The first paper which de Franchis published, based on his thesis, appeared in two parts in 1897. It was entitled Sulla curva luogo dei contatti di ordine k delle curve d'un fascio colle curve di un sistema lineare . This was followed by: Sopra una teoria geometrica delle singolarità di una curva piana (1897); Sulla riduzione degli integrali estesi a varietà (1898) and Riduzione dei fasci di curve piane di genere 2 (1898). After entering the competition for a chair, he was appointed as professor of Algebra and Analytic Geometry at the University of Cagliari in 1905. He moved to the University of Parma in following year on being appointed professor of Projective and Descriptive Geometry. In 1909 he was appointed to the University of Catania where he remained until 1914 when he returned to Palermo on being appointed to succeed Guccia in the chair of Analytic and Projective Geometry.
The Mathematical Circle of Palermo had been led by Guccia who had built it into the largest mathematical association in the world. When he died in 1914 the Mathematical Circle had over 900 members, the majority coming from outside Italy. The leading mathematicians of the day were members, for example in the early years of the 20th century Henri Poincaré, Jacques Hadamard, David Hilbert, Vito Volterra, Federigo Enriques, Guido Castelnuovo, Corrado Segre, Giuseppe Peano, Tullio Levi-Civita, Gregorio Ricci-Curbastro and Luigi Bianchi were members. De Franchis succeeded Guccia taking over production of the Rendiconti, the journal of the Circle, which he directed from 1914 until his death in 1946. However, the Mathematical Circle declined in importance over the following years. This was not the fault of de Franchis and his colleagues but rather a consequence of the difficult international situation due to World War I and its aftermath. Clashes between German and French members of the Mathematical Circle put its very existence in doubt. After World War I ended in 1918, Émile Picard, and other French mathematicians, demanded that all German members of the board of the Mathematical Circle be expelled. It is greatly to de Franchis's credit that he managed to maintain a society containing both French and German members. He was able to get support for his position from many leading mathematicians such as the Germans Edmund Landau and Hermann Weyl, the Frenchmen Maurice Fréchet and Jacques Hadamard, the American George D Birkhoff, and the Italians Gaetano Scorza, Luigi Bianchi, Tullio Levi-Civita, Francesco Severi, and Vito Volterra. The effort involved in keeping the Mathematical Circle a truly international society was huge and the affairs of the Mathematical Circle took up a great deal of de Franchis's time. Not surprisingly, his research output suffered as a result. For example, after writing obituaries of Guccia, de Franchis only published one paper Sulle varietà con infiniti integrali ellittici (1915) and one book Cenni sui determinanti e sulle forme lineari e quadratiche (1919) in the ten years following his appointment to Palermo. The monumental effort made by de Franchis to keep the Mathematical Circle as one of the major mathematical societies in the world was somewhat in vain since, although the Mathematical Circle maintained a high international reputation, it was not well supported by the Italian authorities. De Franchis maintained a strong international presence not only through correspondence but also attending international conferences, for example he attended the International Congress of Mathematicians held at Toronto 11 August to 16 August 1924.
All de Franchis's research contributions are in the area of algebraic geometry, but he was one of the first to use analytic methods in this area. We have seen already that de Franchis's early work studied plane algebraic curves but after 1900 his interests turned more towards global algebraic geometry, working in the main areas of the Italian school. His first major work in this area was Sulla varietà delle coppie di punti di due curve o di una curva algebrica (1903). In this he introduced an important new approach in which he used general properties of surfaces to give information about curves. He used these new methods to prove important results in Sulle corrispondenze algebriche fra due curve which was also published in 1903. The methods which he introduced here became important in his future work, that of Giuseppe Bagnera, and that of Federigo Enriques and Francesco Severi. De Franchis published another significant paper Sulle superficie algebriche le quali contengono un fascio irrazionale di curve in 1905. His work here, involving the existence of Picard integrals of the second kind, was closely related to ideas which were being studied by Castelnuovo and Enriques.
From 1906 to 1909 de Franchis worked in collaboration with Giuseppe Bagnera on the study of irregular surfaces, obtaining fundamental results for the classification of hyperelliptic surfaces. For their outstanding work on the theory of hyperelliptic surfaces de Franchis and Bagnera won the Paris Academy of sciences' Bordin prize in 1909 for their classification of hyperelliptic surfaces. A hyperelliptic surface is an algebraic surface which can be rationally covered by an abelian surface. If not a ruled surface, it is a quotient of an abelian surface by a finite group of automorphisms. There is, however, a rather strange story attached to the 1909 Bordin prize since the same prize was awarded in 1907 to Federigo Enriques and Francesco Severi for classifying hyperelliptic surfaces :-
Strange as it may seem that two couples get two prizes for the same theorem, instead of sharing one, this story is even more complicated, since the first version of the paper by Enriques and Severi was withdrawn after a conversation of Severi with de Franchis, and soon corrected. Bagnera and de Franchis were only a little later, since they had to admit a restriction; their proof however was simpler ...The papers de Franchis wrote with Bagnera over this period are: Sopra le superficie algebriche che hanno le coordinate del punto generico esprimibili con funzioni meromorfe quadruplamente periodiche di due parametri (1907); Sur les surfaces hyperelliptiques (1907); Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti (1908); Sopra le funzioni algebriche che si lasciano risolvere con X,Y,Z funzioni quadruplamente periodiche di due parametri (1909); and Intorno alle superficie regolari di genere zero che ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti (1909).
