Guido Castelnuovo and : Two Personalities, Two Letters Donald Babbitt and Judith Goodstein

he Italian school of letters reflect remarkably the enormous personal- flourished from the latter part of the ity differences between these two giants of Italian nineteenth century through the early . part of the twentieth century. Some of (1865–1952) was born and Tthe main contributors were Luigi Cre- raised in , the son of Enrico Castelnuovo, mona, Eugenio Bertini, , Cor- director of the Scuola Superiore di Commercio and rado Segre, Guido Castelnuovo, , a popular nineteenth-century author of novels and and Francesco Severi. There were, of course, other short stories. He completed his doctor’s degree at important schools of algebraic geometry in other the University of in 1886 under the direc- countries, but the Italian school stood out because tion of Giuseppe Veronese, one of the leading of its unique mathematical style, especially its algebraic geometers of that period. On the advice strong appeal to geometric intuition. Between of Veronese, Castelnuovo spent the following year 1896 and 1900 two members of this school, Guido in on a postgraduate scholarship and then Castelnuovo and Federigo Enriques, developed the spent three years as assistant to geometer Enrico classification of algebraic surfaces, one of the great D’Ovidio at the University of . In 1890, Castel- achievements of algebraic geometry.1 A few years nuovo won a concorso, or national competition, for later (1904–1908), together with Francesco Severi, a new chair of analytical and at they significantly deepened that understanding the University of Rome—an award that was subse- of surfaces. quently withdrawn by the Italian Ministry of Public 2 In this article, we present excerpts from two Instruction on the grounds that the candidate’s letters to , a distinguished al- publications did not match the subject matter cov- gebraic geometer in his own right and a distant ered by the chair, although the ministry had judged relative of : one from Severi in 1932 his work itself to be of higher quality than that of and the other from Castelnuovo in 1938. Severi’s the competition. Thus, Castelnuovo remained in letter provides his frank assessment of his own Turin for another year as D’Ovidio’s assistant, dur- and others’ contributions to algebraic geometry, ing which time he broadened his research interests including those of several of the Italian geometers to include linear systems of curves in a plane and mentioned above. Castelnuovo’s letter discusses the geometry of algebraic surfaces. He won the his collaborations with Enriques and Severi in the next concorso handily, and in 1891, at age twenty- 1904–1908 period and assesses the contributions six, he was appointed to the Rome chair, which he due solely to Severi. The tone and content of the held until his retirement in 1935. A turning point Donald Babbitt is professor of mathematics emeritus at the in Castelnuovo’s scientific life occurred early in his University of California, Los Angeles. His email address is tenure at Rome, in 1892, when Federigo Enriques, [email protected]. a gifted geometer who had earned his degree in Judith Goodstein is university archivist and faculty asso- mathematics at the Scuola Normale in , came to ciate in history at the California Institute of Technology. Rome to attend a course in higher geometry taught Her email address is [email protected]. by , the first occupant of the chair of 1 An accessible account of the classification is given in higher geometry at Bologna and founder of the Ital- [Gray99]. ian school of geometry. Enriques deemed Castel- 2 Translated by Elisa Piccio and edited by the authors. nuovo, only five and a half years his senior, much

800 NOTICES OF THE AMS VOLUME 56, NUMBER 7 more open “[A]bandoning geometric intuition—the only and welcoming means that so far has allowed us to find the way than Cremona, in this tangled territory—would mean extinguish- whose lectures ing the feeble flame that can lead us into the dark completely be- forest” [Cast28]. This may have been a criticism of fuddled him. the then-current “algebraizing of algebraic geom- The two young etry” by Emmy Noether and B. L. van der Waerden mathemati- [Sch07]. cians quickly A student of his, , offers the fol- Mathematics, . became friends lowing sketch of the great man [Parikh91]. and several years later, Castelnuovo was a somewhat distant when Castelnu- fellow, he wouldn’t be chummy with ovo married El- you, he was not that type. He was very bina Enriques, dignified, long beard. He looked like brothers-in- the Moses of Michelangelo. When he law. As they smiled, his face was transformed. But strolled the mostly he was very serious. streets of Rome, Beniamino Segre paints a rather more attractive Fonti iconografiche, Biblioteca Matematica , Department of talking mainly picture, describing Castelnuovo’s house in the Via Guido Castelnuovo, ca. 1890. about algebraic Veneto section of Rome as “modest but welcom- geometry, En- ing”, and as an academic gathering place [Segre54]. riques would update his new friend daily on his progress, while … a center where every Saturday for Castelnuovo listened attentively and offered criti- several years colleagues and students, cal comments. “It is probably not an exaggeration Italians or foreign visitors, gathered for to assert that the theory of algebraic surfaces from friendly conversation on a wide variety the Italian point of view was created during these of subjects; his influence on those pres- conversations,” Castelnuovo notes in a eulogy ent was enormous, with his display delivered at the Accademia Nazionale dei Lincei of calm wisdom, his interest in each following