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1 The Italian Style in Algebraic Geometry

Since the beginning of the twentieth century, historiography has underlined the existence of a School of Algebraic Geometry led by Corrado Segre, a School distinguished by a precise identity, with “its own impress, an Italian one” (Fano 1924–25, 220).Footnote 1 Built up over a time span that coincided with the period of most intense scientific activity of its leader, this identity consists both in a tendecy for the development of particular areas of research: projective hyperspace geometry, geometry on an algebraic curve, geometry of the surfaces, etc., and in the adoption of distinctive methods (synthetic-hyperspace, algebraic, differential-projective), languages and ways of thinking.Footnote 2

The existence and characteristics of the Italian School of Algebraic Geometry were specified by the members of the team and by their Italian colleagues, with singular stability in time and in various circumstances: in obituaries and commemorations of Segre, a few years after his death,Footnote 3 in lectures held in Italy and abroad,Footnote 4 in prefaces to essays and volumes,Footnote 5 in inaugural addresses and courses on Higher Geometry,Footnote 6 etc. This identity was soon recognized by foreigners. Julian L. Coolidge for instance affirmed:

There is a pronounced rise and fall in the tide of mathematical interest in different countries and at different times. […] As for the modern birational geometry, that is almost a monopoly of Italian mathematicians and few others like Macaulay and Snyder, who have been strongly influenced by Italian thought (Coolidge 1927, 352).

Henry F. Baker, in the same line of thought, noticedFootnote 7:

He [Segre] may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraic loci (Baker 1926, 269).

In the first phase of rereading of their activity, in a historical-mathematical retrospective, Italian algebraic geometers underlined not only and not so much the national landmark of their tradition of studies, but above all the international cultural roots and the contributions from exchanges with foreign colleagues.Footnote 8

Recognition of a cosmopolitan dimension in geometry research was weakened, however—and eventually disappeared—during the fascist period, when some members of the School of Segre, with clear ideological intents, stressed the Italianness of their style of researches, to the point of distorting the evolution and identity of the School to which they belonged to make it an emblem of the ‘Latin-Aryan genius’.Footnote 9

Apart from this latter drift, there appears to be no doubt of the fact that Segre made a decisive contribution to the construction of an international identity for the Italian School of Algebraic Geometry, to its definition au fil du temps and to its affirmation at a world level. Some disciples of his rather emphasised that Segre’s scientific activity had been affected—especially in quantitative terms—by the energies that he had devoted to this purpose (Castelnuovo 1924b, 369). In this connection, in the initial phase of his scientific production, between 1882 and 1884, he had written no fewer than 16 articles, and had entertained a wide network of correspondences with mathematicians from various countries like A. Voss, F. Klein, L. Kronecker, O. Schlömilch, G. Darboux, C.F. Geiser, T.A. Hirst, K. Weierstrass, A. Cayley, T. ReyeFootnote 10 and J. RosanesFootnote 11. However after 1887 there followed a less intense period of work. This second period could be properly defined as that of ‘Segre Maestro’ because of the fact that, though continuing to cultivate his personal studies and interests, he devoted a large part of his time and activity to pushing his direct and distance disciples to produce original researches,Footnote 12 which in a sense he ended up considering “as his own.”Footnote 13

In this outlook it thus becomes important to stress that phase in Segre’s life which was entirely unwinded in the Short Century, a phase traditionally somewhat overshadowed by the golden period (1883–1888),Footnote 14 in order to reconstruct the strategies worked out by him to stabilize the identity of his School and ‘give colour and solidity’ to it. A histoire par traces according to Bloch springs, which, investigating some less well-known aspects of Segre’s scientific biography like the epistolary relationships and conversations with foreign colleagues, the publishing activity, the attendance at international congresses of mathematicians, the intentions and actions in the period of the Great War and in the early Twenties, shows that contributions were made to the construction of this identity not only by strictly technical aspects (mathematical ideas, contents and methods), but also by a set of paradigms and socio-cultural behaviours. It was also the sharing of certain civic values and particular stances, proper to many intellectuals of the first generation after the Risorgimento, that helped to make the team of Italian algebraic geometers a real School, internationally recognized, endowed with an appearance and cohesion of intents that went beyond the existence of a common research project, and which remained strong, at least until the divergent political and cultural choices by some members, such as F. Severi, E. Bompiani and F. Conforto, during the fascist dictatorship.Footnote 15

2 Circulation of Knowledge Between Italy and Europe

The first tiele in the construction of an international identity for this School was represented by the efficient network of circulation of texts and readings, put in place by Segre through his masterly teaching, documented by his handwritten Notebooks (Giacardi 2001a, 2013), and through the correspondences with some Italian and foreign colleagues.Footnote 16

In effect, if it is true that the School of Segre identified in the writings of the Italians L. Cremona and E. D’Ovidio, and in some other renowed works by E. Bertini, G. Veronese, G. Battaglini and R. De Paolis, its points de repère, it is likewise undeniable that the cultural horizons of this School were much broader and took up the legacy of a wide range of influences and references to more or less famous foreign authors.

It was Segre himself that looked to international production for a profound renewal of the development of the three fields mainly cultivated by him. Regarding projective hyperspace geometry he referred, from his first works, to the results attained by F. Klein, F.G. Frobenius and K. Weierstrass (Giacardi 2001a, 141, 143–144, 148; Zappulla 2009; Luciano and Roero 2012). In particular, during the period in which he wrote his degree dissertation (spring-summer 1883) and published his first papers (Segre 1883a, b), he ‘reinterpreted’ some recent algebraic results obtained by Weierstrass, for the purpose of giving a geometrical and analytical pattern to projective hyperspaces, as we see from his correspondence with L. Kronecker (16 November 1883, 10 December 1883, 25 December 1883, Annexes 3, 4 and 5), with Weierstrass (28 March 1884, Annex 13) and with A. Cayley (14 May 1884, Annex 14). Aware of the difference of approach in the language adopted by the Italian and German Schools, in those circumstances Segre intensely stressed the utility of geometrically ‘translating’ some elements of the theory of bilinear and quadratic forms, developed by Weierstrass and by Kronecker himself (25 December 1883, Annex 5):

Peut-être ne devrais-je pas dire «interprétation géométrique», car ces mots font penser (et vous ont fait penser à ce que je vois) à un travail qui consiste seulement dans des changements de mots […]. Mais ce n’est pas là ce que j’entendais dire dans ma dernière lettre.

Some years later he again emphasised the true coincidence between the results obtained by Italian and German geometers, in spite of the different languages used, writing to Castelnuovo:

Certain results can perhaps already be found in the analytical works (Kronecker, Weierstrass and alumni), but, because of their style of exposition, which is completely different from ours, comparisons are difficult. (ANL-Castelnuovo, Segre to Castelnuovo, 17 May 1894: “Certe cose poi può darsi si trovino già in lavori analitici (Kronecker, Weierstrass e scolari), ma per il loro modo d’esposizione, completamente diverso dal nostro, i riscontri son difficili”).

Regarding the second favourite field of studies by Segre (the geometry on curves and surfaces), it was instead above all the contacts with Max Nöther that were fundamental,Footnote 17 while for differential geometry and that of imaginary quantities Segre’s letters and conversations with G. Darboux,Footnote 18 with the American E.J. WilczynskiFootnote 19 and with the Czechoslovak E. Čech were important.Footnote 20

Besides these direct contacts with the international community, it must also be remembered that Segre, starting from the third year of his university studies, frequented Turin bookshops looking for texts by famous authors and was wont to consult in the University Library the collections of German journals like the Mathematische Annalen,Footnote 21 the Journal für die reine und angewandte Mathematik Footnote 22 and the Berlin Monatsberichte.Footnote 23

In the first years of his scientific career, there was an evident influence on Segre of the German mathematicians, as can be inferred from his Resoconti di Scritti letti Footnote 24 and from examination of his correspondences with A. Voss, L. Kronecker, O. Schlömilch, C.F. Geiser and K. Weierstrass. It was in this period that Segre, showing a great entrepreneurial spirit, turned to scholars that were already prominent, expounding his research projects, asking for suggestions on how to pursue his studies on complexes of lines and on the geometry of the straight line, and asking them for offprints, essays and books. This allowed him to contextualize his results in an international perspective and thus to enter in his own right the arena of the most authoritative scholars in the discipline.

The helpfulness shown by these foreign mathematicians towards a young researcher aroused Segre’s gratitude and induced him to ask those people that he was accustomed to define his “Maestri for a moment”,Footnote 25 for hospitality for his own articles. Hence his essays on the classification of second-order and fourth-order complexes, on the Kummer surface, on linear complexes, on binary homographies, on curves and ruled algebraic surfaces appeared in the Journal für die reine und angewandte Mathematik (Segre 1884a, c, g), directed by Kronecker, and in Mathematische Annalen (Segre 1883a, b, 1884d, e, 1886a, 1887b, c, 1889a, 1891d), coedited by Klein.Footnote 26 By contrast did not succeed Segre’s aspiration to publish a paper in the German journal Zeitschrift für Mathematik und Physik, edited by O. Schlömilch. He sent his article in French: “Sur les droites qui ont des moments donnés par rapport à des droites fixes”, only later realising that only texts in German were accepted in that periodical. It was Schlömilch himself that forwarded Segre’s paper to Klein, proposing that he publish it in Mathematische Annalen, but in the end it was printed in the Journal für die reine und angewandte Mathematik (Segre 1884c).Footnote 27

As is well known, it was above all Klein that, starting from August 1883, assumed the role of Segre’s ‘distance Maestro’ (Luciano and Roero 2012, 18–27, 81–148). Klein stimulated the young disciple to develop particular lines of research, carefully reread his first works, extended the sphere of his readings and involved him in international publishing initiatives, for instance enrolling him as a reviewer for the Jahrbuch über die Fortschritte der Mathematik. This proposal was immediately welcomed by Segre. In the two years 1883–1885 he wrote 35 reviews of works by Italian mathematicians (with a single exceptionFootnote 28), and his comments were then translated into German by E. Lampe (Togliatti 1963, XII).

Furthermore it was Klein that widened the circle of Segre’s interlocutors, putting him in contact with various European mathematicians (A. Hurwitz, F. Schur, J. Rosanes and T. Reye) that dealt with the same research themes as him,Footnote 29 and inviting Segre to vulgarize his results abroad by sending offprints to foreign colleagues. This suggestion too was attentively followed up by Segre, as is proved by his promotion of his essay on metric geometries of linear complexes and the spheres (Segre 1883–84a), offered to Darboux, Cayley, Geiser, Hirst and Mittag-Leffler.Footnote 30

The capacity of Segre’s to move into the European context of research on algebraic geometry—at first as a ‘distance disciple’ and then as an equal partner—was greatly appreciated at the national level and put him in the condition to become, in turn, a leader for the new generation of Italian geometers. For instance, one of the aspects noticed by the examination board in the competitions for the qualification for university teaching (libera docenza) and for the chairs in Higher Geometry, in which Segre participated (in Turin, Catania and Naples), was precisely the fact that “the excellent young scholar” had built up relationships of scientific collaboration with illustrious foreign mathematicians and yet had an excellent reputation abroad.Footnote 31 Aware of the weight, in evaluation of his curriculum, of the flattering judgments received in the international parterre, a few years later Segre suggested to Castelnuovo that he adopt analogous strategies for himself:

By the way sending some printed copies of Nöther’s letter to the examinatory board would always be a good precaution! (ANL-Castelnuovo, Segre to Castelnuovo, 8 August 1891: “Ad ogni modo l’invio al concorso di alcune copie stampate della lettera di Nöther sarebbe sempre una buona precauzione!”).

Moreover, Castelnuovo, who had been Segre’s assistant in Turin, and then became his friend and confidant, was a witness to the frequent requests for volumes, offprintsFootnote 32 and lithographed polycopies of coursesFootnote 33 that Segre made to his outlander colleagues. Hence, immediately after his death, as one of the principal merits of the Piedmont algebraic geometer he stressed his having enriched “with vital nourishment the culture of the Italian geometric School” through his activity of direction of Italian research and thanks to the international opening of his teaching (Castelnuovo 1924b, 354).

As a matter of fact, the manuscript Notebooks of Segre’s courses in Higher Geometry (1888–1924) gradually grew fuller and fuller of references to foreign scientific literature. For example, as regards his own use of the texts by M. Nöther in his university lectures, Segre wrote to Castelnuovo:

Have you seen the Bericht by Brill and Noether on algebraic functions? I am enthusiastic about it; although, only having received it 2 days ago, I have not been able to examine it thoroughly yet; and although of the Italian style there is only the name (for which the A[Authors] apologise in the preface, and Nöther also apologised to me in the short letter that he sent with the volume). Apart from this lacuna and the one relating to the arithmetic approach, in this treatise there is a historical exposition and critique, minute, meticolous, detailed, of all the theory, inclusive of all the methods and all the connections. If I am not to be disanchanted during the reading (which I do not believe), here there is the true story, which I had been yearning for so long, of this field. If it had come out a few months before it would have saved me – with the chapter on singularities – a large part of the work that I have done to structure my course for this year! (ANL-Castelnuovo, Segre to Castelnuovo, 28 November 1894: “Hai visto il Bericht di Brill e Noether sulle funzioni algebriche? Io ne sono entusiasta; quantunque, avendolo ricevuto da soli due giorni, non abbia ancor potuto esaminarlo a fondo; e quantunque dell’indirizzo italiano non vi sia che il nome (di che gli A[Autori] si scusano nella prefazione, ed il Nöther si scusò pure con me nelle poche righe con cui accompagnò l’invio del volume). Tolta questa lacuna e quella relativa all’indirizzo aritmetico, vi è in quest’opera un’esposizione storica e critica, minuta, coscienziosa, particolareggiata, di tutta quanta la teoria, in tutti i metodi, con tutte le connessioni. Se non incontrerò delusioni nella lettura (il che non credo) vi è qui la vera storia, quale io vagheggiavo da tanto tempo, di questo campo. Se usciva qualche mese prima mi risparmiava – col capitolo sui punti singolari – una gran parte del lavoro che ho fatto per architettare il mio corso di quest’anno!”).

