a fresh look at francesco · pdf filefrancesco severi grew up in poverty in the tus- ......

A Fresh Look atFrancesco SeveriJudith Goodstein and Donald Babbitt

Astudent of Corrado Segre, EugenioBertini, and Federigo Enriques,Francesco Severi is rememberedtoday as one of the key architectsof the Italian school of algebraic

geometry in the first half of the twentieth century.(For a brief discussion of some of his mostimportant contributions to algebraic geometry,see the appendix.) His eagerness to serve theFascist regime after Mussolini became dictator in1925 has cast a long shadow over his name. Eventhough Severi survived multiple investigationsin Italy, rediscovered his Catholic faith, andemphatically denied repeated accusations that hewas anti-Semitic, the stigma has persisted.

Among the first, if not the first, to come toSeveri’s defense in the post-World War II era wasBeniamino Segre, who had lost his own academicchair in Bologna in 1938 in the wake of the govern-ment’s anti-Jewish legislation. Relieved, at Severi’sbehest, of his duties as an editor of Italy’s old-est scientific journal in the wake of the regime’sracial laws, Segre nevertheless insisted that ru-mors of Severi’s “supposed anti-Semitism” were“flimsy” and based on “some misunderstanding”

Judith Goodstein is university archivist emeritus at Cali-

fornia Institute of Technology. Her email address is jrg@

caltech.edu.

Donald Babbitt is professor of mathematics emeritus at

University of California, Los Angeles. His email address is

[email protected].

1References with an∗ can be found at the website: http://

www.JudithGoodstein.com.

DOI: http://dx.doi.org/10.1090/noti881

[Segre 55*].1 Although his tenure as Severi’s as-sistant in Rome from 1927 to 1931 gave him aringside seat as Severi’s politics veered sharplyfrom left to right, Segre took pains to remindItaly’s mathematical and scientific communitiesthat his mentor had once championed Italy’sparliamentary democracy, protested the brutal1924 murder of Giacomo Matteotti (a Socialistdeputy in Parliament), and signed the philosopherBenedetto Croce’s anti-Fascist manifesto—actionsthat forced his resignation in 1925 as rector ofthe University of Rome. These events, Segre oncesaid, had led Severi to have “a profound disagree-ment with the fascist government, which lasted—if even in a form increasingly attenuated—forseveral years, even after having been called totake part in the Academy of Italy” [Segre 63*].And, on that sober note, Segre took leave ofSeveri’s political career under Mussolini.

Our story offers a fuller look at Severi’s politi-cal dossier during the inter-war years and beyond.Primary sources range from the personal corre-spondence of Beniamino Segre and Oscar Zariskito Italian government records of the Fascist pe-riod in the Archivio Centrale dello Stato (ACS) todocuments in the historical archives of the Uni-versity of Rome and the Lincei. The whereaboutsof Severi’s personal papers is a mystery, and it ispossible that he destroyed them. Some mathemati-cians may find a recounting of his activities underFascism uncalled for, preferring to recall only hisconsiderable mathematical legacy. (Again, see theappendix for a nontechnical discussion of someof the most important parts of this legacy.) Others

1064 Notices of the AMS Volume 59, Number 8

http://www.JudithGoodstein.com

http://www.JudithGoodstein.com

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To celebrate fifty years of publishing activity,

Severi offered Zanichelli, the publishing house

with close ties to Federigo Enriques, the

opportunity to publish his collected works, in

multiple volumes. When Zanichelli said no, Severi

enlisted Beniamino Segre’s help in preparing a

selection of his writings instead. Severi chose

this picture of himself for the frontispiece, but

asked Segre to remove the fascist insignia first.

are content to allow the proverbial skeletons inthe closet to remain undisturbed.

In 1989 the Italian geometer Edoardo Vesen-tini raised the subject himself in a talk at theAccademia dei Lincei, which hosted a one-daymeeting on the cultural consequences of the raciallaws in Italy. Speaking about his own discipline,Vesentini said, “Even if we all know that closetsexist and skeletons also, forgotten or hidden, and[even] if we all know that such recognition cannotbe deferred indefinitely” [Ves 90], the thoughtof taking his own teachers and the more seniormembers of the Italian mathematical communityto task fifty years after the Fascist regime issued itsManifesto of Italian Racism seemed unduly harsh.Some colleagues had only recently died, and re-membering them, he told the group, brought back

memories along with “a mixture of admirationand, sometimes, affection.” Fortunately, the math-ematicians of his own generation, those who cameof age after 1945, had at least “been spared theshame of being among those mathematicians” whohad been forced to tell Castelnuovo and Enriques(two Jewish mathematicians) they could not entertheir institution’s math library. Castelnuovo’s lackof bitterness after the war ended, Vesentini noted,had played a crucial role in putting the country’smathematics back on track. “The generosity andfar-sightedness of Castelnuovo,” he added, “mustnot prevent, above all, the events of those yearsfrom being investigated and thoroughly examined…because those events belong to the history ofscience.”

A Taste for Politics

Born in 1879 the youngest of nine children,Francesco Severi grew up in poverty in the Tus-can town of Arezzo, where as a boy he tooka keen interest in politics, following in particu-lar the socialist movement, then on the rise inItaly. After being appointed professor of math-ematics at Padua in 1905, Severi allied himselfwith the left-wing blocco popolare patavino, whichrewarded his allegiance by appointing him presi-dent of the municipal gas and water company. In1910 he officially joined the Socialist Party andwas quickly elected councilor for the communeof Padua and became the Socialist alderman foreducation. When World War I broke out, Severisided with those who urged intervention on theside of Britain and France. He severed connectionswith the Socialist Party, which supported Italianneutrality, and quickly volunteered for militaryservice once Italy entered on the Allied side in1915.

Between 1918 and 1925 Severi flirted with run-ning for political office and received backing fromwar veterans and Socialist-run unions. He servedfor a time as president of the newly formed Na-tional Association of University Professors andbecame rector of the University of Rome on therecommendation of Giovanni Gentile,1 who at thetime was enjoying a brief tenure as minister of pub-lic instruction. It is not entirely clear how Severi’sassociation with Gentile, a philosopher and leadingproponent of fascist ideology, may have affectedhis views. However, several years after Mussolini’s

1An ardent supporter of the fascist government and a

major figure in Italian philosophical circles, Gentile im-

plemented far-reaching and lasting educational reforms

during his time as minister of education (1922–1924). Of-

ten referred to as “the philosopher of fascism”, he wrote

several key political tracts for Mussolini, including “The

Manifesto of the Fascist Intellectuals”, which many leading

fascist writers and artists signed in 1925.

