ABSTRACT
1. The period we reach now is without any doubt the most important of all in the history of algebraic geometry to this day. It is entirely stamped by the work of one man, Bernhard Riemann, one of the greatest mathematicians who ever lived, and also one of those who have had, most profoundly, the perception (or divination) of the essential unity of mathematics. It is quite a paradox that in the work of this prodigious genius, out of which algebraic geometry emerges entirely regenerated, there is almost no mention of algebraic curve; it is from his theory of algebraic functions and their integrals that all of birational geometry of the nineteenth and the beginning of the twentieth century issues. We will see that, without Riemann himself having been clearly conscious of it, two of his other celebrated memoirs, that on the Hypotheses of geometry (i.e., “riemannian spaces”) and that on the zeta function, would open new horizons to modern algebraic geometry from 1920 onward.
