The addition on Jacobian varieties from a geometric viewpoint

TL;DR

This paper provides a geometric interpretation of the group law on Jacobian varieties by extending classical constructions from elliptic curves to more general algebraic curves, including explicit formulas for superelliptic cases.

Contribution

It introduces a geometric framework for understanding the addition law on Jacobian varieties, generalizing chord-and-tangent methods to arbitrary algebraic curves.

Findings

Defines curves that determine divisor sums via intersections

Extends classical elliptic curve methods to superelliptic curves

Provides explicit formulas for divisor addition in specific cases

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.