Change of coordinates (algebraic variety)

Question

I'm studying the change of coordinates in Fulton's Algebraic curves:

Fulton's book is sometimes a little "dry", I'm confused, intuitive speaking what exactly is $F^T$? anyone could give me a concrete example of $T, T'$ and $T''$?

Anyone knows more detailed materials about this stuff?

I really need help.

Thanks a lot.

An example is $T=(2xy, x^2-y^2, x^2+y^2)$. – Aug 28 '13 at 05:13 — , Aug 28 '13 at 05:13
@Hurkyl but $x^2 -y^2$ and $x^2+ y^2$ don't have degree 1. – user42912 Aug 28 '13 at 06:13 — user42912, Aug 28 '13 at 06:13

score 2 · Accepted Answer · answered Aug 28 '13 at 15:07

Let's keep it very simple. Let $T\colon\mathbb A^1\to\mathbb A^1$ be a linear change of coordinates. This means that $T = (T_1)$ where $T_1$ is a linear polynomial in the variables of the domain $\mathbb A^1$. Let the coordinate be $x$. Then we can write $T_1 = ax + b$ for some $a,b$ in our base field. That is $T\colon\mathbb A^1\to \mathbb A^1, x\mapsto ax+b$. Note that this is a composition $T = T''\circ T'$, where $T'$ is multiplication by the $1\times 1$ matrix $[a]$ and $T''$ is the translation $(-) + b$. In this case, $T$ is bijective if and only if $T'$ is, which means that we must have $a\neq 0$.

For any $F$, we have $F^T = F\circ T$. For example, if $F = x^2$ and $T$ is as above, then $F^T = (ax + b)^2.$ Intuitively speaking, $F^T$ is the polynomial function $F$ shifted by the new coordinates determined by $T$. For example, $F$ has a zero at $x=0$, and this zero shifts under $T$ to the zero $-b/a$ of $F^T$. The general case works in the same way.

I've just posted a really similar question http://math.stackexchange.com/questions/478634/help-in-these-really-elementary-definitions-in-algebraic-geometry if you could help me again, I would be really grateful. Thanks again! — user42912, Aug 29 '13 at 01:14

Change of coordinates (algebraic variety)

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