On symmetric products of curves

Abstract

In view of the increasing fashionability of symmetric products these days, we would like to note here that certain proofs in the algebro-geometric literature about curves and their Jacobians gain somewhat in clarity if they are placed on the symmetric product of the curve with itself. At the same time one learns something about the symmetric products. We have in mind Weil's proof (still the most elementary one) of the Castelnuovo-Severi inequality lying at the base of the Riemann hypothesis in function fields, Matsusaka's proof of the intersection relations among the Wi on a Jacobian, and the classical theory of Weierstrass points on a curve. In the first two cases, the proofs given here are essentially the original ones, only cleaned up a little (though we do get a slight sharpening of the inequality as a by-product); in the third case, it is a question of giving the geometric significance of some calculations whose meaning I could never understand.