In Italy, between the end of the 19th and the first half of the 20th century, flourished a school of algebraic geometry whose main authors were Giuseppe Veronese, Corrado Segre, Guido Castelnuovo, Federigo Enriques, and Francesco Severi, a school whose main characterization was its strong appeal to geometric intuition, and it was more and more evidently in contrast with the growing tendency in 20th century mathematics toward more algebraic and rigorous methods.
This school was the final avatar in Italy of the Galilean tradition characterized by a scanty interest toward algebra (a `kinderspiel’!) and a strong interest toward geometry and experimental physics.
This paper reconstructs the interweaving between the cultural adventure of this school and the troubles of Italian culture during fascism, for many great Italian mathematicians were Jewish and were forced to leave Italy before the second world war. In addition, the paper reports on two letters, the first from Severi to Segre, the second from Castelnuovo to Segre, concerning the contributions of the Italian school to the development of algebraic geometry.