An Attack on the Theory of Linear Systems 1950–1956

With the results of his foundational work on varieties and, as he writes in the preface to his Collected Papers, “with the conviction, indelibly impressed in my mind by my Italian teachers, that the theory of algebraic surfaces is the apex of algebraic geometry,” Zariski decided that working out a complete theory of surfaces would be the real test ing ground for his new algebraic tools: “If he could find rigorous proofs of the results in that body of theory and if he could, moreover, extend those results to characteristic p, then it would show that he and Weil had provided proper foundations for the whole theory of algebraic geometry.”¹¹⁵ His work was also an answer to a challenge given by Castelnuovo in 1949: “Will someone come soon to continue the work of the Italian and French School by giving to the theory of algebraic surfaces the perfection that has been reached in the theory of algebraic curves? I hope so, but I doubt it.”

Having developed definitive tools for proving the necessary results, Zariski and Weil were now faced with the question of how to apply these tools. “To do mathe matics efficiently one needs not only powerful rigorous techniques but also an intuitive flexible language—that is to say, an array of definitions of technical terms that readily support both your intuition and your analysis.”¹¹⁷ The basic problem was to find a way of doing algebraic geometry that could use all the powerful tools of Weil and Zariski while capturing the intuitive geometric feel of the old Italian School and it remained an important issue throughout the fifties and sixties.