The addition on Jacobian varieties from a geometric viewpoint

Abstract

We give a geometric interpretation of the group law for Jacobian varieties by extending the geometric construction of chords and tangents on an elliptic curve. For any given algebraic curve $\mathcal{X}$ and reduced divisors $D_1, D_2 \in \mbox{Jac } \mathcal X$, we define curves ${\mathcal{X}}^{\prime }$ and ${\mathcal{X}}^{"}$ such that the intersection $\mathcal{X}\cap {\mathcal{X}}^{\prime }$ determines precisely the divisor $-(D_1+D_2)$ and the intersection $\mathcal{X}\cap {\mathcal{X}}^{"}$ determines $D_1+D_2$. For superelliptic curves such formulas are made explicit.