Enriques’ death in 1946 [Cast47]. The idea expressed, his offering of serene Castelnuovo-Enriques collaboration culminated in and objective opinions and thought- their classification of algebraic surfaces, which has ful advice, his courtly presence, his been hailed as “one of the lasting contributions to genuine modesty. These qualities, even mathematics made by the Italian geometers of a if chastely veiled by a certain reserve, century ago” [Gray99]. made him loved and appreciated by Castelnuovo eventually stopped working ac- everybody: you can be certain that he tively in algebraic geometry. After 1906 he pub- did not have any enemies or detractors. lished only two original papers relating to the Francesco Severi (1879–1961) was a man of a field, including his notable 1921 paper on Abelian very different stripe. His personality is described functions [Cast21]. Although best known outside thus in an obituary in the Journal of the London for his contributions to algebraic geometry, Mathematical Society [Roth63]. Castelnuovo explored other fields, including prob- ability, mathematical pedagogy, and the philosoph- Personal relationships with Severi, ical implications of Einstein’s theories of special however complicated in appearance, and (an interest he shared with were always reducible to two basically Enriques)—lecturing, writing, and publishing on all simple situations: either he had just these topics. Nevertheless, he continued to keenly taken offence or else he was in the follow the developments of algebraic geometry at process of giving it—and quite often home and abroad and made penetrating judgments genuinely unaware that he was doing on them throughout his life. so. Paradoxically, endowed as he was Castelnuovo was an unabashed champion of with even more wit than most of his the role of intuition in the success of the Ital- fellow Tuscans, he showed a childlike ian school. At the 1928 International Congress incapacity either for self-criticism or of Mathematicians held in Bologna, he delivered for cool judgment. Thus he meddled in one of the major addresses, an overview of the politics, whereas it would have been far work in algebraic geometry not just in his own better had he left them alone. country but in Germany, France, and the United Oscar Zariski’s biographer, Carol Parikh, States. At the end of his talk, he issued the follow- describes her subject’s relations with Severi ing warning with regard to its future development: [Parikh91].

AUGUST 2009 NOTICES OF THE AMS 801 A tall heavy man from Tuscany, he geometry, and in lectured in a way that was particularly 1923 he became disquieting to Zariski. Lacking both the rector of the uni- playfulness of Enriques…and the me- versity as well, a Civiltà delle ticulous formality of Castelnuovo, Se- position he re- veri’s dictatorial style seemed designed signed to protest to make it impossible for his students the assassination to distinguish between guesses and of the Socialist assertions, hunches and hypotheses. deputy Giacomo Matteotti by Fas- Outside of mathematics Severi was also cist thugs in June a forceful and disquieting presence. ‘I of the following love you, Zariski, but you don’t love year. me,’ he once said, a surprising state- Severi also ment from a man as vain as he seemed signed (as did his to be. His wild driving was legendary; Rome colleagues oblivious to the pleading of his passen- , Photograph of Francesco Severi previously published in Macchine , 1957 (furnished by authors). gers, he would careen through the hills Castelnuovo, above Rome. Even old age seems not and Tullio Levi- Francesco Severi, 1915, Civita) the phi- to have slowed him down behind the the year that he received losopher Bene- wheel; Zariski remembered with terror the detto Croce’s being driven through Rome by Severi, Mathematics Prize. when Severi was already eighty-one. anti-Fascist man- ifesto in 1925. Severi was born in Arezzo, the last of nine chil- Taking aim at the philosopher Giovanni Gentile’s dren, to a family with deep roots in Tuscany. His manifesto of the Fascist intellectuals published father, Cosimo Severi, a notary who also wrote and ten days earlier, Croce’s manifesto advocated ac- published poetry and hymns, committed suicide ceptance of a universal culture, not one confined when Severi was nine, leaving his widow broke and to a particular political system. Soon after, how- too proud to ask for help raising the four surviv- ever, perhaps because of his outsized ambition, ing children who were still living at home. During Severi began to ingratiate himself with Benito an impoverished childhood, Severi held down a Mussolini’s regime. Severi’s election in 1929 variety of tutoring jobs to help support the fam- to Mussolini’s new Academy of Italy as a last- ily and did not abandon the tutoring until he had minute substitution for Federigo Enriques—En- graduated with a doctorate in pure mathematics riques, now a member of the Rome faculty, had from the University of Turin in 1900. A prodigious not signed either manifesto—thrust Severi into and frenetic worker throughout his life, Severi later the limelight as the regime’s spokesman for Ital- joked with Beniamino Segre, his student, that he ian mathematics. Like Castelnuovo, Levi-Civita, had been “sentenced to a life of hard labor in a and Volterra, Enriques was Jewish, which explains penal colony” [Segre62]. his name’s deletion from the list of candidates At Turin, Severi came under the spell of the forwarded to the Academy’s president-elect. The geometer Corrado Segre and dedicated his first Fascists denied that there was any ban against mathematical work, self-published while he was Jewish members at the outset. In its