# 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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### 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
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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 ...
1 vote
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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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### 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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### 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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### 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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### 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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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 $... 2 votes 1 answer 92 views ### What is the minimal$ 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} $, then the ... 5 votes 0 answers 274 views ### Do algebraic tori have no$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$(the latter ... 4 votes 1 answer 110 views ### Almost toric mutations I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration$f: (M,...
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 \$...