In the context of the Exercise 5.3.D in Vakil's notes, I want to show that there are examples of a reduced graded ring $A$ and a non-radical homogeneous ideal $I$ such that $\text{Proj}(A/I)$ is a reduced scheme.
In particular I am considering, as suggested, $A=k[x_0,\dots,x_n]$ and $I=(x_0^2,x_0 x_1)$. I take the affine open cover given by $U_i\simeq \text{Spec}((A/I)_{x_i})_{\text{deg}=0}$, where $((A/I)_{x_i})_{\text{deg}=0}$ is the $0$-graded piece of $(A/I)_{x_i}$. Then it is enough to show that the $U_i$ are reduced, since those are affine I am left to show that the rings $((A/I)_{x_i})_{\text{deg}=0}$ are reduced.
I tried to do this explicitly but I am stuck. Moreover if I consider $\frac{x_0}{x_2}$ in $((A/I)_{x_2})_{\text{deg}=0}$, this looks to me as a non-zero nilpotent element. Where is my mistake?