fourteen years still an undergraduate, to Segre, calling him an of existence, the Fascist academy never admitted “incomparable teacher”, the one who “trained any Jews to its ranks. Situated directly across the my intellect”, taught him to appreciate “rigorous street from the venerable and anti-Fascist-leaning scientific investigations”, and stirred his “heart to Accademia dei Lincei, its rise was a highly visible the highest filial sentiments” [Sev59]. Severi would first step in the chain of events leading up to the apparently disavow these sentiments by the 1930s, formal annexation of the Lincei in 1939. as we shall see. Beniamino Segre (1903–1977) entered Severi’s He spent several years as an assistant, first in life in 1927, two years after Mussolini had turned Turin with D’Ovidio, then in Bologna with Enriques, Italy into a dictatorship. A native of Turin who and finally in Pisa with the geometer Eugenio Ber- trained as a geometer there (he counted Guido tini, before moving in 1904 to Parma following his Fubini, Gino Fano, and his distant cousin Corrado appointment there as professor of projective and among his teachers), Segre moved briskly through descriptive geometry. A year later, Severi transferred the academic ranks after receiving a doctor’s de- to Padua, where he joined the Socialist Party; there he gree in mathematics in 1923 with a dissertation in remained until the call from the faculty of mathemati- algebraic geometry. After holding several positions cal sciences at Rome came in 1922. At Rome, Severi in Turin (assistant to the chair of rational mechan- taught a variety of courses, from to higher ics; assistant to the chair of analytical, projective,

802 NOTICES OF THE AMS VOLUME 56, NUMBER 7 and descriptive My dear Segre, geometry; and associate of …[T]he general outline [of B. Segre’s analytical math- draft–ed.] is mediocre in several places, ematics at the especially where you talk about alge- Military Acad- braic geometry. emy of Artillery and Engineers), and studying It lacks perspective so that a reader with Élie Cartan who doesn’t know much will not be able and Émile Picard to understand the hierarchy of ideas in Paris on a and names. Rockefeller fel- lowship, in 1927 1) The work of C. Segre has been over- the twenty-four- rated…Segre, for example, did not year-old Segre prove anything major in the field of accepted Severi’s geometry of curves although he did invitation to be- carry out a very significant revision of come his assis- the subject. His contributions to higher tant in Rome. By dimensional projective geometry are then, Segre had overrated when compared, for example, also obtained the with those of Veronese. This exagger- libera docenza, a ated evaluation is probably explained license to teach by your love of him as a disciple… at the university Beniamino Segre in Venice, level, in analytic 2) The work by Veronese is underrated. 1932. Photograph courtesy of Sergio Segre. and projective In Italy he was the true creator of higher geometry. Physically, the two mathematicians—Severi dimensional projective geometry. towered over Segre—could not have been more dis- similar. After meeting both men on a visit to Rome 3) The work of Castelnuovo has been in 1928, W. E. Tisdale of the Rockefeller Foundation overrated as has been that of Enriques, described Severi in his interview log as “a huge, especially when compared to that of bearded man, decidedly teutonic in general ap- [Max] Noether, whose name you have pearance” and Segre as “a nice appearing young completely neglected in your discus- fellow” who spoke decent French and seemed to sion of surfaces. rank high in Severi’s estimation [RFA28]. The self- assured, flamboyant Severi showered attention on 4) My work has been underrated, which his able new assistant, delighted in calling Segre seems odd to me since you were my “my favorite” (a play on the double meaning of student, and, in addition, your affection Segre’s first name in Italian), and cultivated his for your first teacher [Segre] has caused interest in algebraic geometry, the field in which you to overestimate his work. Severi had done his most significant work. In 1931, after four years as Severi’s assistant, Segre won the nationwide concorso for the chair of higher geom- In elaborating on this third point, Severi lists etry at Bologna. By then, he had become Severi’s many of the important things that were known star pupil, his sounding board, the protégé who in algebraic geometry before Castelnuovo and occasionally had to endure his maestro’s harsh Enriques did their work on the classification of editorial judgments (on the occasion described surfaces, some of which were essential to their below, certainly), but he distanced himself from classification. These include the notions of the geo- Severi’s pro-Fascist politics. metric and arithmetic genus of a surface (Cayley, Zeuthen, M. Noether), the Zeuthen-C. Segre invari- ant, the Brill-Noether Theorem, and the work of The Letters Picard on surfaces. He thus suggests that the clas- When Beniamino Segre was appointed to the chair sification of Castelnuovo and Enriques was built in geometry at Bologna in 1931, he was required on the shoulders of giants and that Segre did not to give an inaugural lecture, which he entitled “Ital- appreciate the importance of this fact. Severi ends ian Geometry from Cremona to the Present Day”. this part of his discourse by saying: “And in this He evidently sent a draft to Severi asking for his article you write that before Castelnuovo-Enriques comments. The letter in question, dated January 2, there were only ‘a few developments that had only 1932, is Severi’s response. created difficulties’. Poor Noether!”