Similarly while preparing the 1897–98 course entitled Lezioni sui Gruppi continui di trasformazioni (Segre’s Notebook 11 in Giacardi 2013) Segre confided the difficulties that he met in studying the work by S. LieFootnote 34:

I only deal with studying groups, and drawing up a program for the course. […] As for the method, I sometimes find it difficult to understand Lie’s reasonings and calculus; and I would like to make the treatment clearer. There are also some beautiful theorems in the last chapters of both the 1st and 3rd vols. of Lie’s work. (ANL-Castelnuovo, Segre to Castelnuovo, 22 October 1897: “Non mi occupo d’altro che di studiare i gruppi, e di farmi un programma del corso. […] Quanto al metodo, trovo difficoltà qualche volta a veder chiaro nei ragionamenti e calcoli di Lie; e vorrei rendere la trattazione più luminosa. Vi sono dei bei teoremi anche negli ultimi capitoli sia del 1° che del 3° vol. dell’opera di Lie”).

Indeed, in 1899 reflecting on the use of the papers by David Hilbert for his lectures on Enumerative Geometry, Segre again addressed Castelnuovo for advice:

Tell me if you have ever studied the memoir by Hilbert M[athematische] A[nnalen] 36 “Ü[ber] die Theorie d[er] alg[ebraische] Formen” in which there is a general formula for postulation of an M k of S r for forms of suitable high order. I would like to expound it this year in my course: but I’m held back by the complicatedness of that part in which it is proved that the number of certain relationships is finite (and at most = r + 1): 3rd theorem of that Memoir. Do you have any suggestion to make me known on this subject? (ANL-Castelnuovo, Segre to Castelnuovo, 4 October 1899: “Dimmi se hai mai studiato la memoria di Hilbert M.A. 36 “U. die Theorie d. alg. Formen” in cui si trova una formola generale di postulazione di una M k di S r per forme di un ordine abbastanza alto. Io vorrei esporla quest’anno nel mio corso: ma mi trattiene la complicazione di quella parte in cui si dimostra che è finito (e al più = r + 1) il numero di certe relazioni: teorema 3° di quella Memoria. Hai tu qualche suggerimento da darmi in proposito?”).

Equally numerous were the requests for purchases and exchanges of foreign offprints and volumes, personally made by Segre, and presented from 1907 to 1924 in his capacity of director of the Mathematical Library in Turin.Footnote 35 These acquisitions highly contributed to increasing the scientific collections of the University of Turin (Giacardi and Roero 1999, 442–447).

The range of Segre’s readings, and consequently the international dimension of the exchanges of manuscripts and printed texts, was broadened above all after 1890 when, as an authentic leader, Segre devoted himself systematically to “importing to Italy ideas that had been developing elsewhere.” (Terracini 1961, 11; Fano 1924, 225). Moreover he was one of the first Italian mathematicians to be associated to the Deutsche Mathematiker Vereinigung in 1897,Footnote 36 and from its journal (JDMV) he got news on the scientific activities in universities of Germany and other countries.

As we deduce from examination of his Card Index Footnote 37 and his Library,Footnote 38 Segre was an attentive, untiring and meticulous reader, whose horizons, initially limited to the German area, with the passing of time were extended towards world production, not only in geometry, but also in other disciplines, like analysis, abstract algebra, set theory, foundational studies, mathematical physics and maths education.Footnote 39

In turn, Segre repeatedly encouraged students and colleagues, in both Turin and other places in Italy, to look to international publications to enter into flourishing lines of research and to renew their university teaching. In 1886, for instance, he suggested to Castelnuovo, at the time a student, to read the articles on n-dimensional geometry by W.F. Meyer, W.K. Clifford and H. Schubert, published in Mathematische Annalen and Acta Mathematica,Footnote 40 and he recommended that he conveniently quote them in his works. A few years later he again urged Castelnuovo, who had become a colleague of his, to examine in depht into the M. Nöther’s memoirs, and shared his intention to study F. Neumann’s Vorlesungen.Footnote 41 Similarly Segre advised Loria and Castelnuovo to refer to the international literature for their courses on Higher Geometry in Genoa and Rome, respectively.Footnote 42

Further, during all the period of his direction of the Italian School of Algebraic Geometry, Segre circulated offprints and texts by foreign geometers in ItalyFootnote 43; with his team of coworkers he commented on the most recent bibliographyFootnote 44 and even “researches not yet published”,Footnote 45 and kept him up to date the epistolary contacts of his disciples with foreign authors.Footnote 46

Finally, to Segre’s desire to afford international cultural roots for his School there are connected two publishing enterprises: the Italian translations of Geometrie der Lage by C. von Staudt and F. Klein’s Erlangen Program, commissioned by him from his students M. Pieri and G. Fano (Luciano and Roero 2012, 37–45).

3 Promotion of the Italian Style Abroad

If the construction of a national identity for the School of Algebraic Geometry could not aside from the recourse to sources and comparaisons with international models, reciprocally, for Segre, it was the task of a leader to channel towards other countries the flow of the best contributions by his disciples. This was an action of diffusion of mathematical knowledge that he took on seriously and tenaciously, both in interwining a ramified network of epistolary dialogues and in working inside examining boards for attribution of prestigious prizes like the International Bressa Prize and the Guccia Medal. It was above all this last award, attributed to Severi in the IVth International Congress of Mathematicians (Rome 1908),Footnote 47 that Segre intended to ‘exploit’ for illustrating, at international level, the relevance and value of the researches of his School. So, since 1905 he recommended to Castelnuovo:

If you or your brother-in-law should have an opportunity to write to Severi, you should incite him (as I have already done) to work for the Guccia medal. You know that Valentiner aspires to it. It would now have a flattering meaning for Italy and one corresponding to the leading position that Italy currently holds in Geometry, if an Italian was proclaimed the winner at the Rome congress! It would be enough for Severi to work on the curves existing on a given algebraic surface, profiting by his knowledge of surface geometry, to stick to the proposed theme, without straying too far away from his favourite field! (ANL-Castelnuovo, C. Segre to G. Castelnuovo, 5 November 1905: “Se tu, o tuo cognato, aveste occasione di scrivere a Severi, dovreste eccitarlo (come già feci io) a lavorare per la medaglia Guccia. Sai che vi aspira il Valentiner. Ora avrebbe un significato lusinghiero per l’Italia, e corrispondente al primato che questa ha attualmente in Geometria, il fatto che nel congresso di Roma fosse proclamato vincitore un italiano! Basterebbe che Severi lavorasse su le curve esistenti su una data superficie algebrica, approfittando della sua conoscenza della geometria sulla superficie, per essere nell’ambito del tema proposto, senza allontanarsi troppo dal suo campo preferito!”)

A short time later, returning to the matter, Segre asked his friend:

Speaking of twisted curves, after I wrote to you some time ago telling you to incite Severi to work for the Guccia prize, I wondered whether you yourself or your brother-in-law would not like to compete. Severi has written to me, also recently, that he will think about it. And you? For me it would be a great satisfaction to be able to judge that competition in favour of an Italian! Is it impossible for it to be you? (ANL-Castelnuovo, Segre to Castelnuovo, 25 December 1905: “A proposito di curve sghembe, dopo che qualche tempo fa t’avevo scritto di eccitar Severi a lavorare pel premio Guccia, mi son chiesto se tu stesso o tuo cognato non vorreste concorrere. Severi mi ha scritto, anche recentemente, che vi penserà. E voi? Sarebbe per me una grande soddisfazione il poter giudicare quel concorso in favore di un italiano! È escluso che tu possa esser quello?”).

Above all it is the study of Segre’s correspondence that allows us to retrace his strategy to promote abroad the Italian style in algebraic geometry and to show a clear difference in tone among the relationships he entertained with his outlander colleagues. From the letters of his youthful years (1883–1887) with F. Klein, Kronecker, Weierstrass and others, full of details on his research projects and on their links with international production, he moved on to letters and postcards, whose purpose was to announce to F. Klein, A. Hurwitz, M. Nöther (Luciano and Roero 2012, 25–27, 151–161, 164–165; ANL-Castelnuovo, Segre to Castelnuovo, 25 February 1897 and 10 March 1897), E. PicardFootnote 48 and E.J. WilczynskiFootnote 49 his own important results and, subsequently, those of Castelnuovo, Enriques, Fano, B. Levi and Fubini. For instance, Segre encouraged Castelnuovo to get into epistolary contact with Picard to tell him about the recent essays on the theory of surfaces published by Italians:

I am writing to you from the room of the same [examinations] to congratulate you on the card from Picard (at the appropriate time you will tell me about the noteworthy points in it), and to tell you … that I have nothing to tell you: the recent Italian works on the surfaces, which I would referenced, are only yours and those of Enriques … Perhaps (think about it) it will also be possible to mention the surfaces encountered by Fano in his works, which fit into our line of research. I sent my memoir to Picard some time ago. Before answering him, see in the last issue of the Intermédiaire des mathématiciens the question signed by Poincaré and Automne on curves with trisecant chords; and think whether it is appropriate to mention to Picard that in Italy the question … is not such. (ANL-Castelnuovo, Segre to Castelnuovo, 13 July 1894: “Ti scrivo dalla sala dei medesimi [esami] per congratularmi teco per la letterina del Picard (della quale all’occasione mi comunicherai i punti salienti), e per dirti … che non ho nulla da dirti: i lavori italiani recenti sulle superficie, ai quali io penserei, sono solo i tuoi e quelli di Enriques … Forse (pensaci) si potran nominare anche le superficie incontrate da Fano nei suoi lavori, che rientrano nel nostro indirizzo. La mia Memoria l’ho inviata a suo tempo al Picard. Prima di rispondere a questo, vedi nell’ultimo fascicolo dell’Intermédiaire des mathématiciens la questione firmata Poincaré e Automne sulle curve le cui corde son trisecanti; e pensa se sia opportuno accennare al Picard che in Italia la questione … non è tale”).Footnote 50

Aware of the influence of Castelnuovo and Enriques, Segre did not hide from his friends the satisfaction that he had felt reading the treatise by E. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes (cf. ANL-Castelnuovo, Segre to Castelnuovo, 7 September 1897). On the other hand the gift of the very same volume allowed Segre to face the French colleague on the theme of the resolution of singularities:

I had written a long letter to Picard (…). He answered me very politely recognising the difficulty “… je vous remercie bien d’avoir appelé mon attention sur l’inadvertance que j’ai commise, et que je tâcherai de réparer dans une note du second volume. En réalité, j’ai dans ma rédaction passé, je le reconnais, un peu vite sur ces théorèmes de réduction, au sujet desquels nous n’avons tous aucun dout pour le fond, et j’avais hâte d’arriver à des questions ayant pour moi plus de nouveauté. Si vous, ou vos élèves trouvaient quelque chose de tout-à-fait définitif sur ce sujet, j’en serais bien heureux.” (ANL-Castelnuovo, Segre to Castelnuovo, 22 October 1897: “Avevo scritto al Picard una lunga lettera …. Mi ha risposto molto gentilmente riconoscendo la cosa …”).

