misc/Grothendieck%27s%20Declaration.pdf.)

In 1985 Mumford wrote to Grothendieck: “…theletters that you wrote me are by far the most impor-tant things which explained your ideas and insights.The letters are vivid and clear and unencumberedby the customary style of formal French publication-s…My proposal would be to approach someone witha broad knowledge of your theories…and give…per-mission…I feel sure that such a collection would beextremely useful to the younger generation.” See [2],p. 750. We can be glad that the editors of the presentvolume (clearly with “broad knowledge of your the-ories”) got the permission of J. Malgoire (who hadpower-of-attorney for Grothendieck) to publish (partof) the correspondence of Grothendieck contained inthis volume; see [2], pp. v and xii.

I put this beautiful volume back on my shelf, andI am sure I will consult it again many times.

References[1] D. Mumford, Selected Papers on the Classification of

Varieties and Moduli Spaces, edited and with commen-tary by David Gieseker, George Kempf, Herbert Lange,and Eckart Viehweg, Springer, New York, 2004.

[2] , Selected Papers, Volume II, On Algebraic Geom-etry, including a correspondence with Grothendieck,(Ching-Li Chai, Amnon Neeman, Takahiro Shiota, eds.),Springer, New York-Dordrecht-Heidelberg-London,2010.

[3] P. Samuel, Invariants arithmétiques des courbes degenre 2, d’après Igusa, Sém. Bourbaki 14 (1961/62),no. 228.

[4] J. Tate, The Work of David Mumford, Proceedings ofthe ICM (Vancouver, 1974) Canad. Math. Congress,Montreal, Quebec, 1975, Vol. 1, pp. 11–15.

[5] M. Atiyah and D. Iagolnitzer, Fields Medallists’Lectures, World Scientific Series in 20th Century Math-ematics, Vol. 5, World Scientific Publ. and SingaporeUniv. Press, 1997.

[6] Correspondance Grothendieck-Serre, Pierre Colmezand Jean-Pierre Serre, eds., Société Mathématiquede France, 2001. (Grothendieck-Serre Correspon-dence, Bilingual edition, Pierre Colmez and Jean-PierreSerre (eds.), Catriona MacLean and Leila Schneps(translation), American Mathematical Society, SociétéMathématique de France, 2004).

[7] János Kollár, Book review of David Mumford, Se-lected Papers, Volume I, Notices Amer. Math. Soc. 43(2005), 111–114.

[8] Frans Oort, Book review, Nieuw ArchiefWiskunde 5/8 (September 2007), pp. 225–227,http://www.nieuwarchief.nl/serie5/pdf/naw5-2007-08-3-225.pdf

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