Assume to have a morphism $f: \mathbb{A}^n \to \mathbb{P}^m$, I want to compute the ideal of the the projective closure of $ f(\mathbb{A}^n)$. I think that in general the image of an affine variety trought a morphism $f(x)=(f_1(x), \dots, f_m(x))$ is $f(X)= Z(y_0 -f_0(x), \dots , y_m-f_m(x)) $ where $y_i$ are the coordinates in the target.

In the above situation $f(\mathbb{A}^n)= Z(x_0 -f_0(x), \dots , x_m-f_m(x)) $ and $x \in \mathbb{A}^n $ which is an affine algebraic set and for taking the projective closure I have to homogenize the $ x_i -f_i(x)$. Then the ideal it will be the radical of the homogenized ideal obtained. Is my argument correct? Thank you for the answers!


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