September 2012 Notices of the AMS 1065

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Tullio Levi-Civita and his wife Libera Trevisani,

one of his pupils at Padua, whom he married in

1914, celebrate their arrival in New York City,

1933.

1925 proclamation of a dictatorship—“We are nota ministry; we are not even a government. Weare a regime” [Bo 05], Il Duce (Mussolini’s titlefor himself as “chief” of the fascist movement)told the deputies in Parliament in the spring ofthat year—Severi began sending the Fascist leadera steady stream of articles, several embellishedwith a flowery personal note. Taking his lead fromMussolini’s own political trajectory (Il Duce hadstarted his political career as a socialist), Severiappears to have cast off his socialist political tieswhen they no longer suited his political ambitions.Although he later wrote that his act of defiancein speaking out against Matteotti’s political as-sassination in 1924 had ended forever “his activeparticipation in political life” [Sev 45*], the his-torical record suggests that he wanted to play aprominent role in the intellectual life of the newfascist state. Indeed, Severi used his considerablepolitical connections to promote his advancement.

The Accademia d’Italia

Plans for a new state-sponsored cultural insti-tution, the Reale Accademia d’Italia (The RoyalAcademy of Italy) began to take shape in 1926,and Severi took a keen interest in the slot reserved

for a mathematician. It was an open secret thatanother algebraic geometer from Rome, FederigoEnriques, born into an Italian Jewish family, hadbeen nominated. That Severi had been feudingwith Enriques for several years over a satisfactoryalgebro-geometric proof of the “CompletenessTheorem” (see [B-G 10]) and other matters onlyheightened the stakes for Severi.

The more he thought about his own politicalfuture, the more Severi seems to have becomeconvinced that a Fascist loyalty oath was the an-swer. Speaking, he said, on behalf of “the greatmajority” of professors, in January 1929 Sev-eri dispatched an unsolicited and impassionedmemo to Mussolini himself, urging swift passageof an oath that would serve to indemnify thosewho, like himself, had engaged in “nonorthodoxdemonstrations,” such as signing Croce’s mani-festo. Anything short of that, he added, would“deprive our Universities of most of the bestmathematicians. On the faculty at Rome, almostnone perhaps would remain.”2 Rest assured, Severicontinued, “I have never done anything or criti-cized anyone that could even remotely be inter-preted as contrary to the Regime” while abroad[G-N 05]. Although Severi had not yet becomea party member (he joined the Fascist Party in1932), he had begun to ingratiate himself withBenito Mussolini’s regime.

In mid-February, Severi enlisted his univer-sity colleague and confidant Giovanni Gentile, nolonger minister of public instruction, in his po-litical aspirations. Less emotional, more practicalin tone, Severi’s letter to Gentile began by reit-erating that Italy badly needed a loyalty oath toidentify and isolate university professors hostileto the Fascist regime and to reward those, likehimself, who had resolved their differences withMussolini’s government. “As I already told you inperson,” he reminded Gentile, “within the limitsof my own poor powers, I have done as muchas I could for this purpose, and I have reason tothink that the Head of Government is very welldisposed.” He then went on to instruct Gentileat length about how to manage the other stake-holders in the matter, including the press, thefascist rank-and-file, and various ministries, andwhat they should and should not be told. Only in a

2The loyalty oath required of all university professors was

administered two years later in 1931. Twelve courageous

academicians out of 1,250 refused to sign and lost their

university posts, including Vito Volterra, the undisputed

head of Italy’s school of mathematics before the advent

of Fascism. Relations between Severi and Volterra had al-

ways been rocky, helped along by the fact that Volterra, an

avowed anti-Fascist, had once said to him during a session

at the Lincei, “Algebraic geometry serves no purpose” [Tri67*].


brief postscript, in which Severi informed Gentilethat he had marginally patched up a long-standingquarrel with Enriques over competing textbooks(“but with open [and] utter disgust”), did he men-tion that Enriques’s Academy nomination mightbe in trouble [G-N 93*]. Gentile’s response to thisletter, if there was one, has not survived.

Whether by design or chance, Severi had landedin the right spot if things veered off track forEnriques, which they did. In mid-March, barely onemonth later, the government deleted Enriques’sname from the list of candidates sent to thepresident-elect of the Academy of Italy. In its placeappeared the name Severi. Recalling that period atRome, Giorgio Levi della Vida, at the time directorof the Oriental School, wrote, “The Rector then wasFrancesco Severi, a great mathematician and anenergetic man of action …whose antifascism couldnot resist the seduction of the [Royal] Academyof Italy, and as the first error leads to a secondone easily and a third, [the error] turned intoenthusiastic support for the regime” [LeviV 66*].

One of the many congratulatory telegrams Sev-eri received in 1929 following the government’spublic announcement of his nomination camefrom the differential geometer Tullio Levi-Civita,a colleague at the University of Rome and one ofItaly’s most distinguished mathematicians. In re-sponse, Severi sent Levi-Civita, his friend of morethan twenty years, a letter marked “confiden-tial” from Barcelona, where he was lecturing. Heexpressed “absolute surprise” at his own nomina-tion and remorse over the omission of Levi-Civita’sname. “I certainly would have loved to see you,the strongest among the columns of Italian math-ematics, in the Academy also,” he wrote. “Let ustrust in the future” [Sev 29*], a veiled hint perhapsthat Severi intended to nominate his friend. Laterthat year, in fact, when Severi raised the issue withhis Jewish colleague, Levi-Civita reminded him inwriting of the obstacles (“two distinct objections,stated or implied”) standing in his way: politicsand religion. As the historian Annalisa Capristo,who has written extensively about the exclusionof Jews from Italian academies before and af-ter the racial laws, writes, “In fact, we have noknowledge of other possible ‘objections.’ On thesedelicate questions, however, the two correspon-dents maintained—at least in their exchange ofletters—an understandable discretion” [Cap 03].

The Vatican acted first. In February 1929, Mus-solini’s government and the Vatican had signedthe Lateran Accords, each recognizing the other’ssovereignty as a state, and in April 1929, onlya month after the Reale Accademia d’Italia an-nounced its first thirty members, the Vatican’sAccademia Pontificia delle Scienze elected Levi-Civita and the world-renowned mathematician

Vito Volterra, both born into Jewish families,members of its group. Anti-fascist organizationshad been quick to note the absence of any Ital-ian Jews among the first members of Mussolini’sAcademy, and in fact the fascist academy wasnever to admit any Jews to its ranks [Good 84].The Vatican simply leveled the playing field.

