Intuition and Rigor and Enriques’s Quest

I n the preceding article we have seen that Enriques and, indeed, the whole Italian school of algebraic geometry in the first half of the twentieth centurywere frustratedbyone glaring gap in their theory of algebraic surfaces. This magnificent theory answered essentially all the basic questions about algebraic surfaces and hadbeenconstructedusingpurely geometric tools. But one of its central theorems seemed to defy all their attempts to give it a geometric proof. It had been proven by analytic means by Poincaré with his theory of “normal functions”, so the theory was sound—but this approach was alien to their intuitions. It wasmuch like the need for analysis in proving the prime number theorem before Selberg found his elementary proof. In my own education, I had assumed they were irrevocably stuck, and it was not until I learned of Grothendieck’s theory of schemes and his strong existence theorems for the Picard scheme that I saw that a purely algebrogeometric proof was indeed possible. I say here “algebro-geometric”, not “geometric”, because the first requirement in moving ahead had been the introduction of new algebraic tools into the subject first by Zariski and Weil and subsequently by Serre and Grothendieck. When Professors Babbitt and Goodstein wrote me about Enriques’s work in the 1930s, I realized that the full story was more complex. As I see it now, Enriques must be credited with a nearly complete geometric proof using, as did