The following is stated in this answer on "What is so special about elliptic curves?":

But for these curves there does not exist a nice geometric rule to add points, like in conics and elliptic curves. So we are forced to work in the Jacobian group of these curves, which is not the group of points anymore, but of divisors (which are kind of like polynomials of points, if that makes any sense).

I've got the following questions about those statements:

1. What makes us to work specifically in Jacobian Group and not some other group?
2. We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance $R_1 + (-R_1) = O$ etc. Why is that?

The following is stated in this answer on "What is so special about elliptic curves?":

But for these curves, an excellent geometric rule does not exist to add points, like in conics and elliptic curves. So, we are forced to work in the Jacobian group of these curves, which is not the group of points anymore but of divisors (which are like polynomials of points, if that makes any sense).

I've got the following questions about those statements:

1. What makes us work specifically in the Jacobian Group and not some other group?
2. We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance $R_1 + (-R_1) = O$ etc. Why is that?

This guy here: What is so special about elliptic curves? states this: "But for these curves there does not exist a nice geometric rule to add points, like in conics and elliptic curves. So we are forced to work in the Jacobian group of these curves, which is not the group of points anymore, but of divisors (which are kind of like polynomials of points, if that makes any sense)."

1. What makes us to work specifically in Jacobian Group and not some other group?
2. We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance R1 + (-R1) = O etc. Why is that?

The following is stated in this answer on "What is so special about elliptic curves?":

But for these curves there does not exist a nice geometric rule to add points, like in conics and elliptic curves. So we are forced to work in the Jacobian group of these curves, which is not the group of points anymore, but of divisors (which are kind of like polynomials of points, if that makes any sense).

I've got the following questions about those statements:

1. What makes us to work specifically in Jacobian Group and not some other group?
2. We care in Hyper Elliptic Curves to have points adding to "zero" (for genus = 2). So, for instance $R_1 + (-R_1) = O$ etc. Why is that?