That October the thirty successful candidatesfor the Royal Academy of Italy, including the physi-cist Enrico Fermi, the composer Pietro Mascagni,and Francesco Severi, the sole mathematician,gathered in the Campidoglio, Rome’s city hall,where Prince Francesco Boncompagni Ludovisi,the city’s governor, proclaimed them the aristoc-racy of Italy’s intelligentsia. Clad in a ceremonialsuit embroidered in gold, a stipend of 3,000 lira amonth lining his pocket, and promoted to the rankof “your Excellency”, Severi had been appointedthe regime’s spokesman for Italian mathematics—and in the eyes of Italy’s new rulers, he was. As oneof Severi’s assistants in Rome, Francesco Tricomi,the author of a classic text on integral equations,later recalled, Severi was as “exuberantly fascist ashe had earlier been antifascist—who wanted to beand to a certain extent, was—the ‘boss’ of Italianmathematics in the fascist period” [Tri 67*].

Severi proceeded to play an active role in theaffairs of the fascist academy. Besides diligentlyattending sessions, he nominated candidates formembership, including Levi-Civita in 1933 andagain in 1934, although both he and Levi-Civitaknew of the insurmountable obstacles other Jew-ish candidates had faced. In addition to serving onvarious commissions, including one charged withpurifying the Italian language of foreign words(e.g. cocktail), he presented numerous papers, hisown and others’; and, indeed, when called to ac-count for his activities later, he proudly notedthat he had personally introduced thirteen of thefourteen papers authored by Jewish scientists.However, in 1931, when Mussolini, with the com-plicity of Academy president Guglielmo Marconi,intervened decisively to prevent the Academy fromawarding the first Mussolini Prize in science tothe militantly anti-fascist and Jewish professorof human anatomy Giuseppe Levi, Severi did notspeak up in defense of Levi, nor did he voiceany objection to the government for not followingits own rules [Fabre 08]. (The honor of receivingthe first Mussolini Prize went to a well-known—and Christian—physiologist and explorer in goodstanding with the regime, Filippo de Filippi.)

Severi’s Interactions with Colleagues

In his dealings with others, Severi often cameacross as arrogant, autocratic, and quick to takeoffense. There seemed to be no middle ground: heeither dazzled those around him (“Severi is a man


with bewitching hands, when he does something,it is always splendid” [ScorD 62*], the Sicilianmathematician Gaetano Scorza once remarked) ordemonstrated “a childlike incapacity either forself-criticism or for cool judgment,” as LeonardRoth, the English algebraic geometer who spentthe year 1930–1931 in Rome, famously observed.Worse still, Roth added, “he meddled in politics,whereas it would have been far better had he leftthem alone” [Roth 63].

Mathematical collaborations with Severi couldalso be challenging, recalled Scorza’s son,Giuseppe Scorza Dragoni, who collaborated withSeveri on the second and third volumes of Lezioni

di Analisi , magisterial tomes, published in 1942and 1951. As the younger Scorza recountedin a letter to Segre after Severi’s death, someof the difficulties arose because the Tuscanmathematician always had a hundred differentobligations to attend to. In fact, he had oncespent his entire vacation in Rome without beingable to meet with Severi once, except on thelast day. Scorza also quickly learned that Severiwould not admit to making any mistakes. “Butthis last obstacle, when I realized it,” Scorzacontinued, “I got around easily: instead of sayingwhere there was a mistake, and how it shouldbe fixed, I pretended to not understand and Icontinued to not understand until Severi, furiousby my artfully directed objections and questions,ended by finding the mistakes and [making] thecorrections himself” [ScorD 62*].

Severi believed, according to Roth, “that theworld at large failed to treat him with due consid-eration. For, incredible as it may seem, althoughduring the whole period of his maturity honourswere showered upon him and invitations pouredin, yet he remained forever unsatiated …he seemedmore or less permanently aggrieved” [Roth 63].

Severi’s trip in 1930 to South America is a casein point. During his stay there he gave a series oftechnical lectures to a university crowd in BuenosAires, telling his audience in so many words,according to one listener, that he had come toArgentina “to civilize and teach mathematics” tothe country’s inhabitants. He also ostentatiouslyannounced in several lectures that he disagreedwith Levi-Civita “from several points of view.”Severi’s pronouncements did not sit well with themathematicians in the audience, who knew of Levi-Civita’s pioneering work in differential geometryand admired it. Severi further enraged his hostsby asking point-blank, “If you have made [Frenchmathematician Jacques] Hadamard and Enriquesacademicians [of Argentina’s scientific society, theSociedad Cientifica Argentina], why haven’t youdone the same for me?” With the society’s mem-bers deeply divided over what to make of Severi’s

mathematical reputation, they postponed the de-cision on his election until they could learn moreabout him as a person. The task was entrustedto an old classmate of Levi-Civita’s, Felix Carli, amember of both the Sociedad Cientifica Argentinaand the Accademia di Scienze Fisiche e Matem-atiche di Buenos Aires. “I beg you to give me youropinion of the man” [Carli 30*], he wrote to Levi-Civita. While we lack Levi-Civita’s reply, Severi’ssubstantial list of honors and affiliations does notinclude membership in either of Argentina’s sci-entific organizations, which may suggest a tepidresponse on Levi-Civita’s part.

Serving as Severi’s assistant for four years,Francesco Tricomi, like Roth, had ready answersfor thosewanting toknow whathehadexperiencedin Rome. As a professor and mentor, he reported,Severi knew how to push his students along; onthe pedagogic side, he ranked as an extraordinaryteacher, and Tricomi had found his classroomduties useful, unlike Severi’s other assistants whoconsidered them a “waste of time”. However,Tricomi also thought that Severi behaved as if hewas “the ‘padrone’ [the lord and master], a littleoverbearing…[although] I never had any real clashwith him,” which Tricomi attributed to treatinghis responsibilities as an extension of his recentmilitary service. Later, when Tricomi moved upthe academic ladder and became a professorof mathematics at Florence, he took less kindlyto Severi’s requests, reminding him, politely butfirmly, that “I was no longer under his orders”[Tri 67*].