Besides, during the period of his leadership of the Italian School of Algebraic Geometry Segre recommended to his former students, who had become colleagues of his, that they transmitted their articles abroadFootnote 51; informed them about the quotations of their contributions that he was founding, in international journalsFootnote 52; enlightened about their exchanges with Brill,Footnote 53 Nöther,Footnote 54 RohnFootnote 55 and HurwitzFootnote 56 and was ready to put his disciples in touch with scholars like F. MacaulayFootnote 57 and the Swede A. Wiman.Footnote 58

Moreover, on several occasions Segre made every effort to avoid clashes or controversies between Italian and foreign mathematicians. For instance, in 1903, he suggested to René Baire that he privately point out to Beppo Levi some errors that he had found in notes “Sur la résolution des points singuliers des surfaces algébriques” and “Sur la théorie des fonctions algébriques de deux variables” (Comptes Rendus de l’Académie des Sciences de Paris, 134, 1902, 222–225, 642–644).Footnote 59 In 1906 Segre again acted as an intermediary between H. Lebesgue and B. Levi, trying to abstain from a controversy that would have damaged the international reputation of his School:

Mes théorèmes invoqués par Fatou sont mis en doute actuellement par Beppo Levi dans les Rendiconti dei Lincei. Beppo Levi n’a pas su rétablir quelques raisonnements intermédiaires simples et il s’est cassé le nez sur une faute de rédaction grave que Montel m’a jadis signalée et qu’il est facile de réparer. Naturellement j’ai commencé par rédiger une note où je l’attrapais comme du poisson pourri puis, sur une lettre de Segre, et parce que ce n’est pas le moyen d’acquérir une réputation mondiale que d’attraper ceux qui s’occupent de mes histoires, j’ai été moins dur. (H. Lebesgue to E. Borel, 1 June 1906, in Bru and Dugac 2004, 148–149).Footnote 60

In the context of the international promotion of the Italian geometric style an essential role was also played by Segre’s project, several times postponed and finally abandoned, to publish a volume of Lectures on Algebraic Geometry. It was E. Bertini that suggested that initiative to him, as an ideal way to increase the prestige of the Italian School and to convey knowledge of the production of its members to other countries. Aware of the advantages of similar publishing entreprises, in the summer of 1890 Segre wrote to Castelnuovo:

It is really necessary to thinking about writing treatises, lithographing lectures, extensively popularizing our ideas (I refer to yours and mine, we being perhaps the only ones in Italy, in all modesty, and without offending Peano, that have the right views on the subject). We will resort to Fano, who has already written me a long letter telling me to greet you, and with news on his readings and his researches (on cubic varieties), which always show his activity is prodigious. (ANL-Castelnuovo, Segre to Castelnuovo, 6 July 1890: “Bisogna proprio pensare a far trattati, a litografare lezioni, a divulgare con estensione le nostre idee (parlo di quelle di te e di me, che forse forse siamo i soli in Italia, modestia a parte, e senza offender Peano, che la pensiamo rettamente in proposito). Ricorreremo a Fano, che m’ha già scritto una lunga lettera coll’incarico di salutarti, e con notizie sulle sue letture e sue ricerche (sulle varietà cubiche), che lo mostrano sempre di un’attività prodigiosa.”).

Segre therefore entrusted to his young student the task of compiling the Summaries of the lectures in Higher Geometry in the 1890–91 academic year. It was the famous course devoted to the Introduzione alla geometria sopra un ente algebrico semplicemente infinito (Segre’s Notebook 3, in Giacardi 2013). Nevertheless, starting from the first revision, Segre was not satisfied of the notes by Fano since they were “very careless”Footnote 61 and predicted that they would have required months of patient work, before he could send them to be printed.Footnote 62

For their part, the colleagues Castelnuovo and Bertini urged the time to be shortened, and they offered Segre their help in the work of correction,Footnote 63 with the purpose not to procrastinate the appearance of these Summaries. Reluctant to delegate to others the responsibility of the revision, in the autumn of 1896 Segre went back to the project of editing his Lectures. Fundamental changes were made to the previous plan, for the aim of arriving at a more ambitious publication that would constitute a true treatise, and not only a volume of Summaries. The text should have organically recapitulated his most important and most appreciated monographic courses so as to reach a broad readership. Therefore Segre abandoned the idea of lithography to turn, instead, to well-known and ‘powerful’ publishing houses like Gauthier-Villars or Teubner. Segre also changed the nature of the authorat of these Lectures and the intention to circulate the summaries of a specific university course gave way to a project for a collective publication, that of a ‘master work’, ‘harmonious and original’, the expression of a School, realized thanks to collaboration among Segre, Castelnuovo and Enriques. The project so began to set up in its structure and organization:

Let’s now speak of the grand treatise on Higher Geometry. Castelnuovo will remember that here in Turin we have sometimes spoken about it together as of a work that we would gladly have done together later. Now that a new member, an active one, moves in, the initiative becomes much easier. Besides, I must point out that precisely for Enriques it would be better to defer the enterprise a few more years: because, for good or bad it is, the regulation requires him to prepare titles for his promotion to full professor; so for some time it is better for him to devote his scientific industriousness only to original researches. Then there is the usual difficulty about the publisher: if it were possible to turn to Gauthier-Villars or Teubner, it would be all right! For the rest, I completely approve of the idea, and I believe that among us three it would be possible to write a good treatise, harmonious, original. Whatever the form of collaboration to assume, I already have a sure way to take part in the work: and that is with the summaries (by me) of various courses of mine: that is to say of some general courses on curves, algebraic surfaces, etc. and of special courses on higher singularities, on geometry on a curve, on rational surfaces and linear systems of plane curves, etc. If the undertaking is delayed a few years, I will do some new courses which can also be useful. I must also prepare for the German Encyclopaedia the article on hyperspaces (algebraic varieties), and this can also serve for the respective chapters of the treatise (likewise for the article on algebraic surfaces promised by Guido [Castelnuovo] for the Encyclopaedia). Then I fully approve of a treatise being done by us three and no others: because we three share the same ideas; and others would perhaps upset the harmony of the work. I will be pleased if you write something more to me on this matter. (ANL-Castelnuovo, Segre to Castelnuovo and Enriques, 30 December 1896: “Parliamo ora del grande trattato di Geometria Superiore. Castelnuovo ricorderà che qui a Torino ne abbiam parlato insieme qualche volta come di un lavoro che avremmo fatto volentieri insieme più tardi. Ora che un nuovo elemento, energico, s’introduce, la cosa diventa molto più facile. Per altro fo notare che appunto per Enriques converrebbe differire ancora di qualche anno l’impresa: perché, buono o cattivo che sia, il regolamento vuole che egli si prepari titoli per la sua promozione ad ordinario; sicché per qualche tempo è meglio che egli dedichi la sua operosità scientifica solo a ricerche originali. Vi è poi la solita difficoltà dell’editore: se fosse possibile ricorrere a Gauthier-Villars oppure a Teubner, andrebbe bene! Del resto io approvo completamente l’idea, e credo che fra noi tre si potrebbe fare un buon trattato, armonico, originale. Qualunque sia la forma di collaborazione da adottare, io ho già un modo sicuro di prender parte all’opera: e cioè coi sunti (da me fatti) di vari miei corsi: cioè di qualche corso generale sulle curve, superficie ecc. algebriche, e di corsi speciali sulle singolarità superiori, sulla geometria sopra una curva, sulle superficie razionali e sistemi lineari di curve piane, ecc. Se l’impresa verrà ritardata di qualche anno, qualche nuovo corso farò che potrà pur servire. Inoltre io debbo preparare per l’Enciclopedia tedesca l’articolo sugl’iperspazi (varietà algebriche), e ciò potrà pur servire per capitoli corrispondenti del trattato (analogamente per l’articolo sulle superficie algebriche promesso da Guido [Castelnuovo] all’Enciclopedia). Approvo poi pienamente che il trattato sia di noi tre e non d’altri: perché noi tre siamo in piena uniformità d’idee; ed altri forse turberebbe l’armonia dell’opera. Avrò piacere se mi scriverete altro su questo argomento.”).

Nevertheless, this second project too was postponed, both because of Segre’s recurring hesitations, and because of the fear of damaging Enriques, busy preparing his dossier in view of the competition for the Bologna chair.Footnote 64 In April 1898, however, the project seemed to be becoming concrete and Segre committed himself with Teubner to bring out the treatise in his name, in German:

I am working on that blessed article for the Encyclopaedia. As if this were not enough, I am negotiating with Teubner for the publication of my Vorlesungen on higher geometry, based on some courses that I have already delivered, but enriched, etc. It is another enterprise that will then occupy me for several years, and I hesitated a lot before deciding. (ANL-Volterra, Segre to Volterra, 23 April 1898: “Lavoro attorno a quel benedetto articolo per l’Enciclopedia. Come se questo non bastasse, sto trattando col Teubner per la pubblicazione di mie Vorlesungen di geometria superiore, tolte da alcuni corsi che ho già fatto, ma arricchite, ecc. È un’altra impresa che mi occuperà poi per vari anni, ed ho esitato molto prima di decidermi.”).

The outline of the text, the distribution of the contents and the financial aspects were illustrated and discussed with Castelnuovo and Enriques between 1899 and 1900.Footnote 65 Segre’s Vorlesungen would have chained the classical texts by Salmon-Fiedler (1874) and Clebsch (1876) to the volume on algebraic surfaces, commissioned by Teubner from Castelnuovo and Enriques. The Vorlesungen were intended to offer a balanced overview, extensive but not encyclopaedic, of recent researches on algebraic geometry conducted by Italians.

Against the quite advanced schedule of this ‘collective publication’, it is strange that after 1900 no traces of the manuscript of these Vorlesungen have been spotted, except for a mention of their imminent publication in the Bulletin of the American Mathematical Society Footnote 66 and the advertising announcement which appeared “for several years in the Catalogues of the Teubner publishing house” (see Fig. 2.1; Loria 1924, 13; Terracini 1961, 12).

It is quite possible that other commitments distracted Segre’s attention from this task, above all the one connected to his collaboration, direct and indirect, on the Encyklopädie der mathematische Wissenschaften, a collaboration that can be ascribed to the same strategy of international diffusion of the results attained by the Italian School of Algebraic Geometry that he had put in place.Footnote 67 As far as concerns this aspect, we may remind that the first Italian mathematicians to be asked by F. Klein and F. Meyer to join the authors of the third volume of the Encyklopädie were Castelnuovo, Enriques and Fano, in September 1895.Footnote 68 Segre indeed supervised the progress of the work, from the very beginning, readily advising these ‘disciples’ on the structure to give their own essays, suggesting the suitable content extension of their chapters, and offering to revise himself their manuscripts:

I am pleased that your Bericht is going ahead. Regarding what Meyer has asked Fano for, it seems to me that the latter would do well to agree: of course, provided that the theme is specified better than it is by the Jahrbuch: that is to say “algebraic transformations of algebraic entities in 2 and higher dimensions.” For my part, while I could not myself accept such an assignment, I will willingly help, for what I am capable, whoever does it. As you say, it cannot be a heavy labour! (ANL-Castelnuovo, Segre to Castelnuovo, 23 September 1895: “Ho piacere che il vostro Bericht vada avanti. Quanto a quello che il Meyer domanda al Fano, mi pare che questi farebbe bene ad accettare: s’intende, purché il tema fosse meglio precisato di quel che non sia dal Jahrbuch: vale a dire “trasformazioni algebriche degli enti algebrici a 2 e più dimensioni”. Per parte mia, mentre non potrei accettare io stesso un tal lavoro, ajuterò volontieri, per quel che valgo, chi lo farà. Come tu dici, non dev’essere poi una gran fatica!”).

Taken up with numerous teaching and institutional responsibilities, at last Segre reluctantly forewent the revision of the first article by Castelnuovo and Enriques (ANL-Castelnuovo, Segre to Castelnuovo, 7 January 1896) and, in actual fact, until the first months of 1896 he limited himself to marginally contributing to the Encyklopädie. In May, however, he was entrusted with writing the article on hyperspace geometry. As he announced to Castelnuovo, it was:

about 2 sextodecimos; deadline for presentation 1st of 1899. It seems to me that the general programme of the work is very solid: and I have decided to accept. (ANL-Castelnuovo, Segre to Castelnuovo, 14 May 1896: “circa 2 fogli di stampa; termine per la presentazione 1° del 1899. Mi è parso che il programma complessivo dell’opera sia molto serio: e mi son deciso ad accettare.”).Footnote 69

From that moment on, Segre became in actual fact the Italian ‘delegate’ for the third volume of the Encyklopädie.Footnote 70 In addition to collecting sources for his chapter, to the writing of which he devoted himself for years, with immense zeal,Footnote 71 he was charged by Klein to make contact with other contributors, coordinated the work of the members of his School and, in order to avoid overlaps and omissions, discussed with Fano, Castelnuovo and Enriques the distribution of the contents among the essays on projective hyperspace geometry, on linear differential equations, and on the theory of algebraic surfaces:

For a while I have been collecting sources for my article in the Encyclopaedia. Now I am concerned to specify my task clearly; and, among other things, to have from you and Enriques the assurance that in your article IIIC8 “Algebraische Transformationen und Correspondenzen” which comes after mine, you will also deal with the algebraic transformations of S n (excluding projective ones), that is to say that you, not me, will do what Noether expounded in the M[athematische] A[nnalen] II on such transformations, etc., and Del Pezzo and S. Kantor in some recent works, etc. It seems appropriate to me that that article of yours, as it encompasses together transformations of the plane and space, should also contain those of hyperspaces: and this (I repeat) also because, otherwise, I should speak of hyperspace transformations before those of the plane and of S 3 are explained. I will also have to find out whether Zeuthen in IIIC10 will present the numerative geometry of S n in addition to that of S 3. (ANL-Castelnuovo, Segre to Castelnuovo, 19 February 1898: “Da qualche tempo raccolgo materiali pel mio articolo dell’Enciclopedia. Ora m’importa precisare bene il mio compito; e fra l’altro, avere da te e da Enriques l’assicurazione che nel vostro articolo IIIC8 “Algebraische Transformationen und Correspondenzen” il quale vien dopo del mio, tratterete anche le trasformazioni algebriche di S n (escluse le projettive) cioè farete voi, e non io, quel tanto che su tali trasformazioni dà Noether nei M. A. II, ecc., Del Pezzo e S. Kantor in alcuni recenti lavori, ecc. Mi pare opportuno che quel vostro art[icolo], come abbraccia insieme le trasformazioni del piano e dello spazio, così contenga anche quelle degl’iperspazi: e ciò (ripeto) anche perché, in caso contrario, io dovrei parlare di trasformazioni iperspaziali prima che sian state trattate quelle del piano e di S 3. Dovrò anche informarmi se Zeuthen in IIIC10 darà la geometria numerativa di S n oltre a quella di S 3”).Footnote 72

4 Study Trips and Sojourns

In the activity of a mathematical School an important component is the oral tradition, that is to say the habit of verbal exchanges, which contribute in a decisive way to the collective construction, the sharing and transmission of knowledge between the Maestro and the disciples. From this point of view we can state that the Italian team of algebraic geometers was an authentic School, linked to a very precise local milieu, constituted by the University of Turin and some ‘satellite’ contexts like the Academy of Sciences, in whose meetings Segre assiduously participated, cultural cafes like Giaccardi, Bergia and the American Bar, where the so-called Pitareide met,Footnote 73 and Segre’s house. In his ‘little studio’ various disciples and colleagues were entertained, both Italians and foreigners, cf. (Terracini 1968, 9, 13 and Annexes 29, 30, 31, 49, 57, 58, 71 and 72).

