Let $S$ be the sub-algebra generated by the set $\{x\_1x\_2y\_3y\_4, x\_1x\_3y\_2y\_4, x\_1x\_4y\_2y\_3, x\_2x\_3y\_1y\_4, x\_2x\_4y\_1y\_3, x\_3x\_4y\_1y\_2\}$ of homogeneous polynomials of degree $4$. I need to study the projective variety $Proj(S)$. The relation among the above set of polynomials are $f\_1f\_6=f\_2f\_5=f\_3f\_4$ in the above order. So the projective dimension is $3$. Looks like this is a $3$ dimensional sub-variety of $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ but what is this variety ? 



sub-variety of (P^1)^4
You can think of this ring as the semigroup ring of the semigroup $S$ generated by
$$(1,0,0,1),(-1,0,0,1),(0,1,0,1),(0,-1,0,1),(0,0,1,1),(0,0,-1,1).$$
The above semigroup elements correspond to $f\_1,f\_6,f\_2,f\_5,f\_3,f\_4$ respectively.


I believe that after saturation at the origin the semigroup ring is the polytope ring
of the octahedron. Then the corresponding toric variety has a fan based on the cube of size $2$. This is a rather singular Fano variety of Picard number 5. It has $12$ lines worth of $A\_1$ singularities and $6$ rather complicated Gorenstein singularties.
I don't see it as lying inside $(\mathbb P^1)^4$, because the algebra is not a quotient of the homogeneous ring of the latter, but rather is a subalgebra of it.


