Let $\mathbb Z_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,x_6]}{\langle x_1x_6+x_2x_5-x_3x_4\rangle}\right)^{\mathbb Z_2}\right)$ ?
I have tried to compute the invariant polynomials: I found that $x_3^2$, $x_4^2$, $x_1x_6$, $x_2x_5$, $x_3x_4$, $x_1-x_6$ and $x_2-x_5$ are invariants. But is this a minimal set of generators ?
The relation $x_1x_6+x_2x_5-x_3x_4=0$ is again invariant. Can I say that $$ \left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,x_6]}{\langle x_1x_6+x_2x_5-x_3x_4\rangle}\right)^{\mathbb Z_2}=({\mathbb C[x_1,x_2,x_3,x_4,x_5,x_6]})^{\mathbb Z_2} \, ? $$
What is the projective variety ?