The toric-varieties tag has no usage guidance.

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### Can a toric surface be an elliptic surface?

It is known that a rational elliptic surface is a blow-up of $\mathbb{P}^2$ at 9 points. More precisely it is obtained as the blow-up of the base locus of a pencil of cubic curves in $\mathbb{P}^2$. ...

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### Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say
...

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### Hofer-Zehnder capacity of toric varieties

Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset ...

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### Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...

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### When is a collection of toric subvarieties in $\mathbb P^r$ generic?

Suppose $Z_1,\dots,Z_n \subset \mathbb P^r$ are toric sub-varieties.
By this I mean, there is a fixed co-ordinate system $\mathbb P^r=\{[z_0:\dots:z_r]\}$ and each subvariety is defined by the ...

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384 views

### Are quadric hypersurfaces toric varieties?

A quadric hypersurface (over an algebraically closed field of characteristic zero) in $\mathbb{P}^n$ for $1\leq n\leq 3$ is a toric variety. (Namely, it's isomorphic to ...

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65 views

### Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of ...

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189 views

### On factorization theorem of toric birational morphisms

Let $X_{Σ′}\to X_{Σ}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $Σ′\leq Σ$, i.e. every cone in $Σ′$ is contained in a cone in $Σ$ and ...

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### Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ ...

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### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

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203 views

### To what extent are toric manifolds and principal torus bundles “the same thing”?

I am a little confused by the different definitions for toric manifolds/varieties. Depending on the definition of toric manifolds and principal torus bundles that one chooses, when is a toric manifold ...

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308 views

### How to describe morphisms to a weighted projective space (bundle)?

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. ...

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### Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...

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290 views

### Existence of a morphism between two toric varieties

Does there exist a morphism between the blow-up of $\mathbb{P}^3$ in four general points and $\mathbb{P}^1\times\mathbb{P}^1$? If not why?

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### Stiefel manifolds and “simplicial complex chromated Sitefel manifolds”

Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds
$$
...

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### center of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...

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### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

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188 views

### Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...

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322 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...

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### Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...

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### An example of threefold

Its description is a little bit complicated but it would be great if anyone can give an example.
I tried to construct it as a toric variety (See the previous question) but did not succeed.
I am ...

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### Geodesic rays in a toric variety

Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...

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### Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...

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290 views

### How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

For Calabi-Yau variety $X$ which is a complete intersection
$$
f_1=f_2=\ldots=f_r=0
$$
in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...

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186 views

### Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano.
Thanks a lot.

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### Alexeev's projective torus embeddings

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry.
In chapter 2 in order to obtain an affine toric variety he takes ...

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135 views

### Finite resolution by sums of line bundles on toric varieties

I hope I wasn't searching wrong keywords or overlooking some easy arguments to prove/disprove it. What I'm asking is the following:
Let $X$ be a smooth complete toric variety. $\mathcal F$ a coherent ...

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### A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...

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### Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper
In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...

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### Reduced stabilizers of torus action on toric variety

I hope my question is not too trivial, but unfortunately i'm just starting to study toric varieties.
Let's take $X$ a lattice and $\sigma\subset X^*$ a strongly convex rational polyhedral cone, so ...

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### Singularities induced by the toric ambient spaces

Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...

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### Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...

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### Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...

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### Is the Kähler cone of a toric variety always simplicial?

I am working on a familly of toric varieties which seem to have the following property:
the closure of the Kähler cone is a simplicial cone (and even a smooth cone with respect to the natural ...

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### Computational complexity of deciding isomorphism of rational polyhedral cones

Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...

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### When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?

Let $p_1,...,p_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X_k$ is toric and hence a Mori Dream Space. ...

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### What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...

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### The minimal number of halfspaces to represent a convex but non strongly convex cone

We say a cone at the origin in $R^n$ means that it is an intersection of finitely many halfspaces, i.e.
$$C=\bigcap_{i\in I}H_i,\text{ where }|I|<\infty.$$
A cone is strongly convex if $C\cap ...

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### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

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### Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...

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### Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$.
Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...

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### How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...

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### Does any smooth hypersurface in (C^*)^n admit a smooth normal crossings compactifcation as a hypersurface in a toric variety?

Question 1. Given a Laurent polynomial $f(z_1,\cdots,z_n)$, such that the corresponding zero locus $Z$ of $f$ in $(\mathbb{C}^*)^n$ is smooth, can we find a smooth toric variety $\bar{X}$ (together ...

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### Kahler-Einstein metrics on Toric manifolds are Torus-invariant?

let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus ...

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### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

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### Legal potentials on delzant polytopes

Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$.
Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still ...

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### Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.
In his paper http://arxiv.org/abs/0803.0985 ...

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### Rational surface singularities as Toric varieties

I originally asked basically this question on MSE here. From an appendix in Steinberg's "Conjugacy classes in algebraic groups" I found a list of the rational surface singularities. Equations of types ...

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### Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal ...

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### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...