Beniamino Segre, who became Severi’s assis-tant several years later and who remained closerto him than any of his other pupils, also remem-bered him as “ambitious and pugnacious” [Segre62]. Straining after the events of World War IIto paint an accurate and fair portrait of Severias a man, Segre said, “We need to keep in mindthat Severi had a complex, intense, tormented na-ture, and an exceptional personality,” which Severihimself had described as a “restless energy, whichtakes away my breath and makes me unhappy”[Segre 63*]. All in all, Segre concluded, the appear-ance of fearless confidence that Severi projectedprobably concealed some “mysterious, unjustifiedfear of not being appreciated and loved.” Recall-ing his own dealings with Severi in Rome in theearly 1920s, the American mathematician OscarZariski told an interviewer that Severi once saidto him, “I love you, Zariski, but you don’t love me”[Parikh 91], although the correspondence betweenthe two mathematicians after the war suggests amore complicated relationship.


Public Acts

Unlike Enrico Fermi, his colleague at Rome, whonever talked politics, Severi actively cooperatedwith Mussolini’s government. In an article aptlytitled “Fascismo e Scienza”, published in 1933 inL’Illustrazione Italiana, a popular weekly maga-zine, Severi dismissed those who remained miredin the past (“a wretched preoccupation with con-ventional analytical politics and trifling doctrinesfrom other times”) while extolling, in the nameof patriotism, loyalty to the regime for its goalof restoring the grandeur of Rome and reclaimingItaly’s place in the world of science, starting withmathematics [Sev 33*]. The Italian school of math-ematics was a matter of deep national and personalpride for Severi, who told his readers: “Not every-one knows that…Italy, in fact, occupies one of theforemost places, if not the first, in mathematicsin the world today. Foreign mathematicians ev-erywhere,” he continued, “recognize this and asItalians and as fascists we are legitimately boundto be proud of it.” More importantly, he concluded,“we export many ideas, nay many more than thosewe import.” Even after the war ended, Severi per-sisted in believing that Italian mathematics (andby extension, his own achievements) did not re-ceive the recognition they deserved outside Italy.3

“Thank you for the draft …of [Fabio] Conforto’sreview,” Severi began one letter to Ralph Boas,executive editor of the Mathematical Reviews, inMarch of 1949,

I note, however, that the last lines of theoriginal review, which had expressed asensible opinion of my work, have beendeleted. I would not point this out if I hadnot been very hurt by the way my worksare generally reviewed in MR. This is thefirst time, after a half-century of intense,nonstop work in science, that I find my-self having to express such grievances toa publication. Publishing …Conforto’s finalcomments would perhaps have served toreaffirm that the Directors of MR do notshare the unfavorable attitude of certainreviewers, which is something you kindlymentioned to me and which I certainly be-lieve.…In any case, in your position…youhave connections with many mathemati-cians and frequent opportunities to bringpeople together. Therefore I have faith inyour efforts to restore the reputation ofItalian mathematics, which is alive and ac-tive.…As one [who] loves my country and

3Nor, for that matter, does the recent authoritative refer-

ence treatise The Princeton Companion to Mathematics[Gowers 08] cite the contributions of the Italian school of

algebraic geometers by name.

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Inaugural session of the fascist Academy of

Italy, Farnesina Palace, 1930. Guglielmo Marconi,

in ceremonial uniform, is seated next to

Mussolini. Other academicians gathered around

the table include Francesco Severi, Cesare

Bazzani (first and second from left), and

Francesco Giordani (third from right).

my science…this would distress me greatly,and if I am induced to do something thatcosts me some sacrifice (no one likes to talkabout himself), it is because I believe I haveto do it not so much in my own interest asin defense of my School, which follows mefaithfully and which, like the majority ofItalians, is working to restore our country,after the great disaster, to a high interna-tional level in the intellectual and spiritualrealm, which is the realm where Italy hasalways distinguished itself throughout thecenturies. [Sev 49*]

Enacted in 1938, Italy’s anti-Jewish legislationaltered abruptly Segre’s relationship with Severi.“You already know of the whirlwinds that haveshocked us over the past several months, inflict-ing unspeakable moral pain on us,” Segre wroteto Zariski from Bologna on October 16, the dayafter the racial laws had banished him from theclassroom, expelled him from numerous scientificacademies and organizations, and relieved himof his duties as a managing editor of Annali di

Matematica Pura ed Applicata [Segre 38*]. “Stillmore distressing,” he added, “is the apathetic—ifnot to say hostile—attitude of certain individualsof our common acquaintance,” being careful notto mention anyone by name. That same day, Segresent a letter to Levi-Civita, also a member of thejournal’s editorial board, in which he identifiedSeveri as the chief instigator behind their dis-missal. In it, he wrote: “The initiative was startedby S[everi], who—some time ago—indicated to thePresident of the [organization unidentified] thesituation in the editorial office of the Annali. Said


President then wrote to Dr. [unidentified name],requesting advice and this person left the mat-ter completely up to Severi’s decision” [N 96*].Of the journal’s four editors, three were Jewish(Guido Fubini was also dismissed), leaving Severi,who had joined the editorial board in 1925, asthe journal’s sole editor. “In no other case up tonow has anything similar happened,” he told Levi-Civita. Recalling the event in an article dedicatedto Giovanni Sansone and published many yearslater in the Annali, Severi noted the “deplorabledecree” that left him remaining “alone on thescientific committee of the ‘Annali’, becoming au-tomatically, without my desire, the only managingeditor” [Sev 60]. His colleague Giovanni Sansone,added Severi, had been the very first to share withhim the duties of running the journal.

When the publisher of Springer ordered itseditor, Otto Neugebauer,4 in 1938 to removeLevi-Civita’s name from the masthead of theZentralblatt, the renowned international reviewjournal in mathematics founded by Neugebauer,on the grounds that according to Italy’s racial lawshe was no longer a university professor, Severi’sname quickly appeared in his place. While planningfor the second meeting of the Italian MathematicalUnion in Bologna in 1940, the organizers debatedwhether Levi-Civita, a former member and an ex-pert on general relativity theory, could be invitedto speak on that topic as a guest. Severi is re-ported to have said, “Please! We just got rid ofthat race.”5 The historians of Italian mathematicsGiorgio Israel and Pietro Nastasi, who have writ-ten extensively about science and race in fascistItaly, have also reported that Severi personallyintervened to deny his Jewish colleagues accessto the University of Rome’s mathematics libraryafter the racial laws went into effect.6 Two yearslater, Severi wrote Levi-Civita a friendly, if short,note in which he apologized for not being ableto personally deliver a recent issue of the Annali,“but since I see that it is late, I am sending it to you,

4Neugebauer subsequently resigned from all editorial du-

ties at Springer journals, left Germany, and joined the

Brown faculty in 1939, where he became the founding

editor of the AMS’s new abstracting journal, MR (Mathe-

matical Reviews).5“Per carità, ci siamo appena liberati di quella razza.”

