Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in Jacobian $Pic^0(X)$ and they are distinct? (Since there are 16 2-torsion points, we find all of them.)
My professor says one can prove that $P_i-P_j$ is a 2-torsion points by some argument involving divisor, and he also mentions one can prove $P_i-P_j$ are distinct by using the fact that a genus 2 curve has one degree-2 map to $\mathbb P^1$.
I just have no idea, so I am not able to say more. Could any one help me?