Group law on the neutral component of the Jacobian of a real hyperelliptic curve having many real components

Let C be a real algebraic curve of genus g with at least g real components B1,…,Bg. We give an embedding of C into a blow up in one point of the projective plane. It allows us to describe geometrically the neutral real component Pico(C)o of the Jacobian of C thanks to an isomorphism with the product B1× ×Bg. This induces an explicit geometric description of Pico(C)o in the projective plane, where the group law is given by intersection with curves of genus g.