AUGUST 2009 NOTICES OF THE AMS 803 On his fourth point, he has this to say: twice indicates that Segre took Severi’s comments to heart at least as far as Noether is concerned. It Beginning in 1904 I developed new is easy to conjecture that Segre gave Severi more ideas that untied the Gordian knots attention in his lecture and in this paper than in that had been bound so tightly up to the draft that provoked Severi’s hectoring letter. that time. I myself untied most of them, And here is the excerpt from Castelnuovo’s such as the characterization of irregu- letter to Segre in 1938. It contains comments, per- lar surfaces from both the transcenden- haps solicited by Segre, on a recent paper by Segre tal and algebraic point of view.…Even in the Annali di Matematica Pura ed Applicata setting aside my work on conceptual [Segre38]. The excerpt deals with some histori- clarifications as well as my work on the cal commentary of Segre’s in the preface on the hyperelliptic surfaces with Enriques, work of Castelnuovo, Enriques, and Severi in the which I am willing to do, this does not 1904–1908 period: justify the humiliating description of my status as “arrived” that you have One last comment regarding the his- written on page 12, thus putting my torical issues.…The notion of a con- work and that of Castelnuovo-Segre’s tinuous system [now called an algebraic [Severi may have meant Enriques here– system–ed.] of curves on some special ed.] on two completely different levels. surfaces already appears in some works You need to weigh your words! of Enriques and mine that precede the work of Severi.…In some special cases Also forgotten by you were my Theo- I suggested the definition of the char- rem of the Base [Sev06] and my work acteristic series of a continuous system on the geometry of varieties of higher to Severi. But since this suggestion had dimension. How could you have done been given in an unpublished letter, this when you discuss Italian geometry? and subsequently Severi brilliantly In addition, there is no mention of the developed the idea mentioned in it, it fact that I am the only one among liv- is not useful to make a claim of prior- ing Italian algebraic geometers who has ity here. I only mention this matter to created a school. show you how much caution is needed when you assign scientific priorities in You have also underestimated my con- periods in which the research was often tributions to enumerative geometry. If done in collaboration, or was suggested only you could understand them. All by elders to their more youthful col- of this is not to reproach you, because leagues. It was the good fortune of the you certainly have done your best. Al- Italian school of algebraic geometry to though your mathematical knowledge have this disinterested collaboration is wide, you currently do not have a between 1890 and 1910. But this makes deep enough understanding in the vast it necessary to smooth out certain field of algebraic geometry to allow you overly clean divisions between the work to have a reliable perspective on the of one and the other. subject. But I am also surprised that the comparative evaluations that we What is undoubtedly due to Severi in discussed many times in the past did the period 1904–08 are the following: not have an effect on you even though the theorem that the existence of Picard I was always very conscientious about integrals of the 1st and 2nd kind on being objective. an depends on the irregularity of the surface (1904), a We do not have the original draft that Segre sent theorem that was successively stated to Severi, so we do not know how Severi’s criticisms precisely by both of us; the theory of affected the content of Segre’s inaugural lecture. the algebraic equivalence of curves There is, however, the paper that resulted from the on a surface; and the Theorem of the lecture, which appeared in the Annali di Matema- Base [this has evolved into the Néron- tica in 1932 [Segre33]. In it, one notices that Cre- Severi Theorem–ed.]. That is more than mona is mentioned sixteen times (including twice enough to show his great worth. in footnotes); is mentioned twice; Corrado Segre is mentioned six times, as is Guido The first contribution to which Castelnuovo Castelnuovo (and one footnote); Enriques is men- refers is contained in a beautiful and fundamental tioned a total of seven times (four in footnotes). theorem due collectively to Pierre Humbert, Picard, Severi is mentioned nine times, including four Enriques, Castelnuovo, and Severi concerning the footnotes. The fact that Noether is now mentioned dimension of the space of Picard integrals of the

804 NOTICES OF THE AMS VOLUME 56, NUMBER 7 first and second kind on an algebraic surface F over 1932 letter.) Subsequently, in 1952, Néron [Nér52] the complex numbers C . Its statement requires refined and extended Severi’s Theorem of the Base the notion of the irregularity q of an algebraic and gave a modern (rigorous) proof of it—that is, surface F which is defined to be the nonnegative one acceptable to the Franco-American school of integer pg − pa , where pg is the usual geometric algebraic geometry. genus of F and pa = the arithmetic genus of F (a notion somewhat more difficult to define, which Epilogue we will not do here). The theorem states if F is an The Fascist racial laws enacted in the summer algebraic surface defined over C of irregularity q , of 1938 barred Jewish students from attending then the vector space I1 of Picard integrals of the public schools and universities; Jewish authors first kind has dimension q and the vector space I2 from publishing works under their own names; of Picard integrals of the second kind has dimen- and scores of Jewish academics, including some sion 2q . Picard integrals of the first kind are ones of Italy’s best and brightest mathematicians, whose integrands are closed and regular 1-forms, from teaching. Vito Volterra, the dean of Italian and Picard integrals of the second kind are those mathematics, had already forfeited his position whose integrands are closed 1-forms with only at the University of Rome in 1931, by refusing to polar, as opposed to logarithmic, singularities on sign the Fascist loyalty oath. Guido Castelnuovo F . It is interesting to note that F must be irregular had retired from teaching at Rome in 1935 at the in order to have nontrivial Picard integrals of the age of seventy, capping a career spanning nearly first or second kind. forty-five years in the classroom. But their younger All the mathematicians mentioned above Jewish university colleagues, including Tullio Levi- contributed to the proof of the theorem Civita and Federigo Enriques in Rome, Beppo Levi [Roth63], [Zar34,Ch.6]. The final steps were and Beniamino Segre in Bologna, and Guido Fubini, furnished in 1905 by Severi, who showed that Gino Fano, and Alessandro Terracini in Turin, felt dim(I1 ) ≤ q , dim(I2 ) ≤ 2q , and by Castelnuovo, the full brunt of the racial legislation. Levi and who showed that q ≤ dim(I1 ). Castelnuovo’s Terracini found new jobs in Argentina; Fano emi- proof depended on a technical “Theorem” due to grated to Lausanne, Switzerland; and Fubini went Enriques [Enrq04], based on a suggestion of Severi to Princeton. (in 1904), whose algebro-geometric proof was Beniamino Segre woke up that September to shown later by Severi [Sev21] to be fundamentally find that he had been dismissed from his position flawed.3 Fortunately, Henri Poincaré gave a valid as director of Bologna’s mathematical institute, transcendental proof of the Enriques-Severi asser- relieved of his duties as an editor of Italy’s oldest tion in 1910, so that at least from this date on, the scientific journal, the Annali di Matematica Pura theorem was legitimate. ed Applicata, expelled from numerous scientific The second and third contributions discussed academies and organizations, including the Ital- by Castelnuovo are closely related. Again, they ian Mathematical Union (UMI), of which he had concern an algebraic surface F over C . We need been a founding member, and denied any form to introduce some additional notions. The divi- of compensation. Deeply offended by the anti- sor group of F , Div(F ), is the free Abelian group Semitic legislation, Segre immediately renounced over the integers Z generated by the irreducible his membership in the Fascist Party, reportedly (algebraic) curves on F . Its elements are referred telling Bologna’s rector, Alessandro Ghigi, “Since to as the divisors on F . Following [Mum66], we say his Excellency the Head of the Government has that two curves C1 and C2 on F are algebraically declared that a Jew is not an Italian, I took it as equivalent if they are parametrized by a connected a given that I could no longer wear the fascist algebraic variety S . We denote by Ga (F ) the sub- insignia as it might have been interpreted as an group of Div(F ) generated by divisors of the form insincere gesture” [Finzi94]. Having tried and failed − C0 C1 , where C0 and C1 on F are algebraically to find a position in the United States, Segre took equivalent curves. The Néron-Severi group is the refuge in England in the spring of 1939 with his quotient group Div(F )/Ga (F ). The Theorem of wife and their three children. In September, when the Base in Castelnuovo’s discussion says that the Britain declared war on Germany, he was interned Néron-Severi group is finitely generated. The paper for several months on the Isle of Man as an enemy where the proof appeared (Math. Ann. [Sev06]) was