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.

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.

Oscar Zariski started to work in the 1930s on a more refined theory of birational mappings, incorporating commutative algebra methods.