# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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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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### Showing that a smooth toric variety is a bundle over $\mathbb{C}\mathbb{P}^2$

I'm considering the smooth toric variety
$$\tilde{X}=\frac{\mathbb{C}^5\setminus \{x_3=x_5=0\}\cup \{x_1=x_2=x_4=0\}\cup \{x_1=x_2=x_3=0\}}{G= \{(t_1,t_1,t_1^2t_2,t_1^{-5}t_2^{-3},t_2);t_i\in \mathbb{...

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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 ...

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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 ...

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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 ...

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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-...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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} \...

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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 ...

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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} $?
...

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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?

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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 ...

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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 ...

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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 ...

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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 ...

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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}} $. ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $ \...

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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,-...

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### Tropical limit and Jacobians

Consider a collection of $n$ rational functions $f_i,\ i=1,\dots,n$ in $n$ variables $t_i,\ i=1,\dots,n$.
Let $J^f$ be the rational function defined implicitly by
$$\bigwedge_{i=1}^n \frac{dt_i}{t_i}J^...

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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 ...

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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 ...

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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 ...

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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.
...

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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}^...

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### What sort of spaces show up as intersection complexes of toric degenerations of Calabi-Yau Varieties?

Roughly, a toric degeneration is a proper flat family $f:\mathcal{X}\to D$ of relative dimension $n$ with the properties that $\mathcal{X}_t$ is an irreducible normal Calabi-Yau and $\mathcal{X}_0$ is ...

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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 $\lambda$ ...

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### Seeing $\mathbb{CP}^2 \mathbin\# \overline{\mathbb{CP}^2}$ as a symplectic reduction of different manifolds

I have been reading the paper "Remarks on Lagrangian intersections on toric manifolds" by Abreu and Macarini, which gives several non-displaceability results by avoiding the use of ...

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### Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

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### When is a smooth point of a projective, simplicial, toric variety $ X_{\Sigma} $ compatibly $ F $-split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

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### How to compute the multiplicity of a strongly convex, rational, polyhedral cone $ \sigma $?

In David Cox, John Little and Hal Schenck's book Toric Varieties page 302, Chapter 6, Section 4, Proposition 6.4.4, the authors state the following proposition. If $ \Sigma $ is a simplicial fan of ...

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### When is a wonderful compactification a toric variety?

Given a (projective) hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$, in which we assume the intersection of all hyperplanes is $0$, and a building set $G$, De Concini and Procesi define the ...

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### Line bundles on toric varieties associated to Weyl chamber

I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...

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### What is the subdivision corresponding to the blowup of a toric divisor of a singular toric variety?

Let $X$ be an affine toric variety corresponding to the cone $\sigma$. If $X$ is smooth, blowups of toric strata correspond to star subdivisions of $\sigma$. Suppose that $X$ is singular and let $D \...

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### Does one only need to look at torus invariant curves to calculate the Seshadri constant for a point of a toric variety?

If $ X $ is an irreducible projective variety, $ L $ is a Nef divisor on $ X $, $ x $ is a point of $ X $, and $ \pi: \operatorname{Bl}_{x}(X) \to X $ is the natural projection morphism, then the ...

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### Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety

Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...

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### Doubt in the proof of Mcduff''s method of probes

I have been reading the paper "Displacing Lagrangian toric fibers by probes" by Dusa Mcduff, here is the arxiv link https://arxiv.org/pdf/0904.1686.pdf.
I have a doubt in the proof of lemma $...

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### What is the minimal $ n $ such that the Cox ring of the blow up of a simplicial, $ r $-dimensional, toric variety at $ n $ points in g.p. is not f.g.?

In Shigeru Mukai's paper "Counterexample to Hilbert's 14th Problem for the 3-dimensional Additive Group," Mukai proved that if $ \frac{1}{r+1}+\frac{1}{n-r-1} \le \frac{1}{2} $, then the ...

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### Do algebraic tori have no $H^1$?

If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...

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### Almost toric mutations

I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,...

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### Resolutions of configuration space of the projective line where the complement is of "Tate type"

I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...