That algebraic geometry on the curve was born of dialogues between Segre and Castelnuovo under the porticos of Via Po is not an hyperbole (Conte 1993, 438). The Piedmont geometer, in effect, assigned great importance to conversation, and generally to all those vectors of scientific sociability (private teaching, seminars, reunions, working lunches, etc.) that were set alongside institutional education and, allowing greater freedom of expression and debate between the interlocutors, proved particularly useful in the creative phase of research activity.

Significant in this connection is the role of study trips and sojourns by Italian geometers abroad, and, reciprocally, those of foreigners in Italy. Heir to the tradition of the Grand Tour of the Baroque age and the Enlightenment, the practice of going abroad for a period of master-class is certainly not a prerogative of the School of Algebraic Geometry. Back in the Risorgimento period, mathematicians like C.I. Giulio, L.F. Menabrea, L. Cremona and Q. Sella had grasped the advantage of offering international training to their best alumni. Therefore they had sent them to complete their studies in French and German Ecoles, or in the most dynamic environments from the cultural, industrial, manufacturing, agricultural, mining and technological points of view. In turn, these scholars had known how to exploit to the best the meetings with outstanding scholars visiting our country for work, amusement or health, and had derived fruitful stimuli from conversations with promising scientists.Footnote 74

Having trained in contact with Maestri like F. Faà di Bruno, E. D’Ovidio, F. Siacci and A. Genocchi, who had been capable of treasuring international interplays, Segre entertained the idea of a trip to Germany since 1884. He chose as his privileged destination the University of Göttingen, an emblem, at a world level, of the culture of orality, toward which he would then oriented many of his alumni (Luciano and Roero 2012, 47–49).

In the summer of 1891, accompanied by Gino Loria, Segre stayed in Göttingen a few days (29–30 June and 11 July). However, for the rest of his life, he was to preserve an indelible memory of the ‘School of Klein’, and above all of the ‘luminous conversations’ with the German Maestro. In their itinerary the two Italians touched on Frankfurt am Main (25 June), Berlin (4–8 July), Dresden, Leipzig, Nuremberg (15 July) and finally Munich (16–17 July), before returning homeland on 18 July.Footnote 75

Although partially spoiled by the controversy between G. Peano and G. Veronese, which in the meantime had broken out in the pages of the Rivista di Matematica, the experience of the ‘German journey’ was fully exploited by Segre. He visited libraries and scientific institutes; he assisted, on an invitation from Kronecker and Weierstrass, at a meeting of the Berlin Academy; he attended the Mathematische Verein in Göttingen; he promoted the output of the Italian geometers, and above all that of Castelnuovo, publicizing it to Klein, Nöther and Rohn; and he strengthened the friendship and the scientific collaborations that he had already opened with Reye and Sturm (Boggio 1928, 305). The notes he took in Germany, the cutting-edge literature consulted in the Göttingen Library and the ‘impressions’ of his 1891 trip spangled the letters to Castelnuovo and F. Amodeo, even months after Segres’s return to Turin.Footnote 76

The journey by Segre and Loria on one side consolidated the tradition of the study sojourns in Germany, inaugurated by A. Tonelli (1874–75) and continued by A. Abetti (1876), E. Caporali (1877–78), C. Romaniello (1877–78), S. Pincherle (1878), G. Ricci Curbastro (1878–79), L. Bianchi (1879–80), A. Capelli (1879–80), G. Veronese (1880–81), F. Gerbaldi (1882–83), G. Morera (1883–84), E. Pascal (1888) and R. Marcolongo (1888–89). On the other side it launched the trend of scientific ‘pilgrimages’ to Göttingen. In the ensuing years, numerous Italian mathematicians, went to the ‘School of Klein’: V. Volterra (1891, 1904, 1914), G. Vailati (1899, 1906), Castelnuovo (1903), Enriques (1903), Severi (6–8 January 1937) and E. Bompiani (summer semester 1913). In his conversations with Klein and Hilbert in Göttingen, Enriques, for example, several times dealt with the issues regarding mathematics education and the new developments in algebraic geometry. Klein and Enriques also planned some joint publishing ventures, seeking to establish an agreement between the Teubner and Zanichelli companies, in view of special sale prices of volumes, treatises, textbooks and journals.

In addition to geometers that spent study sojourns in foreign countries on their own accord, there were disciples of Segre’s that—urged by the Maestro—sojourned in Germany and France with ministerial scholarships. Segre for instance carefully monitored the works of the Junta of the Higher Council that assigned posts for master-class abroad. In particular he attended its meetings in the academic year 1893–94, when G. Fano was given a grant to spend the winter semester in Göttingen, and then at the time of his return to Italy.Footnote 77

Moreover the summer holiday periods by members of the School of Segre in famous places in the Alps, frequented by foreign colleagues like Hurwitz, Hilbert and Klein, afforded unexpected opportunities to establish scientific and personal bonds and opened up new scenarios for international promotion of the team.Footnote 78

The relevance of oral exchanges, in the framework of the activity of a mathematical School and for its success on a supranational scale, was fully acknowledged by Segre’s disciples: the homes of the Enriques and Castelnuovo families, in Bologna and Rome, in turn became known for being a ‘cosmopolite theatre of debates’ among the most eminent scholars of the period (Enriques 1983; Parikh 1991; De Benedetti 2001; Linguerri and Simili 2008). And likewise, the charm and brilliant conversations of Fano, Severi, Terracini and B. Levi were to be important in promoting knowledge of Italian algebraic geometry abroad, not only in Great Britain (1923), Japan (1936) and Switzerland (1940), but also in peripheral or developing countries, like Argentina (Luciano 2016).

If many members of Segre’s School crossed the borders to go to foreign countries, no less numerous were the outlanders that chose to stay in Turin to attend Segre’s lectures, or to get acquainted with him in Rome, Ancona, Naples, and Engadina. We can for instance mention C. Hermite in 1892,Footnote 79 F. Mouton in 1895,Footnote 80 F. Lindemann in 1897,Footnote 81 G. Mittag-Leffler in 1899, H.G. Zeuthen in June 1900,Footnote 82 C. Stéphanos, E. Study and J. Coolidge in 1903,Footnote 83 E.J. Wilczynski in 1904 and 1906,Footnote 84 L.W. Dowling in 1907,Footnote 85 V. Snyder in 1922Footnote 86 and E. Čech in 1921–22, alongside with F. Klein, who travelled round Italy several times: in the summer of 1874, in the 1878 Easter holidays, again in March 1899 and in the spring of 1900.Footnote 87

Further, among the professors at the University of Turin, Segre was one of the few that could boast of an international audience of students in his classrooms. As alumni, colleagues and family members recalled, starting from the end of the 19th century:

the fame of his skill as a Maestro went far beyond the borders of our nation, and more or less every year scholars from other countries flocked to listen to his lectures, especially ones from England and North America, who from what they learnt in Italy often drew the inspiration for fine publications (Berzolari 1924, 532).Footnote 88

The first to sojourn in Turin to take Segre’s courses and to carry out studies on higher geometry, under his direction, were Grace Chisholm (1868–1944) and his husband William Henry Young (1863–1942) (Grattan-Guinness 1972, 105–185; Conte and Giacardi in this volume). Having arrived in Turin in October 1898, they stayed until March of 1899 and attended the Higher Geometry course devoted to the Curve algebriche dei vari spazî (Segre’s Notebook in 12 Giacardi 2013). Showing great friendliness, Segre not only integrated his university teaching with private lessons and domestic lectures imparted to them, but on 30 April 1899 presented to the Academy of Sciences two notes, respectively by Chisholm, “Sulla varietà razionale normale di M 4 3 di S 6 rappresentante della trigonometria sferica”, and by Young, “Sulle sizigie che legano le relazioni quadratiche fra le coordinate di retta in S 4”.Footnote 89 Furthermore Segre put the Youngs in contact with Castelnuovo:

Until 18 November we won’t start our lectures. In my audience I will have the Youngs, whom I believe know you. Yesterday I had a visit from the husband, who gave me a brief Note of his, quite good. If you can then send him some offprints of yours (especially on M 3, geometry of the straight line in S 4 , numerative geometry) it will be a good thing. (ANL-Castelnuovo, Segre to Castelnuovo, 23 October 1898: “Fino al 18 Novembre non cominceranno le nostre lezioni. Avrò fra i miei uditori i coniugi Young, che credo ti conoscano. Ieri ho avuto la visita del marito, che mi ha dato una sua breve Nota, abbastanza buonina. Se potrai inviargli poi qualcosa di tuo (specialmente su M 3, geometria della retta in S 4 , questioni numerative) farai bene”).Footnote 90

Chisholm and Young preserved a good memory of their Turin stay, together with deep gratitude for the generosity of the Maestro to them (cf. Chisholm Young to O. Michelli Segre, 19 June 1924, Annex 68).

From the United States there then came Julian Lowell Coolidge (1873–1954) to complete his training under Segre’s guidance. Engaged in a tour of European universities between 1902 and 1904, the American geometer, together with his wife Theresa Reynolds, spent in Turin the winter semester from October 1903 to the spring of 1904. In this period he published the first part of the essay “Les congruences isotropes qui servent à répresenter les fonctions d’une variable complexe” in the Atti of the Academy of Sciences, under the presentation by Segre in the meeting of 20 December 1903 (Struik 1955, 671–672; Hammond et al. 1955; Dauben 1999).Footnote 91 Coolidge’s experience ‘at Segre’s School’ was shortly afterwards described as follows in the article “The Opportunities for Mathematical Study in Italy”:

At the same time such headings as Higher Analysis, Higher Geometry are so comprehensive as to leave to the teacher the greatest discretion in the choice of material. Some fortunate professors give a new course each year, others run through a cycle including a greater or less number of subjects. Americans are sure to find lectures on subjects that will interest them, and they will have the French, rather than the German standard of clearness and elegance. They will also be struck by the eclecticism of the instructor, for Italian mathematicians read widely. I remember being impressed at the beginning of one course of lectures by the fact that the professor put down, as principal works of reference, books in four different languages, and remarked that those of his hearers who could not read English, French and German, must certainly make up the deficiency in the course of the year (Coolidge 1904b, 13).

In fact, that year Segre began his lectures on Applicazioni degli integrali abeliani alla Geometria, giving references to writings by 23 foreign authors, among them W.F. Osgood, W. Wirtinger, A. von Brill and M. Nöther, B. Riemann, K. Weierstrass, A. Clebsch and P. Gordan, F. Neumann, E. Picard, C. Jordan, A. Forsyth, C.A. Briot and C. Bouquet, F. Klein, P. Appell and E. Goursat, H.F. Baker, K. Hensel, G. Landsberg, N. Abel and A. Hurwitz, and only quoting three Italian authors: L. Bianchi, F. Casorati and S. Pincherle (Segre’s Notebook 17: 3–6, in Giacardi 2013).

The synergy between Coolidge and Segre was not interrupted at the end of the Turin sojourn. Although there were no further opportunities for encounters, the two mathematicians continued to correspond,Footnote 92 and in 1924, on Segre’s death, Coolidge evoked his scientific and ethic stature in a long obituary (Coolidge 1927) and with these heartfelt words sent to Segre’s wife:

Je ne me flatte pas que vous vous souviendrez d’un Américain errant qui est arrivé à // Turin avec sa petite famille au moins d’Octobre 1903, pour suivre les cours de l’université, et surtout pour profiter de l’enseignement de votre illustre mari. Pour lui, pourtant, ça a été un évènement d’importance capitale. Non seulement a-t-il trouvé une impulsion scientifique dont il n’a cessé de profiter énormément depuis, mais, chose beaucoup plus précieuse, il a eu le privilège de nouer de liens d’amitié avec son maitre, que chaque année depuis n’a que rendu plus forts. Je ne saurais vous exprimer, madame, ni l’estime que je ressentis pour votre mari comme savant, ni l’affection qui me lié à lui. Toujours je serai fier d’avoir été à la fois de ses élèves et de ses amis (J. Coolidge to O. Michelli Segre, Annex 72).

