On the Geometry of a Theorem of Riemann

Abstract

Let C be a smooth complete algebraic curve. Let I: C-+J be an universal abelian integral of C into its Jacobian J. Furthermore, let I(i): C(i) J be the mapping sending a point c1 + *-+ ci in the ith symmetric product C(i) to I(c1) + *-+ I(ci). Let Wi be the image of I(i) when i is less than the dimension of J. My purpose is to describe the tangent cone of Wi at any point. Let L(w) be the inverse image by I(i) of a point w in J. L(w) is a projective space representing a complete linear system of effective divisors on C. Geometrically, my main result is