Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
320
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Intersection in toric variety
In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension.
On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...
2
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0
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90
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Higher chow groups of affine toric varieties
Let $X$ be an affine toric variety defined over an algebraically closed field $k$ of characteristic zero.
I am trying to use Bloch’s Riemann-Roch Theorem for quasi-projective algebraic schemes in his ...
1
vote
0
answers
80
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Vanishing of chow group of 0-cycles for affine, simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $X$ be an affine, simplicial toric variety over $k$.
If $X$ has dimension one, then it is the affine line over the field $k$, so ...
3
votes
1
answer
137
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A question related to the strong Oda conjecture
A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
1
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0
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135
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Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
1
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0
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71
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Chow ring of simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
2
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0
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132
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
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0
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Coordinate transformation for 3-dimensional simplicial cone in $\mathbb{R}^3$
Let $k$ be an algebraically closed field and let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$.Let $X$ be the affine toric variety over $k$ associated to the cone $\sigma$, i.e. set $X$...
2
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0
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106
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Equivariant line bundles over toric variety
Let $X$ be a projective $n$-dimensional toric variety acted by an algebraic torus $T\simeq \mathbb{C}^{\ast n}$. It is well known that any piecewise linear (and integer in some sense) function on ...
1
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2
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360
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Is this toric variety always smooth?
Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray in $\sigma$.
Let $U_{\rho}$ be defined as $\operatorname{Spec}(k[\...
1
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0
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151
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Is this closed subscheme a toric variety?
Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray of $\sigma$. Say $\rho=\sigma\cap H_m$, where $H_m$ is the plane in $...
0
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0
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59
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existence of moment maps for non-nef toric varieties
The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
6
votes
2
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304
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Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces
I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...
1
vote
1
answer
322
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Three-dimensional analogues of Hirzebruch surfaces
There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
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0
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Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...