# Beniamino Segre

### Quick Info

Born
16 February 1903
Turin, Italy
Died
22 October 1977
Frascati, Italy

Summary
Beniamino Segre was an Italian mathematician who was a major contributor to algebraic geometry and was one of the founders of finite geometry.

### Biography

Beniamino Segre was related to Corrado Segre who was a cousin of Beniamino's mother. Beniamino attended school in Turin and, at the age of sixteen having won a university scholarship, he entered Turin University in 1919. His teachers at Turin University included Guiseppe Peano, Gino Fano, Guido Fubini, Carlo Somigliana and Corrado Segre. Beniamino graduated from Turin in 1923 having written a geometry dissertation on the double curves of symmetroids in $S_{4}$ entitled Genera della curva doppia per la varieta di $S_{4}$ che annulla un determinante simmetrico . His thesis advisor was Corrado Segre. Few mathematicians have published a substantial paper in a totally different area of mathematics from their thesis in the year they graduated but this is exactly what Segre achieved publishing Sul moto sferico vorticoso di un fluide incompressibile (1923). This paper on hydrodynamics studied the origins of anti-cyclones. He was appointed to the post of assistant professor in Turin where he remained until 1926. After studying in Paris with Élie Cartan for the year 1926-27, supported by a Rockefeller scholarship, Segre became Francesco Severi's assistant in Rome.

By 1931 when he was appointed to the Chair of Geometry at the University of Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations. He married Fernanda Coen from Como in 1932, 'a very gentle and sensitive lady'. They had three children and [14]:-
... to the end of their lives the marriage was one of the deepest affection and understanding, and their friends are hardly able to think of either of them without the other.
The Fascist Italian Government passed laws against those of Jewish background. Article 4 of the Royal Decree Law of 5 September 1938 was titled 'Measures for the defence of race in fascist schools' and, after Segre had been identified as Jewish by the University of Bologna, he was expelled from the University on 16 October 1938. By this time he, along with Tullio Levi-Civita, were managing the journal Annali di Matematica and, also in October 1938, both were relieved of their positions (Levi-Civita was also Jewish). Segre, with his wife and three young children, fled to England as refugees. However, this was a particularly difficult time for them as Patrick du Val related [14]:-
[In England they lived for a time in London and Cambridge; but in 1940 he was interned as an enemy alien in the Isle of Man. This was probably the unhappiest period in the family's life, and later they were rarely willing to speak of it at all; during it, while Mrs Segre was living in London with Leonard Roth and his Italian wife, who were old friends, their youngest child died; and when the Andorra Star, carrying prisoners of war and internees to Canada, was sunk in the Atlantic, she had no idea for some time whether her husband was on it or not. However, that phase of hysteria in our country passed off relatively soon, and most of the internees were released. Segre rejoined his family in London and later they returned to Cambridge, but it is perhaps hardly surprising that his list of publications shows a marked gap during the early war years.
Of course, despite the extremely difficult circumstances, Segre continued to devote himself to mathematics, writing the monograph The Non-singular Cubic Surfaces during these years. It was published in 1942 by Oxford University Press. Oscar Zariski writes in a review:-
This monograph contains a most exhaustive study of nonsingular cubic surfaces, especially in the real domain. As the author tells us in the preface, his object is to give a complete theory of what Cayley has called "the complicated and many-sided symmetry" of the relations between the 27 lines of a cubic surface. Such a theory naturally entails a minute analysis of an endless row of details, each of which represents a feature of the complicated pattern of the above-mentioned symmetry relations. This emphasis on details, which is inherent to the subject matter, and the methodical way in which the author goes about his object of exhausting the properties of a cubic surface may frighten away some readers who prefer in mathematics a better balance between the general and the particular, between the abstract and the concrete. But the reader who is willing, so to speak, to live for a while on a cubic surface and to read the book in the spirit in which it has been written will be greatly rewarded by the elegance and ingenuity of the author's treatment of the subject.
Segre was appointed to a teaching post in Manchester with Louis Mordell in 1942. This was a period during which he made exceptional research contributions on algebraic geometry but his interests also broadened, stimulated by discussions with Mordell and Kurt Mahler, to diophantine equations and the arithmetic of algebraic varieties. In 1946 he returned to Bologna succeeding Francesco Severi in Rome in 1950. Edoardo Vesentini, the author of [16] and [17], became Segre's research student soon after he arrived in Rome. He writes [17]:-
Beniamino Segre was very busy at that time (as, in fact, at all times of his life). Besides discharging several academic duties outside the University of Rome, he was teaching an undergraduate course in the Department of Mathematics of the University and an advanced course in the Istituto Nazionale di Alta Matematica.
Segre's output of research papers on geometry and related topics reached nearly 300 not counting a long list of other publications. He gained a high reputation for the quality of his writing as the following extracts from David Bernard Scott's review of Segre's Lezioni di Geometria Moderna Vol I (1948) show:-
The first part of this book is an introduction (of less than 80 pages) to abstract algebra, especially the ideas of groups, rings and, above all, of fields. The presentation, which is completely lucid and yet avoids surfeiting the reader with detail, has clearly been carefully planned. ... The second part deals no less attractively with the foundations of linear projective geometry in an arbitrary field. ... The standards of rigour and accuracy achieved in this work are such as all mathematicians (and not merely algebraic geometers steeped in the traditions of the Italian school) will appreciate.
An English edition, double the length of the original text, was published as Lectures on Modern Geometry in 1961. It is reviewed by Patrick du Val in [15].

Segre's contributions to geometry are many but, particularly in the latter part of his life, he is remembered for his study of geometries over fields other than the complex numbers. He gave a series of three lectures in London in 1950 which were published as Arithmetical questions on algebraic varieties in 1951. Many questions were asked in these lectures about how the results would change if the ground field were different.

By 1955 Segre was concentrating on geometries over a finite field and was producing results which we would now class as combinatorics rather than geometry. He collected many major results into a 100 page paper Le geometrie di Galois (1959) and a further 200 page paper Forme e geometrie hermitiane, con particolare riguardo al caso finito in 1965 was devoted to the case where the order of the ground field is a perfect square.

In [14] it is recounted how many of Segre's publications came from answering questions arising from lectures he attended. Some anecdotes are recounted in [14] about Segre's participation in lectures of others:-
... a lecture by Hodge in Oxford ... ended with Segre and another member of the audience occupying opposite ends of the blackboard and holding forth quite independently. ... a lecture by Severi in Harvard ... was constantly interrupted by Lefschetz in strong disagreement: the situation developed with Segre at the blackboard, firmly explaining what he thought was the resolution of the difference, while Severi and Lefschetz continued to shout each other down in French.
In 1973 Segre reached the age of 70 and retired from his university chair. However, even before this he had semi-retired [14]:-
Neither he nor his wife had much liked living in Rome, preferring their native Piedmont and Lombardy; and some time in the later 1960's they abandoned their flat near the University and settled in Frascati with their two married children in adjacent houses, and lived happily surrounded by grandchildren.
Sadly his wife, who had been so close to him and a companion on all international trips he made in his later years, died suddenly in the autumn of 1976. Segre continued to keep up the high work rate which had marked his whole career, but his wife's death was a blow from which he never recovered. He only survived his wife by a year. As to Segre's character we quote from Giuseppe Tallini [13]:-
He lived for science's sake and taught the sacrifice and complete dedication this requires and what he asked from himself - and it was quite a lot - he asked from others. Sometimes it was difficult to deal with him because of this brusqueness, but he had a great sensitivity and all who worked with him can testify his caring concern. The name of this great man already belongs to the history of mathematics among those who gave much to science and knowledge.
A similar picture is painted by du Val [14]:-
Segre could sometimes seem, especially to anyone meeting him for the first time, conversationally somewhat forbidding, but in the company of friends he could relax very convivially. He was fond of children, and most children liked him at first sight. I know our own children loved him; and I vividly recall an occasion when he and his wife were staying with us, seeing him return from a walk with the children demonstrating a Russian dance all the way up our suburban street, to the astonishment of the neighbours. He was devoted to his own children and grandchildren, and always seemed at his happiest in their company.

### References (show)

1. A Guerraggio and P Nastasi, Italian mathematics between the two World Wars (Birkhäuser Verlag, Basel, Boston, Berlin, 2000).
2. D Babbitt and J Goodstein, Guido Castelnuovo and Francesco Severi: Two personalities, Two letters, Notices Amer. Math. Soc. 56 (7) (2009), 800-808.
3. A Barlotti, Obituary: Beniamino Segre, professor of mathematics 1903-1977, J. Combin. Theory Ser. A 27 (3) (1979), 410-411.
4. D Gallarati, Beniamino Segre (Italian), Atti Accad. Ligure Sci. Lett. 35 (1978), 62-72.
5. R Garnier, Notice nécrologique sur Beniamino Segre, Comptes rendus de l'Académie des Sciences Paris Vie Académique 285 (12-15) (1977), 52-53.
6. J W P Hirschfeld, The 1959 Annali di Matematica paper of Beniamino Segre and its legacy, Combinatorics 2002 in Maratea, J. Geom. 76 (1-2) (2003), 82-94.
7. E Marchionna, Beniamino Segre (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 113 (5-6) (1979), 513-547.
8. E Martinelli, Beniamino Segre (Italian), Ann. Mat. Pura Appl. 116 (4) (1978), i-iii.
9. E Martinelli, Beniamino Segre: his life, his work, Rend. Accad. Naz Sci. XL Mem. Mat. 4 (1979/80), 1-11.
10. G Tallini, Beniamino Segre (1903-1977) (Italian), Archimede 29 (3) (1977), 143-145.
11. G Tallini, In memory of Beniamino Segre, Rend. Mat. 1 (7) (1) (1981), 1-29.
12. G Tallini, Beniamino Segre, Ann. Discrete Math. 18 (1983), 5-12.
13. G Tallini, Beniamino Segre, Acta Arith. 45 (1) (1985), 1-18.
14. P du Val, Beniamino Segre, Bull. London Math. Soc. 11 (2) (1979), 215-235.
15. P du Val, Review: Lectures on Modern Geometry by Beniamino Segre, The Mathematical Gazette 46 (358) (1962), 358-360.
16. E Vesentini, Beniamino Segre (1903-1977), Boll. Un. Mat. Ital. A 15 (5) (3) (1978), 699-714.
17. E Vesentini, Beniamino Segre and Italian geometry, Rend. Mat. Appl. (7) 25 (2) (2005), 185-193.