His teaching skills are noted in :-
In Palermo, de Franchis, as well as being known as an excellent researcher, was known as an excellent teacher. His courses were marked by the clarity and rigour that gives so much quality and aesthetic value to his work. Nevertheless, even in Palermo, de Franchis did not have a sufficient number of students or of collaborators, apart from Bagnera, so he could not, therefore, initiate the formation of his own school of mathematics. Instead, he himself represented the last and perhaps the high point. However, he represented in a completely appropriately way the highest level of achievement reached by the Italian school of algebraic geometry.Towards the end of his career, de Franchis published a number of books in collaboration with Giuseppe Bartolozzi (1905-1982). Bartolozzi was advised by de Franchis who graduated in 1930 after submitting his thesis Sopra una corrispondenza asintotica fra superficie affini equidistanti . Bartolozzi then served as de Franchis's assistant for several years and together they collaborated on the texts: Trigonometria sferica ; Trigonometria piana ; Aritmetica pratica ; Elementi di geometria ; Elementi di algebra ; Elementi di analisi matematica per gli Istituti tecnici ; and Elementi di matematica finanziaria ed attuariale per gli Istituti tecnici commerciali which were all published in 1937. Their final joint text was Nozioni di geometria intuitiva per il ginnasio inferiore (1940). These books were either aimed at students in all levels of secondary school or at students at a Technical Institute. Bartolozzi had experience at teaching in secondary schools and a Technical Institute for, after being de Franchis's assistant at the university, he taught at such institutions for the rest of his career.
The rise of Fascism in Italy proved a disaster for the Mathematical Circle of Palermo. Under de Franchis' leadership the society had struggled, as we pointed out above, through no fault of its leader. He had complained bitterly about the lack of support that the society had received from the Italian authorities although international bodies, such as the Rockefeller Foundation, continued to give strong support. However, the Fascist government, introduced a law in 1935 which limited any Italian society to having less than half as many foreign members as Italian members. This was a severe blow for the Mathematical Circle which had built its fame on its international nature. Pressure was put on the society to remove board members who were "unacceptable" to the Fascist government. Also the Rendiconti could not publish papers from "unacceptable" mathematicians. De Franchis fought hard to maintain the Mathematical Circle at the level that Guccia had raised it to but he fully realised that this was no longer possible. External events also severely damaged the vitality of the school of mathematics at the University of Palermo. De Franchis was greatly saddened by these events over this unfortunate period.
De Franchis was honoured with election to the Accademia Gioenia di Catania (1909), the Accademia Peloritana di Messina (1909), the Accademia delle scienze, lettere ed arti di Palermo (1910), and the Accademia dei Lincei on 15 July 1935. His name is remembered with the de Franchis theorem (sometimes called the de Franchis-Severi theorem) and the Castelnuovo-de Franchis theorem. The first of these was used in an important way by Gerd Faltings in his proof of the Mordell conjecture.
Let us end this biography by using, as a summary of de Franchis's mathematical contributions, the review by Luca Chiantini of the Collected works of Michele de Franchis  published in 1991:-
This book contains a complete collection of the mathematical papers written by M de Franchis (Palermo, 1875-1946), one of the most interesting exponents of the Italian school of algebraic geometry at the beginning of the century. De Franchis's works (after a few early papers devoted to the classification of linear systems on plane curves) are essentially concerned with the study of irregular surfaces, a central subject for the Italian school, with its many related topics (correspondences on curves, cyclic coverings, bundles of holomorphic forms). The fundamental feature of de Franchis's approach consists of a solid mastering of projective techniques joined with a deep sensibility for transcendental methods, which is less evident in the other main mathematicians of the Italian school. De Franchis introduced and used implicitly some of the most important tools of modern algebraic geometry, such as characteristic classes and the Albanese map. The use of both projective and transcendental techniques allowed him to find important improvements in the classification of surfaces; the Castelnuovo-de Franchis theorem (for a surface S of arithmetic genus pa, pa ≤ -1 implies S ruled) is a cornerstone in this theory; de Franchis's approach for the classification of hyperelliptic surfaces set the pattern for Lefschetz's works on general abelian varieties. Some of de Franchis's results seem to suggest still future extensions which can reveal themselves to be useful for modern algebraic geometry.
Article by: J J O'Connor and E F Robertson