Federigo Enriques, Hon.F.R.S.E.

by W V D Hodge

Federigo Enriques, one of the great figures of the Italian school of Mathematics, passed away suddenly during the night of June 12-14, 1946, at the age of seventy-five. During more than fifty years of his life he was a leading figure in the mathematical world, and his influence, both in mathematics and philosophy, had a very great effect, not only on the numerous and brilliant band of mathematicians with whom he was associated, but throughout the whole world of learning.

Born in Leghorn in 1871, he took his degree in Pisa in 1891, when he was a pupil of De Paolis. Soon, however, he came under the influence of Castelnuovo in Rome and of Corrado Segre in Turin, and his interests were quickly focused on algebraic geometry, the subject with which his name is most intimately associated. Brilliant work in the next few years resulted in his appointment, at the age of twenty-five, to the Chair of Projective and Descriptive Geometry at Bologna, which he held until he was promoted to the professorship of Higher Geometry at Rome in 1922. Here he joined Castelnuovo (with whom he was connected by marriage) and Severi, with both of whom he had collaborated with such astounding success for many years. He remained in Rome until the end of his life, but in 1938 he was, like Levi-Civita and many others, suspended from his Chair under the racial laws then introduced. He sustained the calamity with dignity, and when he was reinstated in 1944 he resumed his duties without animosity towards anyone, and to the great delight of his pupils.

Like most of the great Italian mathematicians, Enriques' activities extended into many fields. His mathematical interests lay chiefly in the direction of algebraic geometry and of transformation groups, but in addition to this he was passionately interested in scientific thought in all manner of fields, and also in pedagogy. His lectures on the scientific thought of the early Greeks were most stimulating, and his discussions of nearly forgotten hypotheses of obscure early philosophers revealed a knowledge and sympathy which can hardly be equalled. His Problemi della Scienza reveals a quite exceptional grasp of the most diverse scientific activities, and a mind like a rapier. In addition to the Problemi, he published numerous other books on the philosophy of science: Scienza e razionalismo, Per la Storia della logica, Storia del pensiero scientifico (in collaboration with De Santillana). At the time of his death he was engaged in writing, in collaboration with Mazziotti, a work on the philosophy of Democrates.

Enriques was equally interested in the problems of scientific method and teaching, on which he held pronounced views. He has stated that, in his view, intuition is the aristocratic way of discovery, rigour the plebeian way; and in view of the key position which he occupied in Italian mathematics it is reasonable to suppose that his influence in the Italian school of geometry was not only in the discoveries of that school, but on its methods. His writings on elementary mathematics, the history of mathematics and on scientific method include Questioni riguardanti le matematiche elementari, the Italian edition of Euclid's Elements, and Per la storia delle matematiche. These works are all in addition to his main mathematical activities, and in all he wrote some two hundred and fifty memoirs and papers.

Important as are Enriques' writings on scientific history and method, his name is primarily associated, in the mind of every mathematician, with the birational theory of algebraic surfaces. This subject, which is amongst the greatest achievements of Italian mathematics, and, indeed, of mathematics anywhere, is primarily associated with the names of Enriques, Castelnuovo and Severi, and to Enriques must go the honour of initiating it. The development of the theory of algebraic functions of a complex variable, and its subsequent translation into geometrical language, had resulted in great activity in the field of birational geometry during the eighteen-eighties. In the introduction to his first great memoir on the geometry on surfaces Enriques pointed out that the birational geometry of curves had followed two main directions; one was towards the discovery of birational invariants of curves, and of conditions for the birational equivalence of curves, and the other was towards the study of a curve as a seat of geometry in itself (linear series, etc.). On the other hand, the work being done in the birational geometry of surfaces was almost entirely in the first direction, and there was an urgent need to develop a systematic theory of geometry on a surface. His memoir Ricerche di geometria sulle superficie algebriche (1893) set out to lay the foundations of this theory. All the fundamental notions, so familiar to later geometers, of linear systems, complete systems, adjunction, and so on, are there, but in the development of their properties Enriques confined himself to surfaces which were regular, and of genus greater than zero. For these surfaces he obtained the Riemann-Roch theorem, and, indeed, he may be said to have achieved, in broad outline, at least most of the theory.

