# Tagged Questions

**3**

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### Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...

**5**

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**1**answer

219 views

### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

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**1**answer

222 views

### Kahler-Einstein metrics on Toric manifolds are Torus-invariant?

let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus ...

**1**

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**1**answer

189 views

### Computing rational cohomology of smooth (not necessarily compact) toric varieties

The title pretty much says it all. I am looking for references (lecture notes, books, readable articles, suggestions), preferably example laden, that explain how to compute the rational cohomology of ...

**5**

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**2**answers

2k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...

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**1**answer

688 views

### Cox rings of toric varieties over arbitrary fields

The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous ...