# Tagged Questions

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1answer
54 views

### Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
1answer
58 views

### Finding a toric variety of a cone

I'm trying to find the toric variety associated to the cone $\sigma_0$ which is the region in the real plane with $x\geq 0$ and $y-x\geq 0.$ I found that it's dual cone is $\check{\sigma_0}$ the ...
1answer
86 views

### If a group scheme $G$ operates on another scheme $X$, how do you define orbits?

In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even ...
1answer
51 views

### How to recognize a glueing

This is an exercise in chapter 1 of Fulton's book "Introduction to toric varieties". Let $\Delta$ be the fan consisting of the cones $\sigma_1=\langle e_1, e_2\rangle$ and \$\sigma_2=\langle ...
0answers
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### Toric Variety from a fan with “identical” edges

I have this fan that I've been trying to construct a toric variety from. The problem is, it contains certain edges twice. These are the edges: $$(1,0,1)$$ $$(0,1,1)$$ $$(-1,-1,1)$$ $$(0,0,1)$$ ...
1answer
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### Degree of Toric Divisors

Is it possible to calculate the degree of a toric divisor directly from the fan of the toric variety? If so, how is this done? Or is there some alternative way to calculate the degree of these ...
1answer
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### Is locally free sheaf of finite rank coherent?

Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent? By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that ...
0answers
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### Restriction of locally free sheaf associated projective modules

My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in ...
0answers
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### Transition functions of toric projective bundle (Proposition in [Cox, Toric Varieties])

My reference: David Cox's "Toric Varieties" My question is the proof of Proposition 7.3.3. Proposition 7.3.3. The cones {$\sigma_i$ | $\sigma \in \Sigma$, $i = 0,\dots,r$} and their faces form ...
2answers
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### If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's "On ...
0answers
85 views

### GIT quotient for a certain torus action on an affine space

I'm reading various books and some notes and here is my question. Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by (\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, ...
1answer
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### Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...
0answers
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### Exercise in David Cox “Toric Varieties”

I want to do an exercise in the book Toric Varieties (by David Cox) Exercise 3.3.5. Let $\overline{\phi}:N \rightarrow N'$ be a surjective $\mathbb{Z}$-linear mapping and let $\widehat{\sigma}$ ...
1answer
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### How to find the canonical divisor on a nonsingular toric variety?

I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then  ...
1answer
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### Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
1answer
154 views

### Hodge theory for toric varieties

Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various ...
1answer
67 views

### differential with logarithmic poles

where can I find the computation of the groups $H^i(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^j)$? Moreover, if $D$ is a divisor with normal crossing in $\mathbb{P}^n$, how can I compute the hypercohomology ...
1answer
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### figure out a simple toric variety

consider the plane cones $s=\langle(1,0),(1,n)\rangle$ and $t=\langle(1,0),(1,-n)\rangle$. This produce a toric variety obtained glueing $k[x,xy^n]$ and $k[x,xy^{-n}]$ along $k[x]$. There is a more ...
1answer
213 views

### Cohomology of $\mathcal O_X$ for toric varieties

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases}$$ as for \$\mathbb ...
1answer
205 views

### Equations of a projective toric variety

Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective ...
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