Enriques was born in
Livorno, and brought up in
Pisa, in a
Sephardi Jewish family of
Portuguese descent. His younger brother was zoologist
Paolo Enriques who was also the father of Enzo Enriques Agnoletti and
Anna Maria Enriques Agnoletti. He became a student of
Guido Castelnuovo (who later became his brother-in-law after marrying his sister Elbina), 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. In 1931, he swore allegiance to fascism, and in 1933 he became a member of the PNF. Despite this, he lost his position in 1938, when the
Fascist government enacted the "leggi razziali" (racial laws), 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 (these include
quadrics and
cubic surfaces in projective 3-space) have the simplest geometry.
Quartic surfaces in 3-spaces are now classified (when
non-singular) 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 (so there is a bundle that is
locally trivial actually over a curve less some points). The question of classification is to show that any surface, lying in
projective space of any dimension, is in the birational sense (after
blowing up and blowing down of some curves, that is) accounted for by the models already mentioned. No more than other work in the Italian school would the proofs by Enriques now be counted as complete and
rigorous. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples.
Oscar Zariski started to work in the 1930s on a more refined theory of birational mappings, incorporating
commutative algebra methods. He also began work on the question of the classification for
characteristic p, where new phenomena arise. The schools of Kunihiko Kodaira and
Igor Shafarevich had put Enriques' work on a sound footing by about 1960. == Works ==