Guido Castelnuovo and Francesco Severi: Two Personalities, Two Letters Donald Babbitt and Judith Goodstein he Italian school of algebraic geometry 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 mathematics. part of the twentieth century. Some of Guido Castelnuovo (1865–1952) was born and Tthe main contributors were Luigi Cre- raised in Venice, the son of Enrico Castelnuovo, mona, Eugenio Bertini, Giuseppe Veronese, Cor- director of the Scuola Superiore di Commercio and rado Segre, Guido Castelnuovo, Federigo Enriques, 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 Padua 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 Rome 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 Turin. 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 projective geometry 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 quently withdrawn by the Italian Ministry of Public 2 In this article, we present excerpts from two Instruction on the grounds that the candidate’s letters to Beniamino Segre, a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 Corrado Segre: 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 Pisa, 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 Luigi Cremona, 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, Oscar Zariski, offers the fol- Mathematics, University of Turin. 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 Giuseppe Peano, 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 Italy 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 general relativity (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 , 1957 (furnished by authors). to be. His wild driving was legendary; Rome colleagues oblivious to the pleading of his passen- Vito Volterra, Photograph of Francesco Severi previously published in Macchine 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 Accademia dei Lincei 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.