alien, while his wife and children stayed with the solicited by its editor, Max Noether. (This is per- mathematician Leonard Roth and his wife in Lon- haps why he is mentioned so solicitously in Severi’s don. The youngest daughter fell ill with measles, which turned into blood poisoning during an air 3 Controversy over the proof of this “Theorem”, especially between Enriques and Severi, raged until a valid algebro- raid attack over the city. The hospitals overflowed geometric proof was obtained by Grothendieck in 1960. with emergencies, making it impossible to get Grothendieck’s proof utilized in a nontrivial way non- the little girl admitted in time, and she died early reduced schemes and other powerful machinery due to the next morning. In 1942 the family moved to Cartier and Kodaira and Spencer [Mum66]. Manchester, where Segre taught for several years

AUGUST 2009 NOTICES OF THE AMS 805 before returning to the in National Institute for Higher Mathematics (INDAM), 1946. Four years later, Segre succeeded Severi as also based at “La Sapienza” near the Rome Termini professor of geometry at the University of Rome. train station. Shortly after the liberation, a High In the wake of the 1938 racial laws, Jewish Commission for Sanctions against Fascism was elementary and secondary schools sprang up in established. The commission’s specific charge was Rome and other major Italian cities with the per- to look into allegations of wartime collaboration mission of the authorities, who had banned any against party members who had been accused university-level coursework. In December 1941 of taking an active part in Fascist political life or , 1939 (furnished by authors). Guido Castelnuovo organized a clandestine uni- who had remained loyal to Mussolini after he was versity, recruited a host of professors, including deposed in September 1943. The commissioners himself, and arranged for the students to register initially suspended Severi from university teach- (in absentia) at the privately run Istituto Politecnico ing. He appealed the ruling and received sanzioni di Friburgo, in Switzerland. The ad-hoc university, minori, simple censures involving nothing more as well as the Jewish schools, ceased operations L‘Illustrazione Italiana than a letter placed in his university personnel file. Photograph of Mussolini and Severi previously published in when the Germans occu- When Severi got in touch with Segre again after the pied Rome in September war, he enclosed with his letter a “To whom it may 1943. At the end of the concern” document issued by the Italian Ministry war the students enrolled of Public Instruction stating that the ministry had at the University of Rome, not subjected him “to any sanction provided for by and their transcripts from current legislation on the cleansing of the Italian the Fribourg Polytechnic Civil Service” [MPI46] for Fascist activities. Severi were submitted as evi- also received a clean bill of health from another dence of their advanced commission assigned to examine the behavior of standing. During the occu- former members of the Academy of Italy, which pation, Castelnuovo and concluded that he “had not received from Fascism his wife were sheltered anything more than was his due as a distinguished briefly by Tullio Viola, a scientist.” His “moral rectitude”, the commission young mathematician at added, was never called into question [Sev45]. Rome. The couple later Severi also defended his behavior during the war, took refuge in a religious pointing out to Segre in 1945 that he had worked institute, and when that diligently to save the assets of the local bank in arrangement became too Arezzo, his hometown. However, a committee dangerous, they lived for (including Castelnuovo) that was given the task many months in a small of rebuilding the Lincei, refused to reelect Severi, pensione off the Via and he regained membership only in 1948, after Veneto, using an assumed Mussolini (in front) at the University the government declared a general nationwide name, Cafiero [NM04]. City (Città Universitaria) in Rome to amnesty. Severi, who had lost his position as presi- visit the new building of the Royal After the liberation of dent of INDAM, also recovered that post following Institute of Higher Mathematics. Rome in June 1944, Castel- the amnesty and held it until his death. Severi is next to him on the right. nuovo, then seventy-nine, came out of retirement to Appendix reconstitute Italy’s pre-Fascist scientific organiza- As is well known in the algebraic geometry com- tions. Formally reinstated as professor emeritus munity, there was increasing skepticism among at Rome, he served as general commissioner of algebraic geometers outside the Italian school Italy’s National