Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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4 votes
1 answer
278 views

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
8k 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 ...
2 votes
0 answers
122 views

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 answers
61 views

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 votes
0 answers
397 views

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
232 views

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
24 views

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 vote
0 answers
150 views

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 views

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
238 views

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
201 views

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 votes
0 answers
54 views

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
284 views

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 vote
0 answers
60 views

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 views

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 views

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 views

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 votes
2 answers
296 views

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
167 views

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
116 views

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
47 views

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
2k views

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
86 views

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 votes
0 answers
106 views

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 views

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
136 views

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 vote
0 answers
135 views

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 vote
0 answers
66 views

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 votes
1 answer
47 views

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 views

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
1k views

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 views

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 views

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 votes
0 answers
154 views

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 views

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 views

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 views

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 votes
0 answers
161 views

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 vote
0 answers
55 views

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
136 views

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 vote
0 answers
45 views

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
256 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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. ...

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