Very effective in attracting disciples from America was the fact that between 1904 and 1920 Segre sent the outlines of his courses on Higher Geometry to be published in the Bulletin of the American Mathematical Society. Footnote 93 As a consequence from the United States in 1908 two other students of Segre’s: Charles Herschel Sisam (1879–1964) and Clarence Lemuel Elisha Moore (1876–1931) hastened to Turin to hear his lectures.

An alumnus of V. Snyder, Sisam completed his Ph.D. in 1905 at Cornell University with the thesis Classification of Scrolls of Order Seven Having a Rectilinear Directrix. He then decided to refine his studies in Europe, with post-doc fellowship in Göttingen and Turin, before returning home, to Colorado College, where the rest of his brilliant career was played out. Having reached Piedmont at the beginning of September 1908, Sisam attended Segre’s course Rassegna di concetti e metodi della Geometria moderna (Segre’s Notebook 22, in Giacardi 2013) and was a guest of the professor for private lessons, which proved very stimulating for him, as he confided to Wilczynski:

In his lectures he speaks very distinctly and I have had no difficulty whatever in following them. In his lectures this year he is covering, in a general way, the entire field of Geometry. In our private conferences at his home he is very approachable, makes me feel entirely at liberty to come where I want to, and is very stimulating. I am working on some properties of triply infinite varieties in five dimensions, with references to line geometry. I am very much pleased with the results I have obtained thus far (Sisam to Wilczynski, 12 December 1908, Annex 57).Footnote 94

During the months he spent in Turin, Sisam developed his researches under the guidance of Segre, who presented them at the Academy of Sciences, in the sitting of 5 March 1911.Footnote 95 Between Segre and Sisam a relationship inter pares was maintained, despite the professional, cultural and age difference between the two. On one side, thanks to Segre, Sisam corrected a significant mistake which he, and before him Wilczynski, had made:

In a paper, published in the Bulletin for June, 1904, page 440, Mr. C. H. Sisam gives a proof for a theorem previously enunciated and proved by me in the Mathematische Annalen, volume 58, page 256. Unfortunately, however, he follows me in giving an inexact formulation of the theorem in question. I have used the word self-dual in a more restricted sense than is usual, without having properly called attention to the fact. As others may be misled also, a few words of explanation seem to be in order. A dualistic transformation may have the property of converting a ruled surface into itself without interchanging its generators, so that every generator of the surface is transformed into itself. It is merely of scrolls, for which such a transformation exists, that I wish to assert the theorem that they belong to a non-special linear complex. It is only to this case that Mr. Sisam’s demonstration applies. There actually exist ruled surfaces, self-dual in the general sense, which do not belong to a linear complex. The following example of such surfaces is due to Professor Corrado Segre, who first called my attention to the fact that my theorem was badly formulated (Wilczynski 1904, 8).

At his turn, Segre was prompted by the sodality with Sisam to develop some studies of his own, and expressly thanked the American disciple in the article “Aggiunta alla memoria: Preliminari di una teoria delle varietà luoghi di spazi”:

Mr. C.H. Sisam, of the Universitỳ of Illinois, has kindly pointed out to me that there are exceptions to the theorem (no. 21) enounced in the middle of p. 107 [here at pp. 96–97]. As far as the content of that no. 21 concerns, I also want to emphasize that, since the beginning of last year (1909), Dr. Sisam, who at that time was studying with me in Turin, presented a work of his to me on the V3varieties that satisfy four or more homogeneous linear partial differential equations of order 2. I hope that that research is published soon (Segre 1910b, 346).

In Sisam’s case too, Segre’s role as a Maestro was not limited to the Turin periodFootnote 96 but went on until the twenties. Though having a single opportunity to return to Italy in 1928, as invited lecturer in Bologna, at the VIII International Congress of Mathematicians, Sisam kept in touch with Segre by letter.

By contrast, the recollection by Terracini, that the American Ellis Bagley Stouffer (1884–1965) also spent a study sojourn in Turin as a disciple of Segre’s, is unfounded (Terracini 1968, 13). An alumnus of Wilczynski’s, Stouffer took his Ph.D. in 1911, defending the thesis Invariants of Linear Differential Equations, with Applications to Ruled Surfaces in Five-Dimensional Space. Having become an Instructor in Mathematics at Drake University and the University of Illinois, he pursued most of his career at the University of Kansas, where he was an Assistant Professor (1914–1917), and Associate Professor (1917–1921), and finally a Full Professor and Dean of the Graduate School (1921–1955). An eminent and prolific scholar, Stouffer obtained a fellowship of the John Simon Guggenheim Memorial Foundation in 1926, for Mathematics, reserved for American and Canadian citizens. The grant was to finance a study sojourn of 10 months, beginning from 1 August 1926, to conduct comparative studies on the three general methods of projective differential geometry. Went in company with his wife Anna Lucile Shepard and their children, Stouffer spent most of his time in Italy, where he attended the courses of E. Bompiani at the University of Bologna (Fitch 1928, 331; Ciliberto and Sallent 2012, 156). He also sojourned in Turin, and returned to Italy in 1928, on the occasion of the International Congress of Mathematicians in Bologna, during which he strengthened his contacts with B. Segre, Fano and Severi, but not with Corrado Segre, who passed away four years before.

Quite the contrary, although till now it has not been noticed, the pool of Segre’s American disciples included Clarence Lemuel Elisha Moore (1876–1931). An expert in algebraic and Riemannian geometry, he obtained a Ph.D. at Cornell University, with a thesis on the classification of the surfaces with singularities of the quadratic spherical complex, with Virgil Snyder as advisor. Moore improved his geometric culture by going to study in Göttingen, in Turin with C. Segre, and in Bonn with E. Study. In 1904 he entered the Department of Mathematics of MIT, at first as an instructor, then as an assistant, an associate professor and finally a full professor (Franklin 1933). Reaching Turin in March 1908, Moore attended Segre’s course devoted to Capitoli vari di Geometria della retta (Segre’s Notebook 21 in Giacardi 2013) and entered in friendly relations with the Italian geometer.Footnote 97 For example, Segre advised him and helped him in drafting the report of the IV International Congress of Mathematicians,Footnote 98 which Moore had been entrusted with for the Bulletin of the American Mathematical Society (C.L.E. Moore 1908).

To complete the panorama of Segre’s international disciples, it is also necessary to mention those outlanders that, like René Baire, sojourned in Turin to specialize in other domains of mathematics, and nevertheless assisted to some lectures on Higher Geometry. Among Segre’s other foreign alumni, we can mention David Cytron (1887–1982), a Jew from Bialystok, afterwards naturalized, who came to study in Italy because he was forbidden to do so in his homeland. In the years 1908–1910 he attended two courses of Segre’s (Segre’s Notebooks 22 and 23 in Giacardi 2013) and his lectures at the Teacher Training School (Segre’s Notebook 40 in Giacardi 2013), with good profit. After the degree examination, on 2 July 1910, at which he was given the maximum vote, Cytron became an assistant of B. Levi at the University of Cagliari. Appreciated by Segre for his mathematical talent and diligence, Cytron then devoted himself to finance and trading. Interned with his wife Ida Tyktin, in the province of Chieti, he was detained at Pizzoferrato on 15 October 1940 and subsequently at Villa Santa Maria (10 November 1940–1942) and, after the liberation, returned to Turin.Footnote 99

Finally, from analysis of some notebooks preserved at the Special Library of the Turin Department of Mathematics, in which the alumni were registered that took examinations with Segre in Projective, Descriptive and Higher Geometry, many other names of foreigners have emerged, for whom it is difficult to retrieve information, since often they did not take up a university career.

5 Annali di Matematica: A Journal for the Italian School of Algebraic Geometry?

In the years 1850–1880, many Italian geometers, among them F. Brioschi, L. Cremona, G. Battaglini, E. Beltrami, E. D’Ovidio, E. Bertini, R. De Paolis, L. Bianchi and G. Veronese published essays and articles in foreign journals, first of all Mathematische Annalen, but also Bulletin des Sciences Mathématiques and Journal für die reine und angewandte Mathematik, to reach a broader and more specialized readership, both quantitatively and qualitatively, compared to that afforded by the national context.Footnote 100

This strategy of self-promotion of his own results on an international scale was soon appropriated by Segre, ever since the years of his university studies, thanks, as already mentioned, to the example of D’Ovidio, Faà di Bruno and Genocchi. It was soon effective: it is sufficient to consider the reviews and quotations of Segre’s works, dotted around, from 1883 on, in repertoires like Revue Semestrielle Footnote 101 and Jahrbuch über die Fortschritte der Mathematik,Footnote 102 or the reports of conferences and activities in mathematical societies.Footnote 103

However, as the identity of the Italian School of Algebraic Geometry was consolidated, it became important to have a journal that could confirm, at a national and international level, the ‘command position’ attained by this School. Thus increasing weight was taken on by Segre’s active presence in the editorial board of the Annali di Matematica pura ed applicata Footnote 104—alongside with L. Bianchi, G. Jung and U. Dini—from 1904 onwards.Footnote 105

Publishing activity—to which Segre devoted himself with fervour and with a scrupulousness that was universally appreciated and sometimes feared (Segre to Wilczynski, 16 April 1916 and 2 March 1917, Annexes 59 and 60)Footnote 106—was rather onerous. It was not by chance that in 1922 he declined the invitation to become a member of the editorial board of the Bollettino dell’Unione Matematica Italiana, giving as quite a reason the desire to dedicate himself entirely to Annali (S. Pincherle to C. Segre, 27 January 1923, Annex 64).

As co-editor in chief of this illustrious journal, Segre participated in the rapid changes taking place in the twentieth-century world of mathematics publishing, in the decline of many academic collections and in the resulting process of specialization and internationalization, undertaken by the most successful journals, in Italy above all Rendiconti del Circolo Matematico di Palermo by G.B. Guccia (Brigaglia 2014, 165–178).

Though he continued to choose to publish most of his works in the series of the Atti and Memorie of the Turin Academy of Sciences, or in the Rendiconti of the Lincei Academy, Segre’s attitude regarding the most effective tactic to be adopted for promoting Italian algebraic geometry gradually varied, adapting to the new scientific demands and practices. The letters to F. Klein of the 1880s, where the young Segre and his colleagues and disciples (Veronese, Loria, Amodeo, Fano, Pieri, etc.) asked for hospitality in Mathematische Annalen, certain that this journal could truly afford an international showcase for their production, gave way, in the 1890s, to the correspondences of Segre ‘Maestro’, careful to acquire for Annali di Matematica pura ed applicata the best essays by Castelnuovo, Enriques, Fano, B. Levi, Severi, etc.:

If you are still in time, I would beg you to consider whether, out of national respect, you might say, if you think it is better not to give your important work to the Mathematische Annalen, but instead to Annali di Matematica (or another Italian journal). I am sorry not to have thought about it before, but I believe that you are still in time. For when your discovery has to be quoted why should a foreign periodical be named? Foreigners must get used to reading our collections. About other works, for instance the one that Klein asked you for, I will not say anything. But precisely for this one I would prefer Annali di Matematica, in which it seems that at the moment there is not too much material; so perhaps the delay would not be great. (ANL-Castelnuovo, Segre to Castelnuovo, 26 September 1893: “Se ne sei ancora in tempo, ti pregherei di considerare se, per un riguardo, dirò, nazionale, non ti paja meglio di non dare ai Mathematische Annalen, ma bensì agli Annali di matematica (od altra raccolta italiana) il tuo importante lavoro. Mi rincresce di non averci pensato prima, ma credo che tu sia in tempo ancora. Poiché quando si dovrà citare la tua scoperta si dovrebbe nominare un periodico estero? Gli stranieri s’abituino a leggere le nostre raccolte. Per altre cose, ad esempio per quella che ti chiese il Klein, non dico. Ma per questa proprio preferirei gli Annali di matematica, nei quali pare che ora non vi sia troppa materia; sicché il ritardo forse non sarebbe grande”).Footnote 107

This action was not devoid of effects: after 1891 Segre published no other works in Mathematische Annalen. By contrast, in Annali di Matematica seven articles of his appeared, four papers by Castelnuovo (against two edited in Mathematische Annalen), three essays by Enriques (against five appeared in Mathematische Annalen), and five memoirs by Fano (who only addressed one work to Mathematische Annalen).

Nonetheless, despite Segre’s intentions, the publishing activity continued to be the weakest and most marginal element of his leadership of the Italian School of Algebraic Geometry. On one side we have to recognize, as has often been done (Brigaglia and Ciliberto 1995, 14–16), that the volume XXII of Annali, containing the two famous essays by Bertini “La geometria delle serie lineari sopra una curva piana secondo il metodo algebrico” (Bertini 1894) and by Segre “Introduzione alla geometria sopra un ente algebrico semplicemente infinito” (Segre 1894a), perfectly reflected the interplay in national research on algebraic geometry and the Italian style that characterized it. On the other hand it should not be denied that Annali di Matematica never became the ‘journal of a School’, with a role analogous to this taken on by the Rivista di Matematica, edited by Peano, for the promotion of studies on logic and foundations of mathematics (Roero 2015).