Research Council and president of about the way the Italians did algebraic geometry. its mathematics committee and played a leading In particular, there was concern about the math- role in reviving both organizations. He contributed ematical precision of some of the key definitions in a major way to the rebirth of the Lincei, whose and the logical rigor of the proof of some of the president he became in December 1946, a post important theorems. This was especially the case he held until his death in 1952. Castelnuovo was among members of the Franco-American school of named a Senator for Life in Italy’s parliament in algebraic geometry, beginning with Castelnuovo’s 1949, and soon after his death the building that student Oscar Zariski in the mid-1930s, soon after houses the University of Rome’s (“La Sapienza”) he had written his well-known book on algebraic Institute of Mathematics was named in his honor. surfaces [Zar34]. He was later joined, most promi- From late 1938 until the liberation of Rome, nently, by André Weil, Claude Chevalley, and Pierre Francesco Severi, forever a loyal Fascist, was un- Samuel. The Italian school was very sensitive to deniably Italy’s most prominent mathematician, these criticisms. Francesco Severi, especially, tried especially now that any possible competitors were unsuccessfully to address them. (See, for example, either in hiding or had emigrated. During most [Sev49] and Chevalley’s review in Mathematical of this time, he was the president of the Italian Reviews [Chev52].) There was also a well-known

806 NOTICES OF THE AMS VOLUME 56, NUMBER 7 confrontation between Severi and Weil at the century, when people like Gauss, Abel, 1954 International Congress of Mathematicians Jacobi, Cauchy, and many others rose, at Amsterdam over the rigor of Severi’s theory we certainly worry about the future of of the intersection of subvarieties on a projective our science. variety and rational equivalence. (See, for example, [VdW70].) Posterity has shown that Weil won the One day, sooner or later, the love for argument. the great theories will be born again Below is a translated4 excerpt from the preface and on that day people will read the by Guido Castelnuovo to the posthumous (1949) treatise by Enriques as a report wherein magnum opus of Federigo Enriques on algebraic many gems have been unearthed and surfaces [Enrq49] in which Castelnuovo defends many others wait to be discovered. the more intuitive approach to mathematics of an earlier era—which certainly included Italian Acknowledgments algebraic geometry in its heyday—as opposed to We thank G. Fabre, L. Giacardi, D. Goodstein, the new stress on formal rigor espoused by the S. Lippincott, C. S. Roero, and S. Segre for useful Franco-American school. suggestions in the course of preparing this manu- Will someone come along soon who script for publication. will continue the work of the Italian and French schools [here Castelnuovo References presumably is referring to the French [Brig84] A. Brigaglia, Sur les relations des mathemati- school prior to the dominance of the cians français et italiens au début du XXe siècle, Ca- Franco-American school–ed.]5 who will hiers du séminaire des mathématiques 5 (1984), 21–48. [Cast21] G. Castelnuovo, Sulle funzioni abeliane, Rend. succeed in developing the theory of R. Acc. Lincei (5) 30 (1921), 50–55; 99–103; 195–200; algebraic surfaces that has already been 355–359. accomplished for the theory of alge- [Cast28] ——— , La geometria algebrica e la scuola ita- braic curves? I hope so, but I doubt it… liana, Conferenza tenuta al Congresso Internazionale mathematics has now taken a different dei Matematici di Bologna, Atti, 1 (1928), 191–201. course from that of the past [i.e., the [Cast47] ——— , Commemorazione del socio Federigo nineteenth century]. Fantasy and intu- Enriques, Rend. Acc. Naz. Lincei (8) 2 (1947), 3–21. C. Chevalley ition characterized research then, but [Chev52] , Review of “La géométrie al- now these are treated with suspicion, gébrique italienne. Sa rigeur, ses méthodes, ses problèmes” in Mathematical Reviews, MR0038094 as there is the fear that they could lead (12,353f). to errors. Theories were developed by [Enrq04] F. Enriques, Sulla proprietà caratteristica delle mathematicians to make more precise superficie algebriche irregolari, Rend. Acc. Sci. Bologna many ideas that were already vaguely in 9 (1904–05), 5–13. their mind. It was the exploration of a [Enrq49] ——— , Le Superficie Algebriche, Zanichelli, vast territory seen from a distant shore. Bologna, 1949. In this way such jewels of mathematics [Finzi94] R. Finzi, Leggi razziali e politica accademica: il as the theory of analytic functions, el- caso di Bologna, in Cultura ebraica e cultura scientifica liptic