Let $I,J$ be homogeneous ideals in the algebra of polynomials in $n$ variables over the complex numbers. Let $V(I)$ and $V(J)$ be the affine algebraic varieties that are determined by $I$ and $J$ (not the projective varieties). Suppose that $V(I)$ and $V(J)$ are isomorphic as algebraic varieties. By this I mean that there are polynomial maps $f$ and $g$ from $\mathbb{C}^n$ to itself, such that $f$ restricted to $V(I)$ is a bijection onto $V(J)$, and such that $g$ restricted to $V(J)$ is its inverse.
The question is this: does it follow that there exists a linear map on $\mathbb{C}^n$ that maps $V(I)$ onto $V(J)$?
Thanks to discussions with colleagues (thank you David and Mike), I am quite convinced that if we assume that the origin is the only singular point in $V(I)$ then the answer is yes. Is this true in general?
I think this question is equivalent to the following (see my partial answer below): Is it true that whenever there is an isomorphism between $V(I)$ and $V(J)$, there is also isomorphism that fixes $0$?
