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 ?



proj of an Algebra
You can diagonalise the action on $(x\_1,\ldots,x\_6)$ by the change of variables 
$$(u\_1,\ldots,u\_6):=(x\_1+x\_6,x\_2+x\_5,x\_3,x\_4,x\_2-x\_5,x\_1-x\_6).$$ Then the invariants are $u\_5,u\_6$ and $A\_{ij}=u\_iu\_j$ for $1\leq i \leq j \leq4$. 


Proj of this ring gives $X\subset \mathbb{P}^{11}\_{A\_{11},\ldots,A\_{44},u\_5,u\_6}$ defined by the $2\times 2$ minors of the matrix: 
$$\bigwedge^2\begin{pmatrix} 
A\_{11} & A\_{12} & A\_{13} & A\_{14} \\
A\_{12} & A\_{22} & A\_{23} & A\_{24} \\
A\_{13} & A\_{23} & A\_{33} & A\_{34} \\
A\_{14} & A\_{24} & A\_{34} & A\_{44} \\
\end{pmatrix} = 0$$
and your relation $x\_1x\_6 + x\_2x\_5 - x\_3x\_4=0$ becomes $A\_{11} - u\_6^2 + A\_{22} - u\_5^2 - 4A\_{34} = 0$. (In particular you could use it to eliminate a variable, e.g. $A\_{11}$, to get a variety in $\mathbb{P}^{10}$.)


