I've got to show that any irreducible conic in $\mathbb{P^2(\mathbb{K})}$ where $\mathbb{K}$ is algebraically closed is isomorphic as a quasiprojective variety to the projective line. I wrote the conic as equivalent to the zero locus of $ x^2 +y^2+z^2$ in $\mathbb{P^2(\mathbb{K})}$. I tried to use the standard affine charts to cover this simpler variety and parametrize it but i didn't manage to do that. What am i missing?

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    $\begingroup$ $(\frac{t^2-1}{t^2+1},\frac{2t}{t^2+1})$ is a parametrization of $x^2+y^2-1=0$, so what about $(t^2-s^2:2st:i(t^2+s^2))$, where $i$ is a solution of $u^2+1=0$? $\endgroup$ – Jan-Magnus Økland Jan 19 '18 at 11:54
  • $\begingroup$ And what if $K$ has characteristic 2? $\endgroup$ – quantum Sep 29 at 7:01

For $\mathbb K=\mathbb R$ your specific conic $x^2+y^2+z^2=0$ seems like a poor choice, since it does not contain any points at all.

The general approach I'd suggest is the following: pick three distinct points $A,B,C$ on the conic to establish a projective basis, just as you would pick three points on a projective line to establish a basis there. Also pick a fourth point $P$ on the conic; it does not matter where as long as it is distinct from $A,B,C$. Then you can characterize every point $D$ on the conic via the cross ratio seen from $P$:


The value of this fraction does not depend on $P$, except that for $D\sim P$ you would divide zero by zero. But choosing a different $P$ can remove this singularity. You could treat numerator and denominator as two homogeneous coordinates describing a point $D$ on a projective line.

For the special case of the conic $x^2+y^2-z^2=0$ i.e. the unit circle, with $A=[-1:0:1],B=[1:0:1],C=[0:1:1]$ this corresponds to the tangent half-angle substitution:

$$f(D)=\frac{u}{v} \quad\Leftrightarrow\quad D \sim [v^2-u^2:2uv:v^2+u^2]$$

If you really want a map for $x^2+y^2+z^2=0$ over a field where this has some points to begin with, using those points as $A,B,C,P$ should give you a suitable parametrization, too. Over $\mathbb C$ you could multiply all $z$ coordinates by $i$, as Jan-Magnus Økland suggested in a comment.

See also this answer of mine for some more background, based on the related question of how to parametrize a conic. Also see this question for how to map one conic to another, to see why having an isomorphism for one non-degenerate conic will yields an isomorphism for others as well.

  • $\begingroup$ Yeah, i was really concerned only about the algebraically closed case, but your answer was pretty useful. Thank you. $\endgroup$ – Tommaso Scognamiglio Jan 21 '18 at 9:16

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