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We are working in the projective space $RP^2$. For a general element $x \in RP^2$ we use the notation $x := [(x_0, x_1, x_2)].$ In $RP^2$ we typically choose the line $x_2 = 0$ to be the 'line at infinity'. Then all points in $RP^2$, which are not at infinity, satisfy $x_2 \not = 0$. We can then rewrite all these points in the form $A := [(y_0, y_1, 1)]$, by linearity of the coset. By this identification we can consider the projective point $A$ to be the point $(y_0, y_1)$ in the affine space.

Now suppose that I choose a different line to be the line at infinity. Let $l_1 \subset RP^2$ be given by the condition that $x_0 + x_1 = 0$. In this case, I am very confused on how I can construct affine points and how we can find coordinates for them.

Any help would be greatly appreciated.

LotrFanAndMath
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  • Can you construct an automorphism of $\mathbb P^2$ sending one line to the other? Hint: there will be a linear automorphism. – paul blart math cop Jul 16 '21 at 00:20
  • I think the projective transformation $\phi : RP^2 \to RP^2 : [(x_0, x_1, x_2)^T] \mapsto [\begin{pmatrix} 0 &-1& 1\ 0 &1& 0\ 1 &-1& 0 \end{pmatrix} (x_0, x_1, x_2)^T]
    $

    will send the line $x_2 = 0$ to the line $x_1 + x_2 = 0$. So is it then sufficient to apply this transformation to the points that become affine points by choosing the line $x_2 = 0$ as the line at infinity?

     – LotrFanAndMath Jul 16 '21 at 08:31
  • I haven't checked the actual matrix you gave, but it comes down to pretty simple linear algebra. It's equivalent to ask about a linear automorphism of $\mathbb A^3$ (an actual vector space) sending the plane $x_2=0$ to the plane $x_1 + x_2 = 0$. This is elementary linear algebra which I trust you can do. You can often analyze projective varieties via lifting to the "affine cone." This turns problems about linear varieties into normal linear algebra. – paul blart math cop Jul 16 '21 at 09:10
  • I was able to find the solution. Thank you very much! – LotrFanAndMath Jul 16 '21 at 22:12

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We can simply construct a projective transformation which maps the line $l_1 <-> x_0 + x_1 = 0$ to the line $ l_2 <-> x_2 = 0$, call such a transformation $\phi$. When the line at infinity is $l_2$, we can easily define a bijection between projective points not contained in $l_2$ and affine points, call this bijection $f$. We now find that the bijection $f \circ \phi$ which will give affine coordinates to each projective point in $RP^2$ which is not contained in $l_1$.

LotrFanAndMath
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