The task of retrieving and coordinating the articles to publish in Annali was undertaken by Segre in a fragmentary and occasional way. In actual fact, he limited himself to examining the works that through various vectors were submitted to him for publication, without pursuing any specific cultural line. In his role of leader of a School, Segre did not clearly show he had a preference for Annali. Indeed, he sometimes advised Castelnuovo, Enriques and Fano—for most disparate reasons, not all of a scientific nature—to use the Rendiconti of the Accademia dei Lincei, Memorie di Matematica e di Fisica della Società Italiana delle Scienze detta dei XL, the Giornale di Matematiche ad uso degli studenti delle Università Italiane,Footnote 108 etc. Equally inconsistent it was the strategy pursued by Segre to place his own publications, so much so that it was only in one circumstance, when he thought he was about to complete the essay “Sulla scomposizione dei punti singolari delle superficie algebriche” (Segre 1897a), that he requested Castelnuovo and Enriques to “get the place reserved for him” in Annali, “because he would have been very pleased to appear in their company. In this way, precisely, their School in its entirety would have been represented!” (ANL-Castelnuovo, Segre to Castelnuovo, 9 February 1895: “perché avrei molto piacere di uscire in vostra compagnia. Così appunto sarebbe stata tutta la nostra scuola rappresentata!”).Footnote 109

Besides, continuing to be anchored to ‘old-style’ publishing practices, Segre constantly asked disciples and colleagues to ‘donate’ works to the Turin Academy of Sciences, especially in the periods in which “in our series it seemed that the geometers were silent.” (ANL-Castelnuovo, Segre to Castelnuovo, 21 March 1893 and 25 December 1893). On manifold occasions he ‘reproached’ Castelnuovo in a friendly way for “always turning to the Accademia dei Lincei and not the Turin Academy” for the publication of his most interesting results (ANL-Castelnuovo, Segre to Castelnuovo, 17 April 1894), and reminded him that—as a member of this latter Society—he was ‘morally’ bound to ensure effective collaboration to its periodicals:

I will be pleased to present Enriques’ work, because I like our Academy to publish the important geometric researches that you two are doing. Therefore I need not repeat to you that this year too you must give our Academy some works of yours. (ANL-Castelnuovo, Segre to Castelnuovo, 25 December 1893: “sarò lieto di presentarlo [il lavoro di F. Enriques], perché mi piace che la nostra Accademia pubblichi le importanti ricerche geometriche che si vanno facendo da voi. Non occorre quindi ch’io ti ripeta che anche quest’anno tu devi dare alla nostra Accademia qualche tuo lavoro”).

However, though naive and heterogeneous, Segre’s commitment within the editorial board of Annali di Matematica appears to have been marked by two aspects: the defence, in a Risorgimento and patriotic key, of the Italian language as one of the international languages of mathematics, equal in dignity and breadth of use to French, German and English; and the battle for those journals that, so to speak, had ‘made the history’ of Italian scientific culture. It is therefore not surprising that, though having a full expertise in both French and German, after 1906 Segre did not submit any more work in a foreign language and that as long ago as 1890 he informed Castelnuovo, with a vein of satisfaction, of the possibility of also using Italian for publications abroad:

The problem on the g (1)2 of particular curves that you wrote to me saying you had solved already seems to me very general and important; and you will do well to publish it. If you decide on Mathematische Annalen I must warn you that the editorial board of the latter is really determined also to accept works in Italian (also in the competition of the Berlin Academy for the Steiner prize they now also admit papers in our language). (ANL-Castelnuovo, Segre to Castelnuovo, 13 August 1890: “Il problema sulle g (1)2 di curve particolari che tu mi scrivi di aver risolto mi par già molto generale ed importante; e farai bene a pubblicarlo. Se ti decidi pei Mathematische Annalen t’avverto che la redazione di questi è proprio decisa ad accogliere anche lavori in italiano (anche nel concorso dell’Acc[ademia] di Berlino pel premio Steiner adesso accolgono anche i lavori scritti nella nostra lingua)”).

However, defence of the use of Italian was typical of the period. One need only think that in an other prestigious context too, that of the Accademia dei Lincei Prizes for Mathematics, candidates could only compete with essays written in Italian and Latin. Segre himself, who won the prize in 1898, together with Volterra, was forced to devote the 1895 Christmas holidays to the “very boring occupation” of “summarizing, in Italian” some works of his published in French (ANL-Castelnuovo, Segre to Castelnuovo, 7 January 1896).

In the same line of thought and action was the campaign of international mobilization for the survival of Annali di Matematica, in which Segre was involved in the years after World War One. Though sorry to have to turn to foreigners, which “it is always regrettable to have to ask for money”,Footnote 110 in that circumstance he was willing to sacrifice patriotic pride, whilst it should be guaranteed the life of the journal that, since 1857, had represented the ‘voice’ of Italian mathematical community. To face the serious economic crisis of Annali, Segre asked the Americans for help, expressing his worry over the destiny of the journal to Snyder, at the time in Turin for a study sojourn.Footnote 111 Through the Bulletin of the American Mathematical Society Snyder released an invitation to his American colleagues,Footnote 112 which was echoed by ‘distance disciples’ of Segre’s like C.L. Moore, as well as by many estimators of the Italian geometer (E.B. Stouffer, S. Lefschetz, S. Lipka, etc.).Footnote 113 The excellent results of this subscriptions campaign, an “effect of Segre’s work”,Footnote 114 succeeded in avoiding the ‘death’ of Annali and indeed projected the journal—with the opening of the fourth series, edited by L. Bianchi, C. Segre, S. Pincherle and T. Levi-Civita—towards a “new season of Italian mathematics” (Pincherle to Segre, 27 January and 3 November 1923, Annexes 64 and 67).

6 Segre’s School at International Congresses of Mathematicians

For Mathematical Schools in the 19th and 20th centuries international congresses constituted very propitious occasions to promote their research style. At the same time, they favoured exchanges and interactions among the participants, creating the premises for new collaborations among scholars in different geographical areas or renovating ancient relationships (Albers et al. 1986; Lehto 1998; Curbera 2009).

Segre, probably influenced by Klein, soon appeared aware of the importance of that form of sociability for spreading and vulgarizing his results and projects. In this connection, since 1893, on the occasion of the Evanston Colloquium (28 August–9 September 1893), he commented to Castelnuovo on the usefulness of participating in international conferences:

They write to me from Chicago that I should promote a report from my Italian friends at that congress, especially with “brief critical review of the development during the last 20–25 years of the definite small subdivisions of the science.” Would you like to make such a brief account on geometry of hyperspaces (algebr[aic] entities; proj[ective] geom[etry], etc.) in Italy? I believe that it would cost you little work (it being a matter of briefly characterizing the progress due to the various main contributions, which you already know) and it would be a useful survey. The congress is being held from 21 to 28 August. Let me know if you accept this task. If the answer is affirmative you could then send me your text, authorizing me to retouch it. (ANL-Castelnuovo, Segre to Castelnuovo, 28 June 1893: “Mi scrivono da Chicago di promuovere un resoconto dai miei amici italiani a quel congresso, specialmente con “brief critical reviews of the development during the last 20–25 years of the definite small subdivisions of the science.” Avresti tu voglia di fare un breve rapporto sulla geometria degl’iperspazi (enti algebr.; geom. proj. ecc.) in Italia? Credo che ti costerebbe poca fatica (trattandosi di caratterizzare brevemente i progressi dovuti ai vari principali lavori, che tu già conosci) e sarebbe una cosa utile. Il congresso si tiene dal 21 al 28 d’Agosto. Avvertimi se accetti quest’incarico. In caso affermativo potresti poi inviare a me il tuo scritto con facoltà di ritoccarlo.”).Footnote 115

However, the first opportunity to present themselves collectively, as a true School, came to the Italian geometers from the International Congress of Mathematicians in Zurich (9–11 August 1897). The previous summer, Segre had already told his closest friends of his intention of attending it:

Your letter gave me great pleasure because of the hope that it gives me that you yourself will decide to come to Zurich. […] I believe that if I could not go there, later I would regret it, like an opportunity missed to meet scholars of high value and to attend at special meetings […]. I would devote Saturday evening and part of Sunday to the mathematicians that are already in Zurich. At Göschenen Fano will join me; it seems, instead, that Enriques will come later. (ANL-Volterra, Segre to Volterra, 31 July 1897: “Molto piacere m’ha fatto la tua lettera per la speranza che mi dà che tu ti decida a venire a Zurigo. […] Io credo che se non potessi andarci, dopo ne proverei rammarico, come di un’occasione perduta di vedere uomini di valore, e riunioni singolari […]. La sera di sabato e parte della domenica li dedicherei ai matematici che già si trovano a Zurigo. A Göschenen si unirà a me il Fano; Enriques, invece, pare che verrà più tardi.”).

Regarding personal contacts, the Zurich Congress proved profitable for Segre, who exploited his time there to establish a dense network of partnerships, which he was to preserve for the rest of his life. Besides being acquainted with E. Borel, T. Reye, H. Zeuthen and the American C.A. Scott,Footnote 116 Segre met F. Kraft, who after the end of the Congress went to Italy.Footnote 117 On an invitation from Klein, moreover, Segre was appointed vice-president of the Geometry section, directed by Reye, in which appreciated talks were given by F. Gerbaldi, C. Burali-Forti and G. Fano; Enriques spoke in the Arithmetic and Algebra session; Loria in the Mechanics and Mathematical Physics one, but on a historical theme.Footnote 118 Although C. F. Geiser pressed him to submit a paper, Segre firmly declined the invitation, fearing the judgment of mathematicians of the standing of M. Nöther and F. Klein:

Alongside these there will be, it is true, others for whom my words might perhaps have been of some utility. But meanwhile wouldn’t the former have branded me superficial, wouldn’t they have said that I was expounding ideas that were already known, and partly already developed by me in that article of mine in Rivista di Matematica …? (ANL-Castelnuovo, Segre to Castelnuovo, 12 June 1897: “Accanto a questi vi saranno, è vero, altri pei quali le mie parole potevan essere forse di qualche utilità. Ma intanto quei primi non m’avrebbero tacciato di superficiale, non avrebbero detto che io esponevo cose già note, ed in parte già svolte da me in quel tale mio articolo della Rivista di matematica …?”).

Despite some important defections like those of Castelnuovo, Guccia and P. Del Pezzo,Footnote 119 the Italian delegation in Zurich was quite large, with about twenty participants, including, in addition to the speakers mentioned above, Volterra, F. Brioschi, S. Pincherle, G. Ricci Curbastro, G. Veronese and T. Levi-Civita. In particular, one of the four plenary lectures was delivered by G. Peano, who illustrated to the world scholarly community the second edition of his Formulaire de Mathématiques, showing how this ambitious encyclopaedia had been achieved thanks to the creation of a specific logical-ideographic language (Peano 1898, 299). Segre confided to Volterra the fear that the choice of such a theme might have harmed the image of Italian mathematics abroad:

I want to believe that nothing disagreeable for us will happen there: something comical perhaps, but if there will be cause for laughter, it won’t be such a bad thing! (ANL-Volterra, Segre to Volterra, 31 July 1897: “Io voglio credere che nulla abbia da accadere là di spiacevole per noi: di comico forse sì, ma se vi sarà da ridere, non sarà un gran male!”).

The following October, Segre ‘found on his desk’ a report of the Congress, published by Borel in Revue générale des Sciences pures et appliquées (Borel 1897, 783–789) and regretfully ascertained that the French colleague too had not appreciated Peano’s plenary lecture.Footnote 120

A sort of ‘rivalry’ between the two Turin Mathematical Schools, and consequently between their leaders Segre and Peano, was perceived in a more evident way in Paris at the International Congresses of Philosophy (1–5 August) and of Mathematicians (6–12 August), which opened the Short Century. Segre did not go to Paris but asked Volterra to convey him his impressions, without “waiting to tell him all of them face to face” on his return to Italy, and reminding him:

At present there is a philosophy congress there, with the active participation of the Peanians with their leader: Peano will be speaking of mathematical logic in general, Padoa and Pieri of its applications to arithmetic and geometry. I am warning you about it, so that you can attend this event if you are still in time! (ANL-Volterra, Segre to Volterra, 3 August 1900: “In questi giorni si fa costì un congresso di filosofia, al quale prendon parte attiva i peaniani col loro duce: questi parlando di logica matematica in genere, Padoa e Pieri delle sue applicazioni all’aritmetica e alla geometria. Te ne avverto, affinché tu possa andare se ne sei ancora in tempo!”).

Volterra did not fail to send Segre, from Paris, a letter containing detailed accounts of the two congresses together with an abstract of the famous talk by Hilbert “Mathematische Probleme” (ANL-Volterra, Segre a Volterra, 11 September 1900).

In Paris, the Italian School of Algebraic Geometry was certainly not well represented. Only two of Segre’s alumni presented talks: F. Amodeo, who traced a “Coup d’oeil sur les courbes algébriques au point de vue de la gonalité” (Amodeo 1902), and A. Padoa, who, though having had Segre as the advisor of his degree dissertation, cannot be properly considered a disciple of Segre. Padoa presented two papers, the first one on the theme “Un nouveau système irréductible de postulats pour l’Algèbre” and the second one on a hypothetical-deductive system for Euclidean geometry (Padoa 1902a, b).Footnote 121 To the absences of Segre, Castelnuovo and Enriques, there should be added the circumstance that Veronese spoke in the section devoted to Teaching and methods and that Fano did not present any paper.

