Some elementary mathematics on curves

Abstract

So far our efforts have principally been directed toward getting information about algebraic varieties. But one can regard varieties themselves as “spaces on which one does mathematics.” The reader has met this idea before. For instance one can transfer analysis on ℝ to analysis on the more general real variety ℝ n , getting, for example, multivariable calculus. And in complex analysis one regards the variety ℂ (or more generally ℂ n ) as a carrier of analytic functions; one then studies differentiation and integration of these functions, and so on. One can also do analysis on real differentiable manifolds or complex analytic manifolds. (See, for instance, [Narasimhan], [Spivak], or [Wells].) Such a study in turn often sheds new light on the underlying space; algebraic geometry is no exception in this respect. In such questions, algebraic varieties occupy a special position; they have so much structure that one can transfer to them many ideas not only from analysis, but also from number theory, and these generalizations interconnect with and enrich each other. Transferring differentials and integration to non-singular complex varieties is particularly natural, and in fact in its early days, algebraic geometry was regarded as a part of complex analysis (by Abel (1802–1829), Riemann (1826–1866), Weierstrass (1815–1897), etc.). More recently, attempts to transfer notions from topology and analysis to appropriate generalizations of algebraic varieties have met with varying degrees of success, and have for example already shed new light on some old problems in number theory. In applying algebraic geometry to other fields in this way, one often needs to translate classical geometric results into ring-theoretic terms.