# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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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 ...
• 483
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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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### 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 ...
• 483
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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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• 389
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### How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$

I am given that $k$ is a field and $G$ is the monoid consisting of all monomials $X^iY^j$, where $j$ is between $0$ and $3i$. I am trying to compute the quotient of the monoid algebra $kG$ by the ...
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### A question regarding Corollary 4.12 in Mumford's "Analytic construction of deg. Ab. Var."

Let $S=Spec A$ be the spectrum of an integrally closed, excellent and Noetherian ring. In his paper, David Mumford constructs an $S$-group scheme $G$. He shows that the torsion of $G_s$ is $p$-...
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1 vote
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### A connection between equivariant and non-equivariant cohomology of toric variety

Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$. The $T$-equivariant cohomology $H^*_T(X)$ (with coefficients in a field, say) is an ...
• 19.7k
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### Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$. Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
• 1,027
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### Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold. The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
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### If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?

Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
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### When does the Hirzebruch surface have a nef anticanonical divisor?

Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...
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### Toric Bézier patches

Toric Bézier patches (as described in https://arxiv.org/abs/0706.2116) are maps from a lattice polytope $P$ to the positive part of its associated toric variety $X_P$. While they are not the inverse ...
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### Deformation of toric varieties to complete intersections

I am looking for some systematic study/examples of families of projective complete intersection varieties degenerating to a projective toric variety. In particular, given a projective toric variety, ...
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### How should we think about the algebraic moment map?

My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...
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### When is a toric variety a Poincare duality space?

When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a ...
• 1,413
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### Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
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### When an action on open dense subvariety by an algebraic group extends to variety

A toric variety $X$ over $k$ is a variety which contains an algebraic torus ($T= \mathbb{G}_k^s$) as a dense open subset such that the action of the torus on itself extends to the whole of $X$. Slogan:...
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### the map on Picard groups induced by restriction to a toric subvariety

Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character ...
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### Hodge structure on intersection cohomology of toric varieties

Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the ...
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Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$. ...