Federigo Enriques

Who was Federigo Enriques?

Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.

He was born in Livorno, and brought up in Pisa, in a Sephardi Jewish family of Portuguese descent. He became a student of Guido Castelnuovo, and became an important member of the Italian school of algebraic geometry. He also worked on differential geometry. He collaborated with Castelnuovo, Corrado Segre and Francesco Severi. He had positions at the University of Bologna, and then the University of Rome La Sapienza. He lost his position in 1938, when the Fascist government enacted the "leggi razziali", which in particular banned Jews from holding professorships in Universities.

The Enriques classification, of complex algebraic surfaces up to birational equivalence, was into five main classes, and was background to further work until Kunihiko Kodaira reconsidered the matter in the 1950s. The largest class, in some sense, was that of surfaces of general type: those for which the consideration of differential forms provides linear systems that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surfaces and more generally ruled surfaces have the simplest geometry. Quartic surfaces in 3-spaces are now classified as cases of K3 surfaces; the classical approach was to look at the Kummer surfaces, which are singular at 16 points. Abelian surfaces give rise to Kummer surfaces as quotients. There remains the class of elliptic surfaces, which are fiber bundles over a curve with elliptic curves as fiber, having a finite number of modifications. The question of classification is to show that any surface, lying in projective space of any dimension, is in the birational sense accounted for by the models already mentioned.