It is possible that he limited himself to regular surfaces more readily in the belief, then current, that irregular surfaces, and surfaces where geometric genus was zero, were very special, and belonged to a few well-defined types. But the researches of Enriques and of other Italian geometers soon convinced him that this was not the case - rather were regular surfaces the exception; and so in 1896 he published his memoir Introduzione alla geometria sopra le superficie algebriche, in which, he laid the foundations of the geometry on a quite general algebraic surface. Here the procedure now so familiar was introduced, and all the essential ideas (except that of continuous systems) were considered. He did not get quite so far with the general case as he had gone for regular surfaces in his Ricerche (he did not succeed, for instance, in proving the Riemann-Roch theorem), but he successfully launched the subject on its way, and the succeeding fifteen or twenty years is a period of astounding activity, in which discovery after discovery is recorded in the geometry of surfaces, mainly by Enriques, Castelnuovo, and Severi. It is impossible in this short notice to enumerate the contributions made by Enriques, but it should be mentioned that in 1898 a fundamental memoir by Castelnuovo took the subject further, proving among many other things the Riemann-Roch theorem for irreducible systems (later extended by Severi to reducible systems) and a joint paper by Castelnuovo and Enriques Sopra alcune questioni nella teoria delle superficie algebriche made further advances of the greatest importance, including a proof of the theorem that any surface which is not birationally equivalent to a ruled surface can be transformed birationally into one without exceptional curves (that is, without curves which can be transformed into simple points of a birationally equivalent surface). When Severi introduced the notion of algebraic (nonlinear) systems of curves on a surface a fresh stimulus was provided, and Enriques made important contributions to this theory, culminating in his proof (1904) that the characteristic series of a complete irreducible system of irreducible curves is complete. This proof, however, as well as one given by Severi in 1905, was subsequently found by Severi to be inadequate; but it was an ingenious effort to complete one of the major developments of the theory, though the difficulties turned out to be much greater than Enriques or any of his colleagues could have imagined them to be at the time.

In addition to his fundamental contribution to the general theory of surfaces, Enriques wrote a very large number of papers on special problems, such as the determination of surfaces whose linear genus is p(1) = 1, 2. Among his many contributions of this nature his work on the classification of surfaces which admit of groups of birational transformations into themselves deserves special mention. These form a part of a large series of papers on groups of transformations which he wrote, mainly in collaboration with Fano. These papers have added enormously to our knowledge of the birational transformation of an algebraic variety into itself, and include a classification of the continuous groups of Cremona transformations of a linear space into itself. Another memoir of the greatest importance, written in collaboration with Severi, Sur les surfaces hyperelliptiques, was awarded the Bordin prize in 1907. Generally speaking, it may be said that Enriques' work in the geometry of surfaces was as much devoted to the study of special surfaces, and special classes of surfaces, characterised by assigned values of the main birational transformations, or by groups of self-transformation, or in some such way, as to the general theory, and in the long and complicated story of the theory of surfaces, as developed in Italy, it is impossible to go any distance without coming across a contribution from his pen.

Enriques did not confine his geometrical activities to original researches, but found time to write several textbooks and treatises. His volume Lezioni di Geometria proiettiva was first published in 1898, and subsequently ran through many editions, as well as being translated into several foreign languages. It is regarded by many as the finest Italian textbook on projective geometry - praise indeed, when one recalls the impressive list of works by Italians on the subject. His Teoria geometrica delle equazioni, written in collaboration with Chisini, covers in four volumes the whole theory of curves, algebraic, geometrical and transcendental, and in addition to an exhaustive account of the work of others it includes many original contributions from Enriques. There can be no doubt that this is the greatest of all his books, both in scope and in presentation. In 1932 he published, with Campadelli, volume one of Lezioni sulla teoria delle superficie, a work designed to set out in definite form a systematic account of the theory he had done so much to create. This volume deals with the general theory of linear systems of curves on a surface and with certain problems in the classification of surfaces. The second volume never appeared, but he was working at it when he died. In 1939 a volume Le superficie razionale appeared under the name of F Conforto, but his friends were all aware that it was, in fact, another Enriques publication, the racial laws preventing the appearance of his name on the title-age.

It will have been observed that a very great deal of Enriques' work was done in collaboration. Without the assistance of others he could hardly have written all he did, but there were other and deeper reasons why he preferred to work with others. The method of collaboration was most congenial to a man of his temperament, and, above all, he found in it an ideal means of communicating his enthusiasm to those about him, and many of his younger assistants owe their subsequent successes to their association with him at the beginning of their careers.

Enriques received many honours from universities and academies throughout the world. In addition to honorary doctorates of various universities he was a Fellow of the Accademia dei Lincei, and of the academy of moral and political science of the Institute of France. He was elected an Honorary Fellow of the Royal Society of Edinburgh in 1938.

Federigo Enriques's RSE obituary by W V D Hodge appeared in Royal Society of Edinburgh Year Book 1948 and 1949, 13-16.