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Person: De Franchis, Michele

Michele de Franchis was an Italian mathematician who worked in algebraic geometry.

Mathematical Profile (Excerpt):

• Michele attended the high school in Palermo and, after graduating, continued his education at the University of Palermo.
• He was awarded his laurea in mathematics in 1896, having been advised by Francesco Gerbaldi who had been appointed to the chair of analytic and projective geometry at the University of Palermo in 1890.
• De Franchis was strongly influenced by Giovanni Battista Guccia who was originally from Palermo, but had studied in Rome before returning to Palermo in 1880.
• Guccia, who lectured to de Franchis at the university, set up the Mathematical Circle of Palermo in 1884 making Palermo an important mathematical centre.
• The Mathematical Circle played a large part in de Franchis's mathematical life and the first ten papers he wrote were all published in the Rendiconti of the Mathematical Circle.
• While he was a student, de Franchis became friendly with Giuseppe Bagnera who was appointed as Gerbaldi's assistant in 1893 while studying for his laurea in mathematics.
• 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 Ⓣ(On the locus of points of order k contact of the curves of a bundle of a linear system).
• This was followed by: Sopra una teoria geometrica delle singolarità di una curva piana Ⓣ(On a geometric theory of the singularities of a plane curve) (1897); Sulla riduzione degli integrali estesi a varietà Ⓣ(On reduction of integrals extended to varieties) (1898) and Riduzione dei fasci di curve piane di genere 2 Ⓣ(Reduction of bundles of plane curves of genus two) (1898).
• He moved to the University of Parma in following year on being appointed professor of Projective and Descriptive Geometry.
• When he died in 1914 the Mathematical Circle had over 900 members, the majority coming from outside Italy.
• 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.
• It is greatly to de Franchis's credit that he managed to maintain a society containing both French and German members.
• 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.
• For example, after writing obituaries of Guccia, de Franchis only published one paper Sulle varietà con infiniti integrali ellittici Ⓣ(On infinite varieties with elliptic integrals)(1915) and one book Cenni sui determinanti e sulle forme lineari e quadratiche Ⓣ(Notes on determinants and linear and quadratic forms) (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.
• 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 Ⓣ(On the variety of pairs of two curves or points of an algebraic curve) (1903).
• He used these new methods to prove important results in Sulle corrispondenze algebriche fra due curve Ⓣ(On the algebraic correspondence between two curves) 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 Ⓣ(On the algebraic surface which contain an irrational bundle of curves) 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.
• 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 Ⓣ(On algebraic surfaces that have the coordinates of the generic point expressed with meromorphic quadruply periodic functions of two parameters) (1907); Sur les surfaces hyperelliptiques Ⓣ(On hyperelliptic surfaces) (1907); Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti Ⓣ(The algebraic surfaces which admit a parametric representation using hyperelliptic functions of two arguments) (1908); Sopra le funzioni algebriche che si lasciano risolvere con X,Y,Z funzioni quadruplamente periodiche di due parametri Ⓣ(On the algebraic functions that can be solved with X, Y, Z quadruply periodic functions of two parameters) (1909); and Intorno alle superficie regolari di genere zero che ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti Ⓣ(On the regular neighbourhood of zero generally admitting a parametric representation using hyperelliptic functions of two arguments) (1909).
• 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.
• 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 Ⓣ(On an asymptotic correspondence between similar equidistant surfaces).
• Bartolozzi then served as de Franchis's assistant for several years and together they collaborated on the texts: Trigonometria sferica Ⓣ(Spherical trigonometry); Trigonometria piana Ⓣ(Plane trigonometry); Aritmetica pratica Ⓣ(Practical arithmetic); Elementi di geometria Ⓣ(Elements of Geometry); Elementi di algebra Ⓣ(Introduction to Algebra); Elementi di analisi matematica per gli Istituti tecnici Ⓣ(Elements of mathematical analysis for technical colleges); and Elementi di matematica finanziaria ed attuariale per gli Istituti tecnici commerciali Ⓣ(Elements of financial and actuarial mathematics for commercial technical institutes) which were all published in 1937.
• 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.
• Under de Franchis' leadership the society had struggled, as we pointed out above, through no fault of its leader.
• 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.
• 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.
• 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.
• Some of de Franchis's results seem to suggest still future extensions which can reveal themselves to be useful for modern algebraic geometry.

Born 6 April 1875, Palermo, Sicily, Italy. Died 19 February 1946, Palermo, Sicily, Italy.

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References

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive