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 $X_\Sigma = QProj_{B(\Sigma)}(R)$?


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.

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  • $\begingroup$ Thanks David. I am a physicist learning algebraic geometry (slowly). I am curious as well about non-compact varieties like the total space of $\mathcal O(-2) \to \mathbb {CP}^1$. Is this really just Proj $\mathbb C[x,y,p]$, where the grading is $(1,1,-2)$? $\endgroup$ – James Davidoff Sep 29 '10 at 16:40
  • $\begingroup$ I think you got that right. The irrelevant ideal in this example should be $\langle x,y \rangle$. If you take the irrelevant ideal to be $\langle 0 \rangle$, I think you get Spec $k[x^2/p, xy/p, y^2/p] \cong k[u,v,w]/(uw-v^2)$; this is the blowdown of the zero section in the bundle you want. $\endgroup$ – David E Speyer Sep 29 '10 at 17:26

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