# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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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 $...

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### Linear relations between volume of a polytope and its faces

Let $P$ be a polytope. Is anything known about the set of linear relations that hold between the volumes of the (not-necessarily proper) faces of $P$ as $P$ “varies slightly”? By varies slightly I ...

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### Cohomology of toric blowup

Let $n\geq2$. Let $G$ be a linear automorphisms group of prime order on $\mathbb{C}^n$. We assume that $0$ is the unique fixed point of $G$.
I consider the quotient $\mathbb{C}^n/G$. It is a toric ...

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### Possible volumes of lattice polytopes

All polytopes here are assumed to be convex lattice polytopes.
Given a polytope $P$, set $$v(P):= (\operatorname{vol}(F))_{F\text{ a face of }P},$$ where the volume of a $d$-dimensional polytope $P\...

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### Effective classes in toric Kähler manifolds

In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and ...

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### condition on rational polyhedral cone to guarantee dual cone is homogeneous

Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$).
Definition. The cone $\sigma$ is homogeneous if there are ...

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### A regular sequence in a quotient by a “half lattice” defined by a toric manifold

I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient $$P^{-1}(p) / \mathbb{K}$$ by the ...

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### Isomorphic equivariant sheaves are equivariantly isomorphic on a toric variety

Let $X$ be a toric variety containing the $n$-torus $T\overset{i}{\hookrightarrow} X$. The action of $T$ extends naturally to an action on the sheaf $i_*\mathcal{O}_T$ by
$$(\alpha\cdot f)(x):=f(\...

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### On cohomological algebras related to toric manifolds

I am interested in some cohomological algebras related to toric manifolds. We consider a toric manifold $M$ as a quotient
$$M = P^{-1}(p) / \mathbb{K}, \quad P : \mathbb{C}^n \to \text{Lie}(\mathbb{K})...

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### On the dimension of the cohomology of toric manifolds

Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...

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### Faces of polyhedral cones and open immersions of affine toric schemes

Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.
Let $\sigma\subseteq V$...

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### Closures of torus orbits in flag varieties

Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.
Now, as far as I ...

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### “Reflexive” differentials on Gorenstein affine toric variety

Let $P \subset \mathbb{R}^{n-1}$ be a lattice polytope of dimension $n-1$ and let $\sigma \subset \mathbb{R} \times \mathbb{R}^{n-1}$ be the cone over $1 \times P$.
To the cone $\sigma$, we may ...

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### Spherical varieties as GIT quotients

Let $X$ be a normal projective variety with finitely generated Cox ring. Consider its characteristic space $p:\widehat{X}\rightarrow X$. This means that there is a torus $T$ acting on $\overline{X}=...

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### Principal $G$-bundles on affine toric varieties

Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ ...

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### Fibration of a toric symplectic manifold from a fibration of the moment polytope

This question is regarding fiber bundles, both whose fibers and total space are toric symplectic manifolds. The structure group on the fiber is a subgroup of the structure group of the total space.
...

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### Set theoretic complete intersections in toric varieties

Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety?
Is there an example of a smooth ...

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### Complete intersections in toric varieties

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?

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### How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?

Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. ...

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### Holomorphic line bundles on arbitrary simplicial toric varieties as restrictions

In the question titled "Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$" it was explained how any holomorphic line bundle on $\mathbb P^1 \times \mathbb P^1$ is of the form $\...

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### 2-faces of reflexive Delzant polytopes

Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?
Motivation. I would like more generally to get an answer to the following question:
...

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### Blow-ups of toric varieties

I'm interseted in blow-ups of toric varieties, unfortunately, I don't understand the construction of a blow-up built by a refinement of a fan, if to be more specific, I didn't find any constructions.
...

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### Connection on line bundle over general simplicial toric variety

In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...

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### Can any simplicial toric variety be embedded in a product of projective spaces?

In this question - On a Hirzebruch surface. , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$.
My question is, does such an isomorphism exist ...

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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 ...

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### Number of boundary divisors and colors of a Spherical variety

Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-...

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### Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...

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### Volume of $-K_X$ for a weighted projective variety

Let $X:=\mathbb P(a_0,a_1, \ldots, a_n)$ be a well formed weighted projective variety. Let $-K_X$ be its anticanonical divisor, then how to express its volume ${\rm vol}(-K_X)=(-K_X)^n$ in terms of $...

