This is Vakil 19.11 B, self-study.
Let $E$ be a smooth plane cubic in $\mathbb P^2_{\mathbb C}$. Then $E$ gives us a graded ring $S$. We know $E$ has a non-torsion point $p$. We wish to show that the map $t: E \to E$ given by translation by $p$ cannot come from a map of graded rings $S \to S$ even after regrading.
By putting the curve into Weierstrass form, we find
$$S = \mathbb C[x, y, z]/(zy^2 - x^3 - az^2x - bz^3)$$
We know any map of graded rings has to send $S_n \to S_{dn}$ for some fixed $d > 0$ and for all $n$. We know $t$ is an automorphism since it is a translation. I do not see how to connect all the pieces together to give the result. Strong hints are fine initially.