I'm currently working on understanding the affine cone over the Grassmannian, which according to my paper is given by
$$\text{Spec}(K[p_{ij}^{\pm} : ij \in \binom{\lbrack n \rbrack}{2} ] / I_{2,n}).$$
I know that for the standard projection $p: \mathbb{A}^{n+1} \setminus \{0\} \rightarrow \mathbb{P}^n$ and $Y$ a projective variety given by $Y=V(I)$ for some homogeneous ideal $I \subseteq K[T_0, \dotsc, T_n]$, the affine cone is given by
$$ C(Y)= p^{-1}(Y) \cup \{(0, \dotsc, 0)\}=V_\text{aff}(I) \cup \{(0, \dotsc, 0)\}.$$
So I tried to argue that, as we have $Gr(2,n)=V(I_{2,n})$ and that for a commutative unital ring $R$, the closed subset $V_\text{aff}(I)$ of Spec($R$) may be identified with Spec$(R/I)$, this would provide me with
$$ C(Gr(2,n))=\text{Spec}(K[p_{ij} : ij \in \binom{\lbrack n \rbrack}{2} ] / I_{2,n}) \cup \{(0, \dotsc, 0)\}.$$
My question: How is this equal to the claim? I.e. how do I get the power $\pm 1$? Respectively where is my mistake?
Thank you very much!
Best Answer
In the paper you refer to, they only claim that this is the coordinate ring of the affine cone over the open subset $\operatorname{Gr}_0(2,n)$ of $\operatorname{Gr}(2,n)$, which is defined shortly before by requiring that the coordinates $p_{ij}$ do not vanish. Hence, this is not the coordinate ring of the affine cone over the Grassmannian, which you correctly identified, but the coordinate ring of an open subset of it, namely the open subset that arises by removing the coordinate hyperplanes.