Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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Homology classes of subvarieties of toric varieties

Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety. Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero? Background If $X$ is a Kaehler variety, this is of ...
Piotr Achinger's user avatar
14 votes
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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 ...
Hugh Thomas's user avatar
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Rational points of weighted projective spaces

[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.] Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
Laurent Moret-Bailly's user avatar
12 votes
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A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as $$ v_{m,...
Fabio Tonini's user avatar
11 votes
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Big tangent bundle

Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
jhsiao's user avatar
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Hilbert schemes of points on toric surfaces

Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For ...
ssx's user avatar
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9 votes
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Computing Ext for toric divisors

Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute $$ \...
Karl Schwede's user avatar
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8 votes
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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 ...
evgeny's user avatar
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7 votes
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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 ...
Leo Herr's user avatar
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?

[This is an updated version of https://math.stackexchange.com/questions/4522399/.] Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
Immi's user avatar
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Derived category of toroidal varieties

This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem. A ...
Li Yutong's user avatar
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7 votes
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When should a moment polytope have "smooth" faces?

A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...
Allen Knutson's user avatar
6 votes
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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 ...
algebrachallenged's user avatar
6 votes
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226 views

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) ...
Justin Hilburn's user avatar
6 votes
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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 ...
Kiu's user avatar
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Do simplicial toric varieties have "lots" of base point free linear systems?

Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system? If $X$ is not ...
Anton Geraschenko's user avatar
5 votes
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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 ...
user267839's user avatar
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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 ...
Evan O'Dorney's user avatar
5 votes
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264 views

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 ...
ugosugo's user avatar
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algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
Somatic Custard's user avatar
5 votes
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Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?

Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
მამუკა ჯიბლაძე's user avatar
5 votes
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336 views

Distinguishing ample divisors by minimally intersecting curves on a smooth projective toric variety

My question has an easily formulated generalization, which I will state first. Let $\sigma \subseteq \mathbf{R}^n$ be a full-dimensional strongly convex polyhedral cone. For each lattice point $m \in \...
Mellon's user avatar
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5 votes
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When is vector bundle over toric variety a toric variety?

Is it true that a vector bundle over a toric variety is also a toric variety if and only if it splits? if so, how do we prove it? This seems to be the content of a remark in Oda's Tata's lectures on ...
jj_p's user avatar
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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 $k$ ...
Kiu's user avatar
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127 views

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 ...
R_D's user avatar
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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 ...
Antoine Labelle's user avatar
4 votes
0 answers
107 views

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 ...
Dcoles's user avatar
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4 votes
0 answers
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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 ...
giulio bullsaver's user avatar
4 votes
0 answers
213 views

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 ...
Philip Engel's user avatar
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4 votes
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110 views

Primitive collections as lattice generators for the Mori cone

I am looking at the following theorem from "Toric Varieties" by Cox, Little and Schenk: Theorem 6.4.11: If $X_{\Sigma}$ is a simplicial toric variety, then $\overline{NE}(X_{\Sigma}) = \...
Mellon's user avatar
  • 197
4 votes
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251 views

Quotients of toric varieties

This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...
user313212's user avatar
4 votes
0 answers
141 views

moduli space of toric structures on a fixed toric variety (reference?)

I'm looking for a reference on the following question: Given a fixed toric variety $V/k$, how to describe the moduli space of all toric structures on $V$? In addition to the general question, I ...
Somatic Custard's user avatar
4 votes
0 answers
208 views

Local structure of non-normal toric varieties---possible mistake in "Discriminants, Resultants and Multidimensional Determinants"

I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...
Avi Steiner's user avatar
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4 votes
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158 views

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 $P$ (...
Igor Makhlin's user avatar
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4 votes
0 answers
285 views

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 ...
Qfwfq's user avatar
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4 votes
0 answers
216 views

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 ...
cata's user avatar
  • 337
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269 views

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 ...
Camilo Sarmiento's user avatar
4 votes
0 answers
301 views

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 End(V_{\omega_{i}})\times\prod\...
prochet's user avatar
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4 votes
0 answers
487 views

Intuition about Toroidal Embeddings

I have been trying to understand the very basics of toroidal embeddings, and the definitions on the face of them are not terribly daunting. I've been going with the "locally analytically looks like a ...
Dhruv's user avatar
  • 437
4 votes
0 answers
649 views

Intersection Theory on a toric variety

Hi All, I'm having some trouble understanding a result about calculating $D.C$ on a toric variety. The proposition I am trying to follow is from Cox, Little, Schenck. Either there is a mistake with ...
Robert Garbary's user avatar
3 votes
0 answers
117 views

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 ...
Boris's user avatar
  • 501
3 votes
0 answers
86 views

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 ...
Lineer 's user avatar
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0 answers
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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 ...
Adterram's user avatar
  • 1,361
3 votes
0 answers
128 views

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 ...
Boaz Moerman's user avatar
3 votes
0 answers
102 views

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 ...
EJAS's user avatar
  • 181
3 votes
0 answers
210 views

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 ...
Aidan's user avatar
  • 488
3 votes
0 answers
177 views

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 $...
Asvin's user avatar
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3 votes
0 answers
171 views

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 $p$-...
The Thin Whistler's user avatar
3 votes
0 answers
162 views

Smooth proper varieties over the integers that are not toric

Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric? By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
Asvin's user avatar
  • 7,648
3 votes
0 answers
167 views

Polytope algebra and toric vareties

Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$. For a simple polytope $P$ let us ...
asv's user avatar
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