The story has been passed down from the mathematician

Ugo Amaldi, present at the meeting, to Edoardo Amaldi,

his son, who played a central role in rebuilding Italian

physics after World War II, to Edoardo Vesentini, a distin-

guished algebraic geometer in his own right and former

president of the Lincei, private communication, Tullio G.

Ceccherini-Silberstein to the author (JG), November 11,

2011.6Enzio Martinelli, a pupil of Severi, is the principal source

for this story, private communication, Giorgio Israel to the

author (JG), Oct. 28, 2011.

reserving to myself to come and greet you as soonas it will be possible for me” [Sev 40*]. Historydoes not reveal if Severi kept the appointment.

By the end of the decade, Severi had con-solidated his leadership position within Italy’smathematical community. In spring 1938, Severipetitioned Mussolini to underwrite an Institute ofHigher Mathematics in Rome [INDAM], which wasinaugurated in 1940, with the Duce; the minis-ter for national education, Giuseppe Bottai; anda host of other Fascist dignitaries in attendance.Not surprisingly, Severi also became the institute’sfirst president. In the wake of the racial laws, hehad succeeded Enriques as professor of highergeometry at the University of Rome; he had alsoreplaced Enriques as director of Rome’s school ofpost graduate studies in the history of science.One of the few items still missing from his lengthylist of honors and appointments was member-ship in the Pontifical Academy of Sciences, whoseroster had recently expanded to include RobertMillikan, Max Planck, Erwin Schrödinger, and othernotable scientists. In 1939 an opening occurred.As one of four scientists in the running for theone vacant seat, Severi’s credentials were hard totop. As one academy member wrote to Caltech’sMillikan, highlighting the advantages that Severi’selection would confer on the Pontifical Academy,“[W]e all have a great interest in concentratingour designation on the name of H[is] E[xcellency],Professor Severi, not only because he is a princeamong the mathematicians, and probably the mostprominent in the world in the branch of algebraicgeometry…but on account of his great authorityand his political position in Italy, we are in needof his help as a friendly link between the Italianauthorities and the Pontifical Academy” [Giorgi39*]. One wonders if Severi, who won the election,appreciated the irony of listening to papers pre-sented by its long-time members Levi-Civita andVito Volterra, both of whom had paid dearly forbeing both Jews and anti-fascists in Mussolini’sItaly.

Investigation and Rehabilitation

After the liberation of Rome in June 1944, Italy’snewly installed provisional government estab-lished a High Commission for Sanctions againstFascism to investigate allegations of wartimecollaboration against party members, either fortaking an active part in Fascist political life or forremaining loyal to Mussolini after he was deposedin September 1943. That year, at the behest ofthe Ministry of Public Instruction, the commis-sion took up the case against Severi, the onlymathematician dealt with in this fashion. When itinitially suspended him from university teaching,


effective August 1, 1944, Severi appealed the rul-ing. To the charge that he had been an apologistfor the regime during the war, Severi counteredby saying that he’d given scientific talks in Spainand Portugal, not political speeches, and that hehad made a 1943 trip to Germany for the solepurpose of collecting a medal in connection with aCopernican anniversary. In May 1945, Severi’s sus-pension was annulled and replaced by a sanzioni

minori, a simple censure that involved a letterplaced in his university personnel file, but that didnot prevent him from teaching a course on highergeometry at INDAM during the 1945–46 academicyear [Rog 05].

When pressed on the issue of anti-Semitism,Severi pointed to his repeated efforts on behalf ofLevi-Civita (who had died in 1941) for membershipin the Academy of Italy [Cap 03] and his continuedfriendship with Segre, whom he described ashaving always been his favorite student. Neitherthe central commission nor a separate commissionassigned in 1945 to examine the behavior of formermembers of the Academy of Italy found anythingto condemn in Severi’s personal or professionalconduct. In its report on Severi’s conduct, in fact,the commission noted: “The moral rectitude andgood services of [P]rofessor Severi as a personand as a scientist are beyond discussion and onthe other hand no one could doubt it” [ACS 45*].Faced with a new investigation the following yearinvolving former Fascist academicians, Severi toldSegre, “Naturally, I’ll come out from this painfultrial as immaculate as I have always been” [Sev 46*].

Severi’s prediction proved true but only up toa point. In 1945 a separate committee, appointedby the provisional government to reconstituteItaly’s storied Academy of the Lincei, homed inon members who had belonged to the Academyof Italy. The chairman of that commission wentout of his way to note that the range of Severi’sactivities demonstrated “a marked independenceand courageous conduct to ensure that scienceprevailed over the dominant politics” [ACS 45*].In so many words, as the central commission hadpreviously concluded, Severi “had not receivedfrom Fascism anything more than was his dueas a distinguished scientist.” Nevertheless, thatsummer the committee members unanimouslydecided to purge from the Lincei any memberswho had taken part in a March 1944 meetingheld in German-occupied Florence, regardless ofthe reason behind their attendance. Severi hadbeen present at that meeting, and so he was out.As Guido Castelnuovo, a committee member andJewish mathematician who had gone into hidingduring the war, wrote to Severi that June, “It painsme that it is necessary to take a step of such a

nature with respect to a scientist who has honoredItaly such as you” [Cast 45*].

According to the commission, Severi’s trans-gression had been to accept an invitation from hisold friend Gentile, at the time the Academy of Italypresident, to attend a celebration there in honorof the eighteenth-century political philosopher Gi-anbattista Vico. “I have no reason to correct anypage from the book of my life,” Severi replied, afterreading Castelnuovo’s letter [Sev 45*]. Far fromadmitting any wrongdoing, Severi defended hisvisit, but thought it important to add a postscriptin which he pointed out that “among the impor-tant examples of his behavior with respect to theracial laws,” he had seen to it that Castelnuovo’sbook on the origins of the calculus remained incirculation after 1938.

Gentile was unavailable to shed any furtherlight on the matter, having been assassinated bycommunists in Florence in August 1944 whileriding his bicycle. Four years later, Italy’s ministerof justice, Palmiro Togliatti, declared a sweepingamnesty, and in July 1948, Severi was reelecteda member of the Lincei. Severi, who had lost hisposition as president of INDAM, also recoveredthat post following the amnesty and held it untilhis death.

