I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows:
Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ is a field and $\mathbb P_2(k)$ is the projective space.
There exists (because $GL_3(k)$ acts transitively on $k^3$) a $\phi\in GL_3(k)$ such that $\phi(x_0,y_0,w_0)=(0,0,1)$.
This defines an imbedding:
$$k^2\rightarrow \mathbb P_2(k)$$ $$(x,y)\mapsto \phi^{-1}(x,y,1)$$
with $i_{\phi}{([0,0])}=[x_0,y_0,w_0]$ and $\text{Image}(i_{\phi})=\phi^{-1}(k\times k \times \{1\})$.
Then the inverse $\Phi$ of $i_{\phi}$
$\Phi:\phi^{-1}(k^2\times \{1\})\rightarrow k^2$
defines local coordinates $(x,y)$ around $[x_0,y_0,w_0]$
My questions:
- How can I imagine this imbedding? What does it do?
- Why and how can this imbedding be useful?
- Why we are asking for a $\phi$ such that $\phi(x_0,y_0,w_0)=(0,0,1)$?
- What are local coordinates? Its the first time hearing this. How they are useful?
Thanks in advance!
That depends on how you visualize $\mathbb{P}_k^2$. One way to think of it is that it is an affine plane, i.e. $k^2$, together with an extra (projective) "line at infinity".
To make this precise, one identifies (or maps) a point $(x, y) \in k^2$ with a point $[x, y, 1] \in \mathbb{P}_k^2$. Then the rest of the points of the projective plane are of the form $[x,y,0]$ for some $x, y$. In other words, these are precisely the points satisfying the equation $Z=0$, which is the mentioned extra line. Note that under this map, the point $[0,0,1]$ is somewhat distinguished since it corresponds to the origin of tha affine plane, i.e. $(0,0) \in k^2$.
Now, what is this business with $\Phi$ and $\phi$ about is that one tries to describe the same construction, but "around a point $[x_0, y_0, z_0]$ instead of $[0, 0, 1]$". That is, if one wants to treat the point $[x_0, y_0, z_0]$ as the origin, and not the point $[0, 0, 1]$, this is the way how to alter the construction.
What are these cordinate maps for:
Say you have a projective algebraic curve $$C=\{f(X,Y,Z)=0\}.$$ Then, using the map described above (i.e. the one identifying $(0,0)$ with $[0,0,1]$), one can think of the curve C as the affine plane curve $$C_a=\{(x,y) \in k^2 \;| \;f(x,y,1)=0\}$$ together with several "points at $\infty$" (these correspond to the points of the form $[x,y,0] \in C$, i.e. these are precisely those points of the curve that are not in the image of the chosen map). This way, if you want to see whether the curve $C$ is non-singular at a point $[x,y,1]$, it is enough to do this for the corresponding point on the curve $C_a$, i.e. compute the gradient $$(\frac{\partial f}{\partial X}(x,y,1), \frac{\partial f}{\partial Y}(x,y,1))$$ and decide whether it is $(0,0)$ or not.
The "shift of the origin", i.e. local coordinates around fixed point $[x_0, y_0, y_0]$ can be used if you are, say, interested in the local behaviour at a point $[x_0, y_0, z_0]$ and it for example does not fall into the image of the map above, i.e. $z_0=0$.
It is further useful for mor subtle stuff - say your algebraic curve is singular at a point $[x_0, y_0, z_0]$, and you want to still compute some kind of tangents - for example, if the curve intersects itself at the point $[x_0, y_0, z_0]$, what you are interested in are the tangents of the two "branches" that are intersecting each other. This can be computed/defined, but most naturally it is done for a point $(0,0)$ lying on an affine curve - hence the transformation described in your post. It allows you to treat the point as an origin of affine plane, compute stuff and then translate it back to $\mathbb{P}_k^2$.