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?
Best Answer
The answer to 1) is NO:
For example all hyperplanes $V_+(<l>)\subset \mathbb P^n$ where $l=\sum a_ix_i$ is a non-zero linear form are isomorphic, whereas the ideals $<l>$ are different.
And I'm not sure there is a very reasonable answer to 2) since (for example) the subschemes $V_+(I)$ and $V_+(J)$ might be isomorphic and not even have the same degree:
For example any smooth conic in $\mathbb P^2$, say $V_+(x_0^2+x_1^2+x_2^2)$, is of degree $2$ and is isomorphic to any line, say $V_+(x_0)$, which is of degree $1$.
However I see no obvious relation between the ideals $<x_0^2+x_1^2+x_2^2>$ and $<x_0>$ .