Although Severi emphatically denied being anti-Semitic at the end of the war, he saw evidenceof others “paint[ing] me, especially in the Anglo-Saxon world, as anti-Semitic…[and presuming]racial preconceptions that I have never had,” hewrote in one letter to Segre [Sev 49*], adding, “Andyou and many other Israelites know this well.”A case in point was Oscar Zariski,7 born into aRussian-Jewish family and an algebraic geometerat Harvard, who seemed in no rush to resumecontact with Severi after 1945. “I am very sorrythat he didn’t answer [my letter],” Severi toldSegre,

I had sent him [in 1948] many of my pa-pers including books, together with a verywarm letter…I did the same with [Solomon]Lefschetz, who has already answered…ex-pressing his deep appreciation. No answer,instead from Zariski, to whom I won’t sendanything more from now on, unless things

7Zariski attended high school in Chernigov (Ukraine) and

the University of Kiev (1918–1920). Admitted to the Uni-

versity of Rome in 1921 as a third-year student, Zariski

received his doctor’s degree in 1924 under the supervi-

sion of Castelnuovo. Zariski later said that Castelnuovo

chose a problem that would suit him (Galois theory, solv-

ing by radicals) because he could see that Zariski was not,

at heart, a geometer in the sense of the Italian school of al-

gebraic geometry. He did postgraduate work at Rome on

a Rockefeller fellowship (1924–1926) before going to Johns

Hopkins University in 1927.


change. I know that Zariski had a grudge[against me]; [Tomás Rodriquez] Bachiller,who was there [at Harvard] …wrote to methat he had…explained to Zariski the realfacts and that he seemed to be persuaded.

Does he still perhaps believe in my anti-Semitism, the most disgusting calumny thathas been circulated underground, withoutever having the bravery to pronounce itpublicly, it being so far from the truth? Whywould he keep considering the polemics Ihad with Enriques in the past as a displayof anti-Semitism? Can’t anyone have a boneto pick with a Muslim without being againstMohammed. [Sev48*]

Severi also felt that these accusations had donegreat damage to the reputation of his own fieldof mathematics. As he wrote to Segre in an-other letter, “The attack that has been conductedagainst me has ended by damaging Italian geom-etry abroad and especially in America [Sev 49*].You have only to look, continued Severi, at the“very hostile attitude towards us” by Zariski’sgroup of abstract algebraists, despite the fact thatZariski “formed his outlook (that I always guided)in Italy.”

Severi also found that some French mathemati-cians continued to harbor hard feelings towardhim in the postwar period. Mounting a spiriteddefense of his aging mentor, in 1955 Segre wrote aletter to the French mathematician René Garnier,in which he stated unequivocally that rumors ofSeveri’s anti-Semitism did not square with whathe knew. In the paragraphs that followed, Segrereiterated many of the arguments Severi had usedin the past to defend himself, including Segre’sown interactions with Severi. In his reply, Garniersaid that he was personally satisfied by Segre’snarrative, but others might not be, given that “inParis, there is strong resistance” [Ga 55*].

There is little correspondence between Severiand Zariski in Zariski’s papers, which are depositedin the Harvard University Archives. There is nonefrom before World War II, and the few letters theyexchanged after 1948, when the correspondenceseems to have resumed, at times turned testy. Inresponse to a letter written by Zariski in summer1953, Severi demanded to know why Zariski hadbrought up Hitler and Mussolini: “Do I have per-haps some responsibility in front of those whohad to flee from paradise from the first and didnot find asylum in the second?…Your referencesdemonstrate to me that in certain areas of themathematical world this non benevolent attitudetowards me, from which I have greatly suffered,has not ceased.…On your part I have often hadan impression of coldness, which at times I wasnot expecting. But it could be that it is only a

consequence of your temperament” [Sev 53*]. Sev-eri then turned to mathematical matters. Judgingfrom their later correspondence, the subject nevercame up again.

One year later, when Severi’s name was point-edly omitted from the list of speakers for aninternational symposium on algebraic geometryin 1954, Zariski wrote to the chair of the organiz-ing committee: “I am particularly worried aboutthe omission of the name of Severi. I think thatSeveri deserves a place of honor in any gatheringof algebraic geometers as long as he is able andwilling to attend such a meeting. We must try toavoid hurting the feelings of a man who has doneso much for algebraic geometry” [Zar 54*]. Aninvitation was duly issued to Severi.8

Severi died in Rome on December 8, 1961, atthe age of eighty-two; he remained proud of thefact that, as he once wrote, “I have never recantednor repudiated any of the acts of my life that were,and are, expressions of the strongest attachmentto my country” [Sev 53*].

Appendix: Severi the Mathematician

Francesco Severi, who contributed in fundamentalways to several areas of mathematics in the firsthalf of the twentieth century, is generally acknowl-edged as one of the three great Italian algebraicgeometers of that era, together with Guido Castel-nuovo and Federigo Enriques. A hugely prolificauthor, Severi’s bibliography contains 415 items,including 34 books that range from elementary toadvanced monographs. He was gifted with extra-ordinary geometric intuition; however, when thiswas combined with the Italian algebraic geome-ters’ imprecise mathematical language and theirnotion of what they considered a proof, it ledhim many times to either state a true theorem forwhich it was impossible to convert his proof toan acceptable modern one (e.g., see the article ofJoseph Harris [Ha 86]), or to state as a true theoremone that was “almost true” but which later mathe-maticians had to modify in order to obtain a truetheorem (e.g., see the article by Robert Lazarsfeld[Laz 81]). Ironically, Severi’s own ideas in anothercontext have often been utilized to furnish thecorrect proofs. For an authoritative discussion ofItalian algebraic geometry and algebraic geome-ters, including Severi, between the two world wars,see the book by Aldo Brigaglia and Ciro Ciliberto[B-C 95].

In 1949 Severi presented a paper at the Colloquede géométrie algébrique, held in Liège [Sev 49],entitled “La géométrie algébrique italienne: sarigueur, ses méthodes, ses problèmes”. In it he

8It was at this symposium where the famous exchange with

André Weil took place (see next page).


defended the Italian approach to what consti-tuted a satisfactory proof. Here are two samplesof what Severi had to say. In his opening para-graph he writes: “For many years a legend hasrun through a certain part, fortunately limited, ofthe mathematical community that Italian algebraicgeometry, while being ingenious and rich in im-portant results, has not yet attained the necessaryrigor”. He then goes on to discuss “substantialrigor”, i.e., the sense of rigor of the Italian school,and “formal rigor,” i.e., the sense of rigor of theFranco-American school of André Weil and OscarZariski.