The plenary lecture by Volterra himself (Volterra 1902), which Segre had hoped “would have been very beautiful and would have doubly honoured the name of Italy both for the value of the eminent scholars mentioned and for the expertise and the competence of the speaker” (ANL-Volterra, Segre to Volterra, 3 August 1900) proved a little disappointing for him. In fact Volterra concentrated on the study traditions opened up by E. Betti, F. Brioschi and F. Casorati and only marginally hinted at the evolution of Italian geometry.

While the congresses in Paris, and above all that of Philosophy, for Peano’s ‘phalanx’ represented the key moment of its affirmation, for the Italian algebraic geometers the apex of internationalism was reached in 1904. During the third Congress, which was held in Heidelberg from 8 to 13 August of that year, Segre, together with F. Morley, chaired the Geometry session (10 August), with talks by F.S. Macaulay, C. Guichard, E. Study, F. Meyer, K. Rohn and G. Scheffers and he took part, together with L. Automne, in the relevant debates. He was also entrusted with one of the four plenary lectures, on an indication by F. Klein. The topic chosen, “La Geometria d’oggidì e i suoi legami coll’Analisi”, gave Segre an excellent opportunity to recap the progress of the researches of the Italian School and to highlight the style that marked them:

A whole Italian school of geometers recognizes its starting point in the Memoir by Brill and Noether! Those concepts became even more fertile when, thanks precisely to this school, they took on a more abstract and more general character, being referred to algebraic curves, especially with the methodical introduction of the important notion of the sum of two linear series (corresponding to that of product in the field of rationality defined by an algebraic irrational). With these tools Castelnuovo obtained major new results on algebraic curves, for example regarding the issue of postulation, which I have already mentioned. More important still is the way in which it has been possible to apply that theory, or to extend it, by analogy, to surface geometry! (Segre 1905, 115).

Preparing the lecture, which Segre held in Italian on Saturday 13 August, for him was a source of some worry and constant commitment, during the summer stay in the Swiss Alps. From Airolo, where some colleagues joined him (Guccia and G. Morera), he updated his wife on the revisions that he was broughting to the text, on the decision not to have it printed before the Congress and on the programme of the events in which he would participate in Heidelberg (cf. Segre to O. Michelli Segre, letters from 17 July to 8 August 1904, Annexes 32–39). His presence at the Congress, together with Castelnuovo, Loria and Fano, certainly allowed the team of algebraic geometers to broaden the horizon of their scientific links. We nevertheless notice the unusual fact that, against the large number of members of this School (four out of eleven Italians that attended the Congress), only Segre gave a plenary lecture and only Loria a talk, not on a geometrical theme but on a historical one, as he had already done in Zurich in 1897.

From correspondences and other testimonies we infer that in Heidelberg Segre made conversations with Klein, M. Nöther, G. Mittag-Leffler, E. von Weber, H. Zeuthen, A. von Brill, P. Stäckel, S. Dickstein, E. Study and E. Wilczynski.Footnote 122 Further, to his wife he confided:

This is the great pleasure of conferences: to entertain with so many people that one only knew through their works, and to talk together of so many topics (8 August 1904, Annex 39).

1904 was also a year of very great importance for Segre from the point of view of the internationality of publications. In this connection, in June the Bulletin of the AMS published the English translation, by John Wesley Young (1879–1932), of the article “Sulle investigazioni geometriche, Osservazioni dirette ai miei studenti”, which, as is well known, had been at the origin of a bitter controversy with Peano in Rivista di Matematica (Segre 1891a). The English version “On Some Tendencies in Geometrical Investigations” of his ‘old’ textFootnote 123—whose salient points were taken up in the Heidelberg plenary lecture—gave Segre an opportunity to illustrate at an international arena how he conceived the role of Maestro, giving details of the modalities with which he structured the higher teachings and with which he trained young scholars towards original production. The fundamental convictions on the balance between intuition and rigor, on the alternation of methods and on the distinction between elementary mathematics, higher mathematics and elementary mathematics from an advanced standpoint, expressed by Segre in the 1904 article, and strongly permeated by the ideas of Klein, echoed the assumptions that he had defended since long time in his lectures at the university and at the Teacher Training School.Footnote 124 Filtered by the network of exchanges intertwined by Segre and his disciples with D. E. Smith and others American scholars, sensitive to mathematical instruction and education, these instances achieved an international resonance even before the publication of Segre’s Notes for the lectures at the Teacher Training School (Tricomi 1940). This led foreign mathematicians to perceive Segre’s School as a team that not only shared a particular research project but also precise assumptions and convictions regarding teaching and methodology.

Further the opportunity to publish the article “On Some Tendencies in Geometrical Investigations” (Segre 1904) in the United States was grasped by Segre to illustrate the changes—compared to 1891—in the conditions of research on and teaching of higher geometry in Italy, and to maintain that, precisely thanks to the more recent studies on the foundations of mathematics, it was now possible to consider as universally accepted the abstract and logical-deductive character of geometry:

Since the appearance of the present paper multi-dimensional geometry has spread more and more, so that now (among mathematicians!) its opponents have become rare, who at one time were so common. […] In regard to the foundations of geometry, the books by Pasch and by Peano, and since the publication of this article the book by Veronese and the papers of Pieri, Hilbert, and others have led mathematicians in recent years more and more to consider geometry from an abstract, purely logical or deductive point of view, detaching it entirely from every physical consideration. […] Following this method the points of a space of 4, or 5, … dimensions are treated as above stated in the same way as those of S3, the system of postulates being slightly modified (Segre 1904, 459, 462, 463).

If for Segre the Heidelberg Congress marked the acme of his international prestige, the next symposium, held in Rome in April 1908Footnote 125 had as its protagonists above all his disciples. Castelnuovo was the General Secretary and Fano the deputy-secretary; Enriques, B. Levi and Severi delivered plenary lectures and talks (Enriques 1909; Levi 1909; Severi 1909). With the exception of Severi, nevertheless, none of the members of Segre’s School spoke about algebraic geometry or higher geometry themes.

Entrusted, with L. Bianchi, with organizing and introducing the Geometry session, Segre proposed sending a best wishes telegram to his friend and colleague Reye, who had not been able to come to Rome for family reasons, and suggested that the meetings of 7 and 10 April should be chaired, respectively, by two ‘great Maestri’ from abroad: the Danish Hieronymus Georg Zeuthen and the German Issai Schur. However, the Geometry session was almost entirely animated by foreigners. There were talks by the French J. Andrade, the Croatian V. Varićak, the Dane Zeuthen, the Romanian G. Tzitzeika and the Ukrainian G. F. Pfeiffer; the only Italian to lecture was D. Montesano.

In Rome Segre’s School saw its importance fully recognized, in the international arena, in relation to three main aspects: the awarding to Severi of the Guccia Medal, the praise of the contributions of this team to the Encyklopädie der mathematischen Wissenshaften and the recognition of the Italian tradition in algebraic geometry by Volterra in his plenary lecture “Le matematiche in Italia nella seconda metà del secolo XIX” (Volterra 1909).

Regarding the first aspect, Segre, who had been charged with delivering the final report for the attribution of the Guccia Medal, succeeded in “capturing the attention of the immense public” present,Footnote 126 particularly underlining the connections between Severi’s researches on geometry on algebraic surfaces, the algebraic-geometrical methods of Enriques and Castelnuovo and the transcendent ones of Picard (Segre 1909, 212). He also listened with satisfaction to the presentation of the Encyklopädie der mathematischen Wissenshaften by W. von Dyck, who, speaking of the third volume, emphasised the input by Italian geometers as follows:

I would like to particularly single out another field, that of algebraic curves and surfaces and their integrals, in connection with the Analysis situs. The latter first arose in Germany, and then in Italy, through the problems promoted by the life work of Cremona, recently the rivalry of French and Italian geometers has operated successfully and – I refer to the latest presentation by Mr Segre about the Guccia Prize – produced new studies, rich in surprising results. We owe to this intense interest in geometric researches if the volume of the Encyclopaedia devoted to geometry benefited from the excellent collaboration of our Italian colleagues (von Dyck 1909, 128).

Lastly, the Italian style in geometrical research was “honourably” mentioned (Segre to his wife, 6–7 April 1908, Annex 54) by Volterra, who, referring to his friend Segre, in his lectio magistralis affirmed:

The further development of these studies in Italy and the new direction that they have taken is mainly to the credit of Segre with the first line of his researches, and to him there should be added Del Pezzo, Fano and others. Then, in the second phase of his scientific career, in which he drawn on the great essay of Noether, Segre was responsible for the beginning of that patrimony of works with which Castelnuovo, Enriques, Severi and De Franchis achieved their important results on the theory of the surfaces, the most recent of which are connected to the discoveries of Picard on algebraic functions and hence are in the framework of the theory of functions (Volterra 1909, 63–64).

The Rome Congress was also very fruitful from the point of view of conversations and exchanges with foreign colleagues. In addition to seeing Borel, Nöther and Mittag-Leffler again, Segre on that occasion got to know Henri Poincaré and met the young mathematician Emmy Nöther, Max Nöther’s daughter.Footnote 127 By contrast, he regretfully noticed the absence of many American correspondents of his, including Coolidge, Wilczynski, Sisam and Stouffer.

In the ensuing years the participation of Segre’s School in International Congresses of Mathematicians became increasingly episodic. The team continued to be represented by a remarkable group of members at the Cambridge Congress (22–28 August 1912). The new ranks of scholars like F. Severi, E. Bompiani and A. Terracini were present alongside the algebraic geometers of the first generation: Castelnuovo and Enriques. Fano and Segre, although enrolled, did not attend the conference. However, in Cambridge the themes of the talks reflected the new interests, not of a geometric type, cultivated by some members of Segre’s équipe.

Enriques, for instance, presented a talk “Sul significato della critica dei principii nello sviluppo delle Matematiche” (Enriques 1913, 67–79) and Castelnuovo—as one of the Italian delegates of ICMI—limited his activity to the session on mathematics education (Giacardi and Furinghetti 2008). The first Geometry meeting (23 August), chaired by H.F. Baker, elected Bompiani as Assistant Secretary, together with W. Blaschke. As had already happened in Rome, the session was dominated by foreigners: L. E. J. Brouwer, F. Morley, L.P. Eisenhart, E. Neville, M. Brückner, C. Stéphanos and A. Martin. The only Italian speaker was Bompiani, who dealt with a topic in line with the favourite research interests of Segre, like projective hyperspace geometry:

In a series of works published from 1906 to 1910 Prof. Segre resumed this branch of hyperspace geometry and took it to a high degree of perfection. The contribution made to it, under Segre’s impulse, by young geometers in Italy and abroad, can therefore be characterized as truly Italian (Bompiani 1913, 23).

At the following session too (24 August), chaired by Severi, there were interventions by W. Esson, M. Grassmann, P.H. Schoute, E. Kasner and G. Tzitzeica, while there were no Italian lecturers.

The presence of Segre’s School, seen as a community at International Congresses of Mathematicians was further reduced after the Great War. Indeed, Segre refused to take part in the Strasburg conference (22–30 September 1920), an event that was only deemed international ‘in name’, seeing the exclusion of German, Austrian, Hungarian and Bulgarian mathematicians. Agreeing with Segre’s stance, the School of Algebraic Geometry deserted the conference en masse, submitting no paper (cf. Klein to Enriques, 13 August 1920 and Enriques to Klein, 18 January 1921, in Luciano and Roero 2012, 214–217).

After Segre’s death, in a context that was radically changed from both the cultural and the political points of view, it was Severi and Castelnuovo that at the International Congresses in Toronto (11–16 August 1924), Bologna (3–10 September 1928) and Zurich (5–12 September 1932) represented the Italian School and inherited that directional role that SegreFootnote 128 had maintained for thirty years:

Nous devons cet esprit à nos maitres italiens Cremona, Betti, Bertini, Veronese, Segre, aux savants allemands Riemann, Clebsch, Klein, Brill et Noether, au danois Zeuthen, aux anglais Cayley, Sylvester et Salmon, et aux travaux, si profondément géométriques dans leur esprit, des analystes français, de Galois à Poincaré, à Picard, à Painlevé, à Humbert. Comme le peu que j’ai pu faire dans la science est le fruit de l’enseignement savant et passionné de mon maitre direct, Corrado Segre, que la mort nous a prématurément ravi, le 18 mai dernier, qu’il me soit permis d’envoyer à son souvenir les hommages du disciple affectionné et reconnaissant et ceux, bien plus hauts, du Congrès (Severi 1929, 154).

A current of thought that was different from Cremona’s and, through Klein, spread in our country between 1880 and 1990, led projective geometry to be extended to hyperspaces. Giuseppe Veronese and Corrado Segre were the greatest representatives of this trend. Segre in particular, an eclectic spirit, an insuperable Maestro, prematurely taken away from our affection and our admiration, foresaw the applications that could be made of hyperspace geometry to the theory of algebraic curves (Castelnuovo 1929, 192).