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### When are these definitions of “toric variety” equivalent?

Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, ...

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### Toric Desingularization Algorithms

There are certainly many algorithms to desingularize toric varieties (e.g https://arxiv.org/pdf/math/0411340.pdf). I would imagine in analogy with desingularizing surfaces these all involve blowing up ...

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### Examples of small covers that are not real toric manifolds

Definition of quasi toric manifolds :
The action of $(S^1)^n$ on $\Bbb C^n$ by pointwise multiplication is called the standard representation. Given a manifold $M^{2n}$ with an $(S^1)^n$-action, a ...

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### Is the Chow ring of a wonderful model for a hyperplane arrangement isomorphic to the singular cohomology ring?

In the article "Hodge theory for combinatorial geometries" by Adiprasito, Huh and Katz, it it claimed in the proof of theorem 5.12 that there is a Chow equivalence between the de Concini-Processi ...

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### When is the blow-up of a closed sub-variety of a toric variety a toric variety?

If $ X $ is a toric variety and one has a closed sub-variety $ Y \subseteq X $, is the blow-up $ \operatorname{Bl}_{Y}(X) $ a toric variety as well? I suspect not, but wanted to check with other ...

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### Are rational varieties symplectically rationally connected?

Was it proven already that smooth rational complex projecitve varieties are symplectically rationally connected? I.e. some GW invariant with two point insertions is non zero. What about smooth toric ...

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### When does a discrepant toric resolution induce a crepant resolution of a subvariety?

Let $Y$ be a complete intersection in a complete simplicial toric variety $X_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X_\Sigma)$. Suppose that $\phi:X_{\widehat{\Sigma}}\...

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### Resolving $\mathbb Z_n$ action on $\mathbb C^2$

Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...

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### Relative canonical divisor associated to toric morphism induced by refinement of fan

Let $\phi:X'\to X$ be a morphism between toric varieties $X=X(\Delta), X'=X'(\Delta')$, induced by a refinement $\Delta'$ of $\Delta$. This refinement is obtained from a sequence of stellar ...

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### Understanding projective space as fibrations of tori over spaces with boundaries

The toric manifold $\mathbb{CP}^1$ can be understood as a circle fibration over an interval $I$, with the circles having zero radius at the boundaries of the interval. How does one generalize this ...

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### Why are the toric fibers of a toric manifold Lagrangian submanifolds?

How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general ...

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### toroidal compactifications of modulis spaces of ppav's

Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...

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### Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety

It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X_\Delta$.
My question is whether there is a relationship between the number of maximal cones ...

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### Toric variety defined by the Weyl orbit of a minuscule weight

Let $\Phi$ be a (reduced, crystallographic) root system with Weyl group $\mathcal{W}$, and $p$ a (nonzero) minuscule weight for $\Phi$: its orbit $\mathcal{W}p$ is the set of vertices of a convex ...

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### Application of toric varieties for problems that do not mention them

I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?
To give ...

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### Is there a “fundamental theorem of toric geometry”?

I have some questions about toric geometry.
1) Can any toric variety (I mean not necessarily smooth or projective, ...) be constructed from a fan ?
2) Suppose $T_1$ and $T_2$ are toric varieties ...

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### Moment map for complete flags variety

Let $M:=U(n)/T^n$ be a complete flag variety, where $U(n)$ is an unitary group and $T^n \simeq (S^1)^n$ consists of its diagonal matrices. I have heard the following construction of a symplectic ...

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### The boundary of toric varieties

Let $\mathcal{X}$ be a toric variety, with $T$ a torus embedded as an open set
in $\mathcal{X}$ (and where the algebraic action of $T$ extends to $\mathcal{X}$). As I am not a toric specialist at all, ...

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### Equivariant derived category versus graded derived category

Everything here has the Zariski topology.
Let $T=(\Bbb{C}^*)^d$, and define an action of $T$ on $\Bbb{C}^n$ by $$t\cdot x=(t^{\mathbf{a}_1}x_1,\ldots, t^{\mathbf{a}_n}x_n).$$ Here $\mathbf{a}_1,\...

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### How to calculate the top Chern class of a “functorial” vector bundle on a moduli space of sheaves?

Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...

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### Descent of flatness from algebras to monoids II

This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...

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### Taking powers of polytopes

I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known.
Imagine this situation where I want to think of such a thing : say ...