Mumford, David (Author)

Notices of the American Mathematical Society

Volume: 58, no. 2

Issue: 2

Pages: 250-260

Publication date: 2011

Language: English

Publication Date: 2011

Edition Details: Part of a special issue on the history of algebra

Edition Details: Part of a special issue on the history of algebra

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In the preceding article we have seen that Enriques and, indeed, the whole Italian school of algebraic geometry in the rst half of the twentieth century were frustrated by one glaring gap in their theory of algebraic surfaces. This magni cent theory answered essentially all the basic questions about algebraic surfaces and had been constructed using purely 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,1 so the theory was sound---but this approach was alien to their intuitions. It was much 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 rst requirement in moving ahead had been the introduction of new algebraic tools into the subject rst by Zariski and Weil and subsequently by Serre and Grothendieck.

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