On the neutral component of the Jacobian of a real algebraic curve having many components

Abstract

Let C be a real algebraic curve of genus g ≥ 1 having at least g real components. We show that there is an embedding of C into ▪ 2 g as a curve of degree 3 g which induces a group structure on a connected component X of the set of effective divisors on C of degree g. Moreover, after having chosen a base point O ϵ X, there is a natural isomorphism of X onto the neutral real component of the Jacobian of C. This furnishes an explicit description of the group structure on the neutral real component of the Jacobian of a real algebraic curve of genus g ≥ 1 having many real components. If g = 1, one recovers the geometric description of the group structure on the neutral real component of a real elliptic curve.