Later in his talk (p. 41), while discussing arecent letter from “an eminent foreign geometer”(not named, but almost certainly Zariski), he says:“Personally, I believe that our methods, when care-fully analyzed, give the same sense of assuranceas the purely algebraic methods.” Here by “alge-braic methods” he seems to be referring to theFranco-American school’s methods of proof. Forthe response of the “formal proof” community,see the critical and very interesting review ofthis paper by Claude Chevalley in Mathematical

Reviews, 1951 [MR 0038094; 12, 353f].What might be correct to say is that Severi, per-

haps more than any other major mathematician ofhis day, stated more true theorems whose proofswere “irreparable” by modern standards or “al-most true” theorems that required modificationsto make them true or that were just plain false“theorems”. But sorting through his triumphs andmissteps is a story for another day.

Some of Severi’s Main Contributions to

Algebraic Geometry

Theorem of the Base. Severi’s most cited work isalmost certainly his Theorem of the Base, whichhe proved early in his career and is now usuallyreferred to as the Néron-Severi Theorem [Sev 06].

It was one of the major results of early twenti-eth century algebraic geometry over the complexnumbers C, and the paper itself was explicitlysolicited for the Math. Annalen by Max Noether.The theorem says that a certain important abeliangroup NS(F) naturally associated with an algebraicsurface F over C is finitely generated. Specifically,NS(F) is the quotient of the free group generatedby the irreducible curves (divisors) on F by thesubgroup of divisors algebraically equivalent tothe zero divisor. Néron [Ner 52] showed that thetheorem also holds for surfaces over any field,which is the reason that his name is also attachedto it. For more detail on the theorem and Severi’scontribution, see [Zar 71, Ch. 5.6].Contribution to the Proof of the Fundamental Theo-

rem of Smooth Surfaces over C. Both Severi [Sev 05]

and Castelnuovo made related and key contribu-tions to the proof of the Fundamental Theoremof Smooth Surfaces over C, one of the mostbeautiful and deepest contributions of early twen-tieth century algebraic geometry. The theoremstates the equality of two apparently different bi-rational invariants associated with the surface F :the dimension of the space of closed holomorphic1-forms on F (called Picard integrals of the firstkind) and the irregularity q ≡ pg − pa of F , wherepg is the geometric genus and pa is the arith-

metic genus of F . See, e.g., [Kl 05, pp. 243–244]for a more thorough discussion of this theorem,including the relative contributions of the twomathematicians.Principle of the Conservation of Number and Enu-

merative Geometry . Severi in 1912 made an ex-haustive analysis of the validity of the Principleof the Conservation of Number, a very importanttool in enumerative geometry [Sev 12]. However,he did this utilizing the foundational language andtools available in 1912. Further progress required amore refined foundational language, which indeedevolved in the following decades and consequentlyled to significant additional progress in enumer-ative geometry and Schubert’s calculus (Hilbert’s15th problem). See [Kl 76] for a definitive history,including Severi’s contributions.Rational Equivalence of Algebraic Cycles and Inter-

section Theory on Smooth Projective Varieties over

C. Severi’s contributions here consist both of deepinsights and serious missteps. He first introducedthe notions of rational equivalence of algebraiccycles and studied its relationship to the inter-section of algebraic cycles [Sev 33]. Early on, hebecame aware of difficulties with his definitionof rational equivalence and its relationship to theintersection theory of algebraic cycles, and as aresult, he continued to modify his definitions astime went on. At the 1954 International Confer-ence of Mathematicians (as related by B. L. van derWaerden), Severi presented a new exposition of histheory, which Andre Weil famously (and correctly)criticized in the discussion session. Subsequently,in 1970 van der Waerden [Waer 70]9 showed to hissatisfaction that Severi’s definitions and resultscould in fact be adjusted to meet the modern stan-dards of rigor. Precise and satisfactory definitionsof rational equivalence in the rigorous languageof the “French” school were given independently.See, e.g., Claude Chevalley [Chev 58]. To summa-rize the significance of Severi’s contributions torational equivalence and intersection theory, we

9There were many exchanges over the years between Sev-

eri and van der Waerden concerning various aspects of

algebraic geometry, including rational equivalence. For a

description of some of these interactions, see the fascinat-

ing article by Norbert Schappacher [Sch 07].


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Benito Mussolini and Francesco Severi (on the

right) in the University of Rome’s mathematics

library, 1939.

turn to William Fulton [Ful 98, pp. 25–26], who,in his authoritative work on the subject, wrote:“It would be unfortunate, however, if Severi’s pi-oneering work in this area were forgotten; andif incompleteness and the presence of errors aregrounds for ignoring Severi’s work, few of thesubsequent papers on rational equivalence wouldsurvive.”

Some Severi Conjectures

In addition to the multitude of theorems Sev-eri claimed to have proved, he also made manyinteresting conjectures, some true, some false,including the examples below. Both of theseconjectures have been settled—one as true, theother false—in two well-known papers of Kodaira[Kod 52] and Mumford [Mum 69].

Conjecture 1. Let V be a smooth irreducible pro-jective variety of dimension n. In [Sev 09], Severiconjectured that

pa = gn − gn−1 + gn−2 − ·· · (−1)n−1g1,

where pa is the arithmetic genus of V and gj isthe dimension of the vector space of holomorphicj-forms on V . This was proved by Kodaira in 1954.

Mumford proved the following conjecture ofSeveri to be false.

Conjecture 2. Let F be a smooth surface over C.Then the group of 0-cycles modulo rational equiv-alence is finite dimensional.

Ironically, paraphrasing Mumford, the methodof disproof of this conjecture is based almost en-tirely on Severi’s systems of equivalence.

For an excellent discussion of this conjectureand several other related topics that Severi ex-plored, see the article by A. Brigaglia, C. Ciliberto,and Claudio Pedrini [B-C-P 04]. Interestingly, theydescribe some of Severi’s work, starting with his

theory of rational equivalence in 1932, that hasproved to be relevant to the study of motives, animportant topic of current interest in algebraicgeometry.

