Polynomials and ideals of $k[X_1, \cdots, X_n]$ resp. $k[X_1, \cdots, X_n, Z]$ are considered, with $k$ an arbitrary field. Only classical algebraic geometry is considered (no schemes).
Definitions: $f^*$ is the homogenization of $f$ relatively to $Z$; $I^*$ is the ideal generated by $\{ f^* | f \in I \}$; $V^* = V(I(V)^*)$.
Question 1: Is $\sqrt{(I^*)} = (\sqrt{I})^*$ true ?
Question 2: Is $V(I)^* = V(I^*)$ true if $k$ is algebraically closed ? Or do only their affine parts ($Z = 1$) agree ?
Question 3: Is there a counter-example of the latter if $k$ is not algebraically closed (but possibly infinite) ?
I do not necessarily need detailed proofs, but I would like to know if those claims are correct or not (possibly with some hints or references).