The group law for the Jacobi variety of plane curves

We extend the group law of curves of degree three by chords and tangents to the Jacobi variety of plane curves of degree n>4 by replacing points by point groups and lines by algebraic curves. The curves are nonsingular or have simple singularities. In the cases n=4,5,6 we have a close analogy to the case n=3. We describe an algorithm using Groebner bases.