The group law on the cubic curve and the Jacobian variety

The cubic curve in the projective plane (also known as an elliptic curve) is an extremely interesting object in mathematics because it is also a group. There are many ways to introduce the group structure. In this video, we introduce the most natural one via the group structure on its Picard group. We show that the set of points on the cubic curve is in natural bijection with the degree zero invertible sheaves so the group structure on the Picard group induces a group structure on the cubic curve and the geometry on the cubic curve induces a geometry on the Picard group, or at least the subgroup corresponding to degree 0 invertible sheaves. The latter is known as the Jacobian variety.