In the light of this overview, it seems pertinent to affirm that Segre’s School only partially succeeded in exploiting (and to a lesser extent with respect to the ‘rival’ School of Peano) Mathematical Congresses as showcases to build and validate its own identity at an international level, and to give resonance to the Italian style in algebraic geometry. It was mainly two elements that prevented the maximum profit from being derived from this strategy: in the first place, the decision of several members of Segre’s School not to present papers, or to devote their talks to themes that were not strictly mathematical, but rather historical, philosophical or methodological; in the second place, the fact that Segre himself was not able, and/or did not choose to coordinate the involvement of his disciples in the various Congresses, and often was not informed on the intentions of Castelnuovo, Enriques and Fano.Footnote 129 Very effective, by contrast, was the policy of exchanges developed by the Italian geometers within these symposia, a policy that led them to build up a network of long-lasting relationships destined to be maintained until the First World War.

7 Interventionism and Pacifism: Segre’s School and the First World War

The web of supranational partnerships shaped by Segre’s School underwent an abrupt, though temporary, interruption following the outbreak of the First World War. Even in the months around the Sarajevo assassination, the letters from Castelnuovo and Enriques to their German colleagues document a rich agenda of commitments, regarding both research and teaching. Castelnuovo and Klein, for example, planned to send out a questionnaire of the International Commission on Mathematical Instruction devoted to the training of secondary school teachers; the final report was to be presented by Loria, the Italian delegate, at the 1916 Stockholm International Congress of Mathematicians (Castelnuovo to Klein, 3 March 1914, in Luciano and Roero 2012, 208–209). At the same time, several members of Segre’s School, like Castelnuovo, Enriques, Loria and Segre himself, were dealing with the translation and correction of the proofs of their chapters for the Encyklopädie der mathematische Wissenshaften, while Klein was already thinking about recruiting yet another Italian, Berzolari, for the essay on transformations and correspondences (Klein to Castelnuovo, 4 March 1915 and Castelnuovo to Klein, 10 March 1915 (Luciano and Roero 2012, 209–213).

However, even before Italy entered the war, the contacts became increasingly difficult: a part of the Italian mathematical community—and first of all Volterra—took up interventionist stances, while postal censorship hindered the circulation of letters and books. Klein signed the Aufruf an die Kulturwelt, denying the war crimes committed by German army in Belgium, and this alienated him from many of his ‘distance disciples’.Footnote 130

In this situation, Segre put in place a series of concrete initiatives, to limit the effects of ostracism towards colleagues from the central powers. The pacifist beliefs of the Turin algebraic geometer—which dated back to his youth, though never paraded—led him to keep in touch with his friends Zeuthen and Reye, for example painfully assisting at the epilogue of the life of Reye, who died shortly after, having repaired to Würzburg at his daughter’s home:

On the eve of our entry into war, I received a postcard, dated “Strassburg Els., 18-5-15”, approved by the German censorship, which said: Lieber Freund und College, Bewahren Sie mir Ihre freundschaftlichen Gesinnungen, wie ich die meinigen Ihnen bewahren werde, auch wenn Italien, wie ich fürchte, in den Weltkrieg hineingerissen wird. Herzlich grüsst Sie Ihr. Th. Reye. This kind act, which touched me and shows the delicacy of feeling of our dear departed colleague, came to my mind when, some days ago, from the President of the Academy I received an invitation to commemorate Theodor Reye (…) whom I began to admire when I was a student, reading his classic Geometrie der Lage; and with whom I did not wait for enter into a scientific relationship, and also a personal one. I was able to appreciate not only his value as a mathematician, but also the real goodness of the man: a true gentleman! […] Strasburg being occupied by the French, Reye and his wife in March 1919 were expelled from that city, in which they had lived no less than 47 years, without any respect for their advanced age (Segre 1922, 492–493).

Further, in his quality of national member of the Turin Academy of Sciences, Segre made every possible effort to send the volumes of the academic collections to hostile or neutral countries, as to the Sweden through Mittag-Leffler.Footnote 131 Moreover, as the dean of the Turin Faculty of Sciences, in the war years he actively worked for students who were enlisted to help them in catching up on their examinations. He also interceded so that lecturers engaged at the front in research activity of military engineering, meteorology or ballistics would receive the instruments and books necessary for their studies (ANL-Volterra, Segre to Volterra, 13 January 1917). As far as this latter aspect is concerned, Segre insisted that the Special Library of Mathematics should go on receiving German publications, as they were “essential for scientific institutes”, despite the customs block on commodities coming from the Central Empires (Giacardi and Roero 1999, 444). Finally, in relation to an extraordinary Agenda, concerning participation in chairs by foreign scholars, voted by the Faculties of Letters and Sciences in Rome and by that of Letters and Philosophy in Turin, Segre affirmed, in one of the last meetings held while he was dean:

Personally [I] do not feel to wholly adhere to the votes formulated in the said Agenda. They are delicate matters, which it is necessary to deal with and decide on serenely, and ones which should not be subjected to deliberations taken irrationally, moved by feeling alone, as happens in some manifestations occasioned by the present war.Footnote 132 (“[Segre] personalmente, non si sente di aderire in tutto e per tutto ai voti formulati in detti Ordini del Giorno. Sono questioni gravi, che occorre trattare e decidere con serenità, e alle quali non convengono deliberazioni prese affannosamente, mosse dal solo sentimento, come avviene in talune manifestazioni occasionate dalla guerra presente.”).

The ensuing discussion saw many colleagues in the Faculty align with Segre’s opinion. In particular, D’Ovidio declared he was “favourable to admitting full Professors of those countries which granted full reciprocity of treatment.”Footnote 133

These various actions caused Segre to be branded filo-German, and to be criticised by some foreign and Italian scholars (Mazliak and Tazzioli 2009, 23; Aubin and Goldstein 2014, 189–192).

Actually, his conduct in the war years cannot be easily liquidated, since it took on distinct tones according to the role that he was playing. In this connection, if on one side Segre—as a private citizen—contributed to charity initiatives for ex-combatants and disabled war veterans,Footnote 134 on the other side—as a functionary—deemed it right to applaud the heroism of students and colleagues fighting on the battlegrounds.Footnote 135 As a scientist, he did not betray the ideals of brotherhood and cosmopolitism, which he had believed in since his youth, when he read the volumes by A. Thiers on the history of the French Revolution,Footnote 136 or when he attended the free course on Criticism of socialistic doctrines held by Salvatore Cognetti de Martiis. As a man, Segre always respected the deeds of those co-workers that, faithful to the ideals of the Risorgimento, ‘made the nation illustrious with their mind and arms’ (Luciano 2013, 307–309, 335–345). Therefore it is not surprising that to his friend Volterra he freely manifested patriotic enthusiasms, even ending letters with the exclamation: “Long live Italy!” (ANL-Volterra, Segre to Volterra, 27 May 1915). Nor does it appear contradictory that Segre acted in favour of A. Terracini and M. PiconeFootnote 137 so that the young mathematicians could receive the authorizations, the books and the sources indispensable to lead their studies on mountain artillery, “to the advantage of our war, which is at the core of our thoughts”.Footnote 138

Segre’s international relationships and those of his disciples took on a new impulse after the end of the war. He revived his links with Hilbert and Klein, expressing regret at not having been able to participate in their jubilees (Segre to Hilbert, 20 October 1919 and Segre to Klein, 24 February 1921, Luciano and Roero 2012, 213, 218). Further, in the inaugural lecture for the 1919–20 academic yearFootnote 139 he expressed his feelings with these words full of pathos:

More than twenty years ago, a work dedicated to the subject of the war that caused a great stir, by the Russian De Bloch, contained these prophetic words: “It is impossible to guarantee that the emperor Wilhelm II, in one of those fits of passion, vibrant with partiality, that he is wont to have, may not be capable of violating a treaty and taking on himself the responsibility … of provoking a war, whose consequences are impossible to predict.” Others too foresaw the huge tragedy that was to break out. But humanity has not been able to prevent it. There is a great difference between ‘foresee’ and ‘prevent’! […] And yet, through uncertainties and corrections, Science progresses; it becomes more and more capable of foreseeing; and to its practitioners it appears more and more beautiful. O young people, about to take up new studies, acquire new knowledge, or competences that will serve you in life; and you, who are distant, and to whom our thought always went out in these years, full of love and gratitude, students that have fought gloriously for our great mission, and have won! There, the day has come when humanity, freed of the arrogant, can start works of peace again with greater safety than it ever had. And we will be able in these rooms, without that shadow of remorse that during the war we seemed to feel, to resume all together to cultivate Science: not only that which is applied to bringing material wellbeing to men, but also that other kind of Science whose only aim is the satisfaction of our spirit. And Science will give you—allow me, in concluding, this prevision—the highest, the purest joys! (Segre 1918–19, 11, 24).

It was, however, above all in the context of learned Societies, that Segre continued to be the messenger of convictions and actions of authentic supranational landmark. He did this, for instance, when in 1922 he told Pincherle that he would resign from UMI if ostracism persisted towards mathematicians from Central Empires.Footnote 140Segre’s internationalism also shines through the penetrating commemorations of Reye (22 April 1922), Zeuthen,Footnote 141 Schwarz and Nöther, who were “glories not only of Germany, but of the whole civilized world”.Footnote 142 In these he charmingly illustrated to what extent their works had conditioned the evolution of Italian algebraic geometry. Lastly, we shall remember, what Segre wrote to Klein, following the publication of his Gesammelte Mathematische Abhandlungen, which he presented at the Accademia dei Lincei together with Castelnuovo and Enriques:

After so many vicissitudes, after so many sorrows, I have always preserved for you the affection and the veneration that I started to profess when I was a university student, when I grew excited reading your geometric works. What a deep influence those readings had on my ideas at that time! Now, cutting out the pages of this volume, and seeing those papers again, I seem to feel once again the freshness and the limpidity of those impressions. You were my Maestro, though we were so distant from one another! (Segre to Klein, 24 February 1921, in Luciano and Roero 2012, 218).Footnote 143

8 Conclusion

Speaking of the flourishing School of Luigi Cremona, in 1930 Castelnuovo stated:

Now since the School goes beyond the value of the man and the importance of a given discipline, to affect all scientific activity, it is worth saying a few words on the subject. All of you know what difficulties we meet in our Latin countries, which are prevalently individualist, in constituting a scientific School, that is to say a reunion or I would almost say a family of people collaborating in developing and pursuing a well defined project of research. But you also understand what advantages the School brings with itself. In scientific respects it offers the means accelerating and deepening the exploration of a given field, penetrating every facet of it, illuminating it from various perspectives. But the School also brings advantages as regards individuals, since it makes it possible to exploit in the most effective way the various aptitudes, and also to treasure the work of mediocre scholars, who, if guided, can perform useful services, while if they are abandoned to their own devices they tend to encumber science with contributions of little or no value. Now, to create a School the worth of the Maestro is not sufficient, nor is it sufficient that he knows how to trace out such a vast plan of researches as to go beyond his own working capacity. It is also necessary that he succeed in communicating his passion and his faith to his disciples and in demanding and directing their collaboration (Castelnuovo 1930, 615).

This description of what it means the membership to a mathematical School, of what advantages and detriments it introduces in the collective work, but above all of what the prerogatives are that a Maestro has to have for creating and directing it, is well suited to the case of the team of Italian algebraic geometers.

The question of the pertinence and effectiveness of the interpretative category of ‘School’ in relation to the group that Segre gathered around himself has already been discussed (Luciano and Roero 2012). However, it was still necessary to explore more closely the extent and the nature of the exchanges that involved the members of Segre’s School, according to the recent literature on the topic ‘internationality and science’ in the Belle époque.Footnote 144 In light of new archival sources it appears clear that Segre contributed in an essential way to the construction of a specific identity for his School and to the promotion of its image abroad, not only with writings and lectures, but also through other forms of sociability.

The various aspects of twentieth-century internationalism were not, of course, equally present in his scientific activity. If the breadth of Segre’s cultural horizons really was supranational, also thanks to his training in a context like the Turinese one, permeated from the time of the Risorgimento by influences of French and German science, his international vocation was weaker from the publishing point of view. In this sphere his patriotic feelings sometimes ran over, above all in the 20th century, into a policy of almost nationalistic promotion of Italian journals.

Likewise, if it is true that Segre did not reproduce the stereotype of the ‘Jew with a suitcase in his hands’ (Heims 1980),Footnote 145 nonetheless he was capable of developing a strategy of internationalism that was very effective for his School, even in the short term, fully exploiting the dimension of orality, and in particular conversations with his ‘distance Maestri’ and then with his ‘distance disciples’, during trips, study sojourns and international congresses.

The Short Century may have seen Segre less prolific on the research front. On the other hand, it marked the moment of his greatest commitment to crystallizing the identity of his School, not only in the milieu of Old Europe, but also in that of developing countries, from Americas to Scandinavia and Poland.Footnote 146

From 1890 on, and even more from 1904, Segre would have devoted his best energies and much of his time to drive abroad the Italian style in algebraic geometry, recognizing—in the evolution of his School—a shining example of the law of rapprochement highlighted by Klein, according to which:

The development of mathematical Schools, subject to alternations of progress and decadence in the limits of a nation, is revivified passing from one nation to another, almost making the spirit of the world participate more amply in the common work (Frajese, but Enriques 1938, 181).

Fig. 2.1
figure 1

Teubner’s announcement of Segre’s Vorlesungen with autograph corrections by Segre (UTo-ACS)

Full size image