Segre, Castelnuovo, Enriques: Missing Links

At the end of the 1880s, Segre guided Castelnuovo’s research towards the geometry of algebraic curves, introducing Castelnuovo, whose earlier studies had been focused on n-dimensional projective geometry, to birational geometry, which is the starting point of the Italian school of algebraic geometry. After graduating and attending one post-graduate year at the University of Pisa, Enriques got in touch with Segre, aiming to spend one year in Turin. He was attracted by the reputation of the young master and, perhaps, by the mathematical environment of Turin University, which was particularly lively in those years. Contrary to his expectations, Enriques was sent to Rome, where in the meantime Castelnuovo had moved. He began to study the birational geometry of algebraic surfaces under Castelnuovo’s direct supervision. Segre followed Enriques’s first results but he was committed to finding a rigorous proof of the theorem of resolution of singularities of algebraic surfaces. His article on singularities was concluded in December 1896 and in the following year his student Beppo Levi completed the proof of the resolution theorem. Meanwhile, Enriques had laid the foundations of the general theory of linear systems of curves on algebraic surfaces and Castelnuovo had proved his famous rationality criterion. The link between Segre, Castelnuovo and Enriques could have turned into a scientific partnership: at the end of 1896, the three geometers planned to collect their results in a general treatise on the theory of algebraic varieties. However, this treatise was never realised.