when is the ideal $I(\overline{V})$ of the projectivization of an affine variety generated by the homogenization of the generator of $I(V)$ Let $k$ be a field and $V \subset \mathbb{A}^n(k)$  an affine variety defined by polynomials $f_1,\cdots, f_k$. Under which conditions the ideal $I(\overline{V})$ of the projective completion $\overline{V}\subset \mathbb{P}^n(k)$  of $V$ is genereted by the homogenizations of $f_1,\cdots, f_k$ ?
What are the examples where it is not true?

It turns out that if the $f_i$ form a Grobner basis for $I(V)$, then the homogenizations of the $f_i$ will generate $I(\overline{V})$. You can find more information about this in Eisenbud's book on Commutative Algebra in chapter 15.10.5.
As for an example where homogenizing the generators of the ideal doesn't give the correct result, we can look at the twisted cubic. The set $V=\{(a,a^2,a^3)\in\mathbb{A}^3\}$ is an affine variety defined by the equations $y-x^2=0$ and $z-xy=0$ (they generate $I(V)$). The homogenizations of these generators are $wy-x^2$ and $wz-xy$. However, the variety in $\mathbb{P}^3$ defined by these two equations consists of $\overline{V}$ and the line defined by $w=y=0$ (look in the $U_w$ chart for example to see that the line $w=y=0$ 'at infinity' is an extra component).
