Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
315
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4
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Intrinsic definition of a cone in a normal fan
Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...
18
votes
2
answers
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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 ...
2
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0
answers
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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 ...
1
vote
0
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61
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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 ...
12
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0
answers
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Rational points of weighted projective spaces
[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.]
Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
1
vote
1
answer
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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
vote
0
answers
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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 $X$...
1
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0
answers
150
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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 $...
2
votes
0
answers
92
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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
vote
1
answer
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Secondary fan and KN strata
Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...
3
votes
1
answer
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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 ...
0
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0
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54
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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 ...
1
vote
1
answer
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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 $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
1
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0
answers
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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 ...
2
votes
0
answers
115
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Abstract definition of hypertoric varieties
I'm reading Proudfoot's survey on hypertoric varieties. In Section 1.4 he mentioned such a conjecture:
Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its ...
0
votes
0
answers
76
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How to compute the higher G-theory of the weighted projective space $\mathbb{P}(1,1,m)$ using Mayer-Vietoris sequence?
Let $k$ be an algebraically closed field of characteristic zero.
Let $m$ be a positive integer and let $X$ be the weighted projective space $\mathbb{P}(1,1,m)$ over the field $k$.I am trying to ...
1
vote
0
answers
113
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Computing $G$-theory for a 3-dimensional affine simplicial toric variety
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be the cone in $\mathbb{R}^3$ generated by $e_1,2e_1+e_2,e_1+2e_2+3e_3$.
Then it is easy to check that $\sigma$ is a 3-...
6
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2
answers
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Irreducibility of Gelfand-Serganova strata
To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
4
votes
0
answers
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K-theory of toric varieties
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. Is there a good presentation for the K-theory ring $K_0(X)$ in terms of the corresponding fan, analogous to the presentation of the Chow ...
3
votes
0
answers
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Computing Grothendieck group of coherent sheaves of affine toric 3-fold from a simplicial cone
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$. Let $X=\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$ be the ...
2
votes
0
answers
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Log discrepancy of a toric exceptional divisor
Let $X$ be a toric singularity determined by a single cone. By taking a partition of the cone we may get a toric resolution of $X$.
Question: How can we compute the log discrepancies of the ...
8
votes
1
answer
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
3
votes
0
answers
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How can one determine the fan of a toric Weil divisor of a complete toric variety?
It's well known that there is a so-called cone-orbit correspondence between the cones in the fan of a toric variety and the orbits of the T-action on the toric variety. My question aims to understand ...
4
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0
answers
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Reference Request: Classification of spherical varieties by "Weyl group invariant fans"
Apologies in advance for the vague question.
Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...
3
votes
0
answers
53
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Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
4
votes
1
answer
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Finitely generated section ring of Mori dream spaces
Set-up: We work over $\mathbb{C}$. Let $X$ be a Mori dream space. Define, following Hu-Keel, the Cox ring of $X$ as the multisection ring
$$\text{Cox}(X)=\bigoplus_{(m_1\ldots,m_k)\in \mathbb{N}^k} \...
1
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0
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References/applications/context for certain polytopes
First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
1
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0
answers
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Zariski Cancellation and Toric Varieties, why isn't this affine variety toric?
The Zariski cancellation problem asks the following. If $ Y $ is a variety such that $ Y \times \mathbb{A}^{1}_{k} \cong \mathbb{A}^{n+1}_{k} $, then is $ Y $ isomorphic to $ \mathbb{A}^{n}_{k} $?
...
0
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1
answer
47
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Holomorphic cyclic action on smooth toric manifold extends to C^* action?
Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?
2
votes
0
answers
91
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Minimal model program for toroidal pairs
Suppose $(X, \Delta)$ be a toroidal pair over $Z$ where $f:(X, \Delta) \rightarrow (Z, \Delta_Z)$ is a toroidal morphism (see https://arxiv.org/pdf/alg-geom/9707012.pdf sections 1.2, 1.3 for the ...
8
votes
1
answer
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Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
0
votes
1
answer
163
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How to compute the $G$-theory of the weighted projective space $\mathbb{P}(1,1,2)$?
Let $k$ be an algebraically closed field of characteristic zero. Let $\Sigma$ be the fan in $\mathbb{R}^2$ consisting of three cones, cone generated by $e_1,e_2$,cone generated by $e_2,-e_1-2e_2$ and ...
2
votes
0
answers
73
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How to estimate the locus of non-zero cohomology for a equivariant toric reflexive sheaf, with a Klyachko description
I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of ...
7
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0
answers
154
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
3
votes
0
answers
127
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Inverse image Weil divisor on a toric variety as a Cartier divisor
Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
2
votes
1
answer
178
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Is a toric variety over a field of positive characteristic complete if and only if the support is all of $ N_{\mathbb{R}} $?
In Cox, Little and Schenck's book Toric Varieties they show that a toric variety $ X_{\Sigma} $ over a field of characteristic zero is complete if and only if the support is all of $ N_{\mathbb{R}} $. ...
2
votes
0
answers
92
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Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
5
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0
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Strong factorisation conjecture for toric varieties
In this survey is remarked (see page 6 after Example 1.12) that to prove the
Conjecture 1.10 (Strong factorisation). Let $\phi: X \dashrightarrow Y $ be a birational map
between two quasi-projective ...
1
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0
answers
55
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Singularities of toric pairs
Suppose $(X,B)$ is a log canonical pair and $f: X \rightarrow Y$ an equidimensional toroidal contraction such that every component of $B$ is $f$ -horizontal. Let $\Gamma$ denote the reduced ...
4
votes
1
answer
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Approximation of convex bodies by polytopes corresponding to smooth toric varieties
Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
1
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0
answers
45
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Is there some kind of construction of a "canonical unirational variety" like the one for toric varieties?
Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many ...
7
votes
0
answers
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Cohomology of fibers of a morphism of a blowup of affine space
Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the ...
4
votes
2
answers
3k
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Clarification on the definition of a quotient singularity
I am working on the quotient construction of a simplicial toric variety as described in chapter 5 of this book. I have tried the following two examples -
The fan $\Delta$ in $\mathbb R^2$ consists of ...
2
votes
2
answers
149
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If $ Z $ is an $ n $-dimensional, projective variety, containing $ \mathbb{G}_{m}^{n} $, what is the obstruction to $ Z $ being toric?
Let $ Z $ be an $ n $-dimensional, projective, variety, over a field of arbitrary characteristic and let $ \iota: \mathbb G_{m}^{n} \to Z $ be a morphism such that for any $ z \in Z $, the fibre $ \...
2
votes
0
answers
165
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How to compute the $G$-theory of this simplicial toric surface?
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-...
7
votes
3
answers
1k
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Has anyone researched additive analogues of toric geometry in characteristic zero?
One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...
1
vote
0
answers
59
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Polytope of a projected toric variety
I was looking for such a result in the book by Cox, Little and Schenck but I'm not able to find a proper reference.
All of the following requirements are tacitly assumed to be in the projective ...
2
votes
0
answers
200
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Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
2
votes
1
answer
201
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Why are symplectic toric varieties projective?
Let $X$ be a symplectic toric manifold meaning a compact symplectic manifold $(X, \omega)$ with $\dim{X} = 2n$ equipped with a Hamiltonian action of a maximal-dimension torus $\mathbb{T} = (\mathbb{S}^...
1
vote
0
answers
54
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Facets of polytopes and toric morphisms
To every convex lattice polytope $P$ is associated a toric variety $X_P$, which can be realized as a projective variety.
Consider a facet $f$ of $P$, i.e. a codimension one boundary of the polytope.
...