Is a (quasi)projective toric variety (Q)Proj of its homogeneous coordinate ring?

Question

This is really two questions. First, consider a normal toric variety $X_\Sigma$. Its homogeneous coordinate ring $$R=\mathbb C[x_1,...,x_{|\Sigma(1)|}]$$ is graded by $A_{n-1}(X)$. In analogy with projective space, I guess that there is an analogue of the Proj construction: homogeneous ideals of $R$ not contained in the Stanley-Reisner ideal $B(\Sigma)$ of the fan $\Sigma$.

If $X_\Sigma$ is projective, is $X_\Sigma = Proj_{B(\Sigma)}(R)$?

Assuming this is true, I am curious about the case when $X_\Sigma$ is quasi-projective; $R$ has non-trivial elements in "negative" degree.

If $X_\Sigma$ is quasi-projective, is there a analogue of Proj with $X_\Sigma = QProj_{B(\Sigma)}(R)$?

Answer 1

No variant is necessary, $X_{\Sigma}$ is $\mathrm{Proj}_{B(\Sigma)} (R)$. Note: I'm assuming you already understand how this construction works in the projective case, so that I can jump in and start working an example.

Let's work through the example of $\mathbb{P}^2$ with a point deleted. The corresponding fan has three rays, in directions $e_1= (1,0)$, $e_2 = (0,1)$ and $e_3 = (-1, -1)$ and with two dimensional faces $\mathrm{Span}(e_1, e_3)$ and $\mathrm{Span}(e_2, e_3)$. The ring $R$ is $k[x,y,z]$, with $x$, $y$ and $z$ corresponding to $e_1$, $e_2$ and $e_3$. The grading is that $x$, $y$ and $z$ are in degree $1$. The ideal $B(\Sigma)$ is $\langle x, y \rangle$.

So, the points of $k[x,y,z]$ correspond to those hemogenous primes of $k[x,y,z]$ which do not contain $\langle x,y \rangle$. Sure enough, that's $\mathbb{P}^2$ with a point removed! Note that we don't change the ring or the grading $R$, both of which are determined by the set of rays of the fan. We just change the irrelevant ideal $B(\Sigma)$.

See David Cox's paper The Homogeneous Coordinate Ring of a Toric Variety for the full generalities.