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Based on this great answer of my own post: Change of coordinates (algebraic variety) which the answerer really give me concrete examples of the $T$ of the second paragraph. I'm with the same kind of doubts in the first paragraph.

I understood that $F^T$ is the polynomial after the change of coordinates, but I didn't understand what is $T^{-1}(V)$, is it just a notation or is the pre-image of $V$?

I'm trying to found a concrete example of $V, I^T, V^T$:

Let $F\in k[X_1,X_2]$, (in this example $k=\mathbb R$) given by $F=X_1^2+X_2^2-1$ and the change of coordinates be given by $T(X_1,X_2)=(2X_1,2X_2)$, then we have:

$F^T=4X_1^2+4X^2_2-1$ and in our example, $V=V(F)$, then

$I^T$ is the ideal generated by $\{F^T=4X_1^2+4X^2_2-1\}$, since $I=I(V)=I(V(F))=F$, by definition.

Then $V(I^T)$ is the points which vanish $F^T$, so $V(I^T)$ is the circle around the origin with radius $1/2$.

My calculations in these examples are correct? these definitions are really elementary, I need to understand very well if I want to pursue in this field.

I really need help.

Thanks a lot

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user42912
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    Your calculations look mostly fine to me, though you should not say $I=F$. $I$ is an ideal, with generator $F$. And yes, $T^{-1}(V)$ means the inverse image of $V$ under the map $T$. – Andrew Aug 29 '13 at 15:08
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    @Andrew thank you for your remarks! – user42912 Aug 29 '13 at 16:23

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