We work over the complex numbers.
If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ Proj$(A/I)$.
My question is about how many different ideals can define the same scheme, up to isomorphism.
1) Suppose that $I$ and $J$ are homogeneous ideals of $A$ such that $V_+(I)$ and $V_+(J)$ are isomorphic as closed subschemes of projective space. Is $I=J$?
2) Suppose now that $V_+(I)$ and $V_+(J)$ are isomorphic as varieties, but not necessarily as closed subschemes. How are $I$ and $J$ related?