Federigo Enriques ’ s Quest to Prove the “ Completeness Theorem ”

T he golden age of the Italian school of algebraic geometry began with Antonio Luigi GaudenzioGiuseppeCremona and included among its main contributors Enrico Castelnuovo, Federigo Enriques, and Francesco Severi. The Italian school spanned nearly a century, from the unification of Italy in 1861 to Enriques’s posthumously published postWorld War II monograph on algebraic surfaces [Enrq 49]. In the 1890s Enriques, a mathematician who once quipped that “intuition is the aristocratic way of discovery, rigour the plebian way” [Hodge 48], and his colleague and future brother-in-law Castelnuovo began their monumental work on the birational theory of algebraic surfaces over the complex numbers C. Severi joined them in this effort a few years later. Broadly speaking, the aspect of algebraic surfaces that will concern us here can be traced back to Rudolf Friedrich Alfred Clebsch, Arthur Cayley, and Max Noether, who in 1868–1875 introduced two different genera for characterizing an algebraic surface F . The first (proposed by Clebsch) was the dimension of the space of algebraic regular (i.e., without poles or, over C, holomorphic) 2-forms on F , a direct analog of the genus of an algebraic curve C, i.e., the dimension of the space of algebraic regular 1-forms on C. Shortly after, Arthur Cayley, seeking an easier way to calculate Clebsch’s genus, introduced an expression in the degree and characteristics of the singularities of a generic projection of the surface to P3 which he hoped gave the samenumber. But, in fact, it turned out that the second could be negative, unlike the