The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.
This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold language by Kunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, the Shafarevich school and others by around 1960.
Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity".
Reference to the Mathematics Genealogy Project shows that, in terms of Italian doctorates, the real productivity of the school began with Guido Castelnuovo and Federigo Enriques.
The roll of honour of the school includes the following other Italians: Giacomo Albanese, Eugenio Bertini, Luigi Campedelli, Oscar Chisini, Michele De Franchis, Pasquale del Pezzo, Beniamino Segre, Francesco Severi, Guido Zappa (with contributions also from Gino Fano, Carlo Rosati, Giuseppe Torelli, Giuseppe Veronese).
Elsewhere it involved H. F. Baker and Patrick du Val (UK), Arthur Byron Coble (USA), Georges Humbert and Charles Émile Picard (France), Lucien Godeaux (Belgium), Hermann Schubert and Max Noether, and later Oscar Zariski (United States), Erich Kähler (Germany), H. G. Zeuthen (Denmark).
This has been a favorite study during the last fifty years, of the Italian geometers, and they have also made contributions of great beauty to a similar theory of surfaces and of “Varieties” of higher dimensions.