Acknowledgments

We thank Maryse Bowers, who translated keypassages of Severi’s letter to Boas; William Fultonfor his helpful comments; Elisa Piccio, AnnalisaCapristo, andGiorgioFabre for their friendship; IvoBiagianti and Italo Farnetani for useful discussionsin Arezzo; Francesca Rosa for archival research inRome and Michelle Frankfort for tracking downdocuments in the Zariski Papers; Rita Zanatta,Susanna Panetta, Alessandro Romanello, CharlotteErwin, and the staffs at the Archivio Centrale delloStato (EUR), the Archivio Storico della Sapienza,Università di Roma, the Istituto storico italiano peril medioevo, the Accademia Nazionale dei Lincei,Harvard University Archives, and the CaliforniaInstitute of Technology Archives for their helpand permission to quote from materials in theirrepositories. Heidi Aspaturian provided valuableeditorial advice and assistance.

References[B-G 10] D. Babbitt and J. Goodstein, Federigo

Enriques’s quest to prove the “Complete-ness Theorem”, Notices Amer. Math. Soc. 58

(2010), 240–248.[B-C 95] A. Brigaglia and C. Ciliberto, Italian Alge-

braic Geometry between the Two World Wars,transl. from the Italian by J. Duflot, Queen’sUniversity, Kingston, 1995.

[B-C-P 04] A. Brigaglia, C. Ciliberto and C. Pedrini,The Italian school of algebraic geometry andAbel’s legacy, in The Legacy of Niels Henrik

Abel, Olav A. Laudal and Ragni Piene, eds.,Springer-Verlag, Berlin/Heidelberg, 2004, pp.295–347.

[Bo 05] R. J. B. Bosworth, Mussolini’s Italy: Life under

the Fascist Dictatorship, 1915–1945, PenguinGroup, New York, 2005.

[Cap 03] A. Capristo, Tullio Levi-Civita e l’Accademiad’Italia, La Rassegna mensile di Israel 69

(2003), 237–56.[Chev 58] C. Chevalley, Les classes d’équivalence ra-

tionelle I, II, Séminaire C. Chevalley 2e année,in Anneaux de Chow et Applications, ÉcoleNormale Supérieure, Paris, 1958.

[Fabre 08] G. Fabre, I volenterosi collaboratori di Mus-solini, Un caso di Antisemitismo del 1931,Quaderni di storia 68 (2008), 89–122.

[Good 84] J. Goodstein, The rise and fall of VitoVolterra’s world, J. Hist. Ideas 45 (1984),607–617.

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to Mathematics, Princeton University Press,Princeton, 2008.

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World Wars, Birkhäuser Verlag, Basel, 2005.


[Ful 98] W. Fulton, Intersection Theory, 2nd ed.,Springer-Verlag, New York, 1998.

[Ha 86] J. Harris, On the Severi problem, Invent.

Math. 84 (1986), 445–461.[Kl 76] S. L. Kleiman, Rigorous foundation of

Schubert’s enumerative calculus, Proc. ofSymposia in Pure Math. 28, Amer. Math. Soc.,Providence, RI, 1976, pp. 445–482.

[Kl 05] , The Picard scheme, Math. Surveys andMonographs 123, Amer. Math. Soc., 2005, pp.237–331.

[Kod 52] K. Kodaira, Arithmetic genera of algebraicvarieties, Proc. Nat. Acad. Sci. U.S.A. 38

(1952), 527–533.[Laz 81] R. Lazarsfeld, Excessive intersection of

divisors, Comp. Mathematica 43 (1981),281–296.

[Mum 69] D. Mumford, Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto 9 (1969),195–204.

[Nér 52] A. Néron, La théorie de la base pour lesdiviseurs sur les variétés algébriques, Deux-

ième Colloque de Géométrie Algébrique Liège,Masson & Cie, Paris, 1952, pp. 119-126.

[Parikh 91] C. Parikh, The Unreal Life of Oscar Zariski,Academic Press, Boston, 1991.

[Rog 05] Gino Roghi, “Materiale per una Storiadell’Istituto Nazionale di Alta Matematica dal1939 al 2003”, serie La Matematica nella Soci-

età e nella Cultura, Bollettino U.M.I., Bologna,Serie VIII, Vol. VIII-A, Dicembre 2005/2.

[Roth 63] L. Roth, Francesco Severi, J. London Math.

Soc. 38 (1963), 282–307.[Sch 07] N. Schappacher, A historical sketch of B.

L. van der Waerden’s work in algebraicgeometry: 1926–1946, Jeremy J. Gray andKaren Hunger Parshall, eds., in Episodes in

the History of Modern Algebra (1800–1950),American Math. Soc., Providence, 2007, pp.245–283.

[Segre 62] , L’opera scientifica di Francesco Sev-eri, Rend. Mat. (5) 21 (1962), 524–584.

[Sev 05] F. Severi, Sulla differenza fra i numeridegli integrali di Picard della prima e dellaseconda specie appartenenti ad una super-ficie irregolare, Atti Acc.Torino 40 (1905),288–296.

[Sev 06] , Sulla totalità delle curve algebrichetracciate sopra una superficie algebrica,Math. Annalen 62 (1906), 194–225.

[Sev 09] , Fondamenti per la geometria sulla va-rietà algebriche, Rend. del Circolo Matematico

di Palermo 28 (1909), 33–87.[Sev 12] , Sul principio della conservazione del

numero, Rend. del Circolo Matematico di

Palermo 33 (1912), 313–327.[Sev 33] , Über die Grundlagen der algebrais-

chen Geometrie, Abhandlungen aus dem

math. Seminar der Hamsburgischen Univer-

sität 9 (1933), 335–364.[Sev 49] , La géométrie algébrique italienne: sa

rigour, ses méthodes, ses problèmes, in Col-

loque de géométrie algébrique, Liège, 1949,Misson et Cie, Paris, 1950, pp. 9–55.

[Sev 60] , Fondamenti della Geometria sulle va-rietà algebriche, Annali di Matematica, 49

(1960), 283–298.[Waer 70] B. L. van der Waerden, The theory of

equivalence systems of cycles on a vari-ety, 1971 Symposia Mathematica, INDAM,Rome, 1969/70), Academic Press, London,pp. 255–262.

[Ves 90] E. Vesentini, Il caso della Matematica, Conse-quenze culturali delle leggi razziali in Italia,Accademia Nazionale dei Lincei, Rome 84

(1990), 97–105.[Zar 71] O. Zariski, Algebraic Surfaces (second sup-

plemented ed.), Springer-Verlag, New York-Heidelberg, 1971.

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