I am trying to understand varieties $X’$ that are obtained by twisting a variety $X$ over a field $k$ by a cocycle $\gamma\in H^1(k^{sep}/k, Aut(X))$.
To make this feel concrete, I would like to find a defining equation in $\mathbb{P}^3$ for the form of $X=(\mathbb{P}^1\times\mathbb{P}^1)/\mathbb{R}$ where the cocycle is given by mapping $Aut(\mathbb{C}/ \mathbb{R})$ to the automorphism of X that swaps fibers.
I know a priori that this variety is the Weil restriction of $\mathbb{P}^1_\mathbb{C}$ to $\mathbb{R}$, so I should expect to get something like a sphere.
Here is my attempt so far:
First I use a segre embedding $[x_1:y_1;x_1:y_1]\mapsto [x_1x_2:x_1y_2:y_1x_2:y_1y_2]$ to embed into $\mathbb{P}^3$. It is clear that this gives the variety $V(bc-ad)$ in $\mathbb{P}^3=\mathbb{P}(a:b:c:d)$ and that switching fibers in $X$ amounts to swapping $b$ and $c$.
This means that the real points of $X’$ are determined(as a subvariety of $\mathbb{P}^3$) by the equations
$bc-ad=0$
$a=\lambda \overline{a}$
$b=\lambda \overline{c}$
$c=\lambda \overline{b}$
$d=\lambda \overline{d}$
where $\lambda\in \mathbb{C}^*$.
I’ve tried working with these equations, but I’m not really sure where to go from here, as these equations are not even algebraic.