functions, and Abelian functions in Italia (Antonio Di Meo, ed.), Editori Riuniti, 1994. [Gray99] J. Gray, The classification of algebraic surfaces were created during this past century. by Castelnuovo and Enriques, Mathematical Intel- Nowadays there is more interest in the ligencer 21 (1999), 59–66. road that leads to a field of exploration [MPI46] F. Severi to B. Segre, January 10, 1946, with enclo- rather than to the field itself. And this sure, The Beniamino Segre Papers, Private Collection. tendency will not be short-lived, as we [Mum66] D. Mumford, Lectures on Curves on an Alge- can also see in other fields such as in braic Surface, Princeton University Press, 1966. music and in the arts, where fantasy [Nér52] A. Néron, La théorie de la base pour les diviseurs is banned and where the technique or sur les variétés algébrique, Colloque de Géométrie Algébrique, Liège, 1952. the way of expression is more interest- [NM04] R. Natalini and M. Mattaliano, Colloquio con ing than the work itself. It would be : la fantasia e la memoria, Lettera an exaggeration to extend these pessi- matematica PRISTEM 52 (2004), 4–7. mistic judgments to the evolution that [Parikh91] C. Parikh, The Unreal Life of Oscar Zariski, mathematics is undergoing nowadays, Academic Press, 1991. but if we compare these fifty years to [RFA28] Rockefeller Foundation Archives, Record Group the corresponding years of the last 12.1 Officer’s Diaries, box.139, folder 3. [Roth63] L. Roth, Francesco Severi, J. London Math. Soc. 4 Translated by the authors. 38 (1963), 282–307. N. Schappacher 5For an interesting discussion of the relations between the [Sch07] , A historical sketch of B. Italian and French schools in the early twentieth century, L. van der Waerden’s work in algebraic geome- see [Brig84]. try: 1926–1946. Episodes in the History of Modern

AUGUST 2009 NOTICES OF THE AMS 807 (1800–1950) (Jeremy J. Gray and Karen Par- About the Cover shall, eds.), History of Mathematics, 32, Amer. Math. The complexity of factorization Soc., 245–283, Providence, RI, 2007. [Segre33] B. Segre, La geometria in Italia, dal Cremona ai This month’s cover was produced by a mild giorni nostri, Ann. di Mat. (4) 11 (1933), 1–16. variant of Eratosthenes’ sieve.The fascination [Segre38] ——— , Un teorema fondamentale della geo- with the mixture of determinism and random- metria sulle superficie algebriche ed il principio di ness associated with factorization has lasted spezzamento, Ann. di Mat. Pura Appl. (4) 17 (1938), over 2,000 years. As the WHAT IS … article by 107–126. Friedlander and Iwaniec (page 817) reminds [Segre54] ——— , Onoranze alla memoria di Guido Castel- us, optimism still reigns in this field. nuovo, Univ. Roma. Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5) 13 (1954), 19–41. —Bill Casselman, Graphics Editor [Segre62] ——— , Francesco Severi (13 aprile 1879–8 ([email protected]) dicembre 1961), Ann. di Mat. Pura Appl. (4) 61 (1963), i-xxxvi. [Sev06] F. Severi, Sulla totalità delle curve algebriche tracciate sopra una superficie algebrica, Math. Ann. 62 (1906). a [Sev21] ——— , Sulla teoria degl’integrali semplici di 1 specie appartenenti ad una superficie algebrica (7 Notes), Rend. R. Acc. Lincei. (5) 30 (1921), 163–167, 204–208, 231–235, 276–280, 296–301, 328–332, 365–367. [Sev45] F. Severi to B. Segre, October 15, 1945, The Be- niamino Segre Papers, Private Collection. [Sev49] F. Severi, La géométrie algébrique italienne. Sa rigeur, ses méthodes, ses problèmes, Colloque de Géométrie Algébrique, Liège, 1949, Masson et Cie, Paris, 1950. [Sev59] ——— , Dalla scienza alla fede, Edizioni Pro Civi- tate Christiana, 1959. [VdW70] B. L. Van der Waerden, The theory of equiva- lence systems of cycles on a variety, 1971 Symposia Mathematics, Vol. V (INDAM, Rome, 1969/70), pp. 255–262,Academic Press, London. [Zar34] O. Zariski, Algebraic Surfaces (second supple- AMERICAN MATHEMATICAL SOCIETY mented edition), Springer-Verlag, 1971, New York– Heidelberg.

Lessons in TEXTBOOK Geometry I. Plane Geometry Jacques Hadamard This internationally renowned geometry text goes well beyond the basics of plane Euclidean geometry, in a way that is both pleas- ingly classic and surprisingly modern. This book estab- lished the tradition of teaching geometry throughout the world. Much of its value lies in more than 450 problems, whose solutions open worlds to the engaged reader. 2008; 330 pages; Hardcover; ISBN: 978-0-8218-4367-3; List US$59; AMS members US$47; Order code MBK/57

1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email: [email protected]. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA

For many more publications of interest, visit the AMS Bookstore

AMS BOOKSTORE www.ams.org/bookstore

808 NOTICES OF THE AMS VOLUME 56, NUMBER 7