Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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Gromov-Witten invariants of cocharacter closures in toric varieties

$\require{AMScd}$ Let $X$ be a toric projective variety with dense algebraic torus $\iota:(\mathbb{C}^\times)^n \to X$, and let $u:\mathbb{C}^\times \to X$ be a cocharacter, by which I mean a map ...
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When are two complex Tori biholomorphic

Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which is isomorphic to $\mathbb{R}^{2g}$ is real vector space. An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a ...
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Isomorphic Jacobians for different choices of basis of $1$-forms

In Otto Forster's Lectures on Riemann Surfaces on page 170 Jacobi Variety is defined in 21.6: Suppose $X$ is a compact Riemann surface of genus $g$ and $ \omega_1,..., \omega_g $ is a basis of $\Omega ...
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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
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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 \...
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Explicit formula for the moment map of toric manifold

Let $P$ be a Delzant polytope in $M\otimes{\mathbb R}\cong \mathbb R^n$, and it is well-known that we can associate to it a toric manifold $X=X_P$ with the moment map $\pi: X\to P$. I would like to ...
Hang's user avatar
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Relationship between fans and root data

A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum. A toric variety is described by combinatorial information called a fan. Both ...
Ian Teixeira's user avatar
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Coordinate-symmetric convex polytopes with equal Erhart (quasi)-polynomials

Recall that given a nondegenerate polytope $P \subset \mathbb{R}^n$ which is the convex set of some vectors with integral coordinates, the Erhart polynomial $p_P(t)$ a polynomial such that $p_P(t)$ ...
skr's user avatar
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Low rank approximation

Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
anjan samanta's user avatar
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Is toroidalization local?

Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...
Hu Zhengyu's user avatar
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Connected components of a codimension one fiber for a finite morphism

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do ...
Hu Zhengyu's user avatar
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Log canonical centers of toric (and toroidal) varieties

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question: Is any lc center of $(X,B)$ an irreducible component of an intersection ...
Hu Zhengyu's user avatar
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Birational model of a log smooth pair

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...
Hu Zhengyu's user avatar
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Groebner basis of a toric ideal

I know about toric ideals that it is a sort of binomial ideal i.e. generated by $x^u - x^v$, where $Au = Av $ ( A is the associated matrix). So by finding all integer solutions of $AX = 0$, can we ...
anjan samanta's user avatar
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Equivariant cohomology algebra of toric variety

Let $X$ be a complex projective and smooth toric variety of complex dimension $n$. It is acted by the real torus $T=(S^1)^n$. Is it true that the $T$-equivariant cohomology $H^*_T(X,\mathbb{Z})$ ...
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On two different descriptions of Delzant polytopes

I have seen two different ways of describing a Delzant polytope: From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ ...
BrianT's user avatar
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(Implemented) algorithm for Hodge numbers

Let $X$ be a smooth projective toric variety. Do any of the math computer algebra systems have an algorithm implemented to compute the Hodge numbers of a generic complete intersection in $X$? Say in ...
Philip Engel's user avatar
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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
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Lines on a toric cubic surface with a line of nodes

Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...
UserUser's user avatar
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Local toric varieties and tropicalization

Let $K$ be a valued field, and consider the ring $R=K((x_1,\dots,x_m))$ of formal Laurent series. This is "the germ of the torus at $0$". Is there a theory of "local toric varieties" where $R$ ...
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Is the boundary divisor of a smooth projective toric variety an snc divisor?

Let $X$ be a smooth toric projective variety. Let $T$ be the big torus acting on $X$. Let $D=X\backslash T$ be the boundary divisor. Question 1. Will $D_i$ be a smooth toric projective variety for ...
Sheng Meng's user avatar
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Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
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Polynomial size embeddings of toric varieties from polytopes?

Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan. $X_P$ is always projective, because the collection of characters corresponding to the points $\...
Elle Najt's user avatar
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A characterisation of faces of rational polyhedral cones

This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
Fred Rohrer's user avatar
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Sections of Cartier divisors on toric varieties

Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring $$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$ Define $\deg(x_{\rho}) = D_{\rho}$. Now, take a divisor $D = \...
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Toric Fan for the Du Val's singularities D_n and E_n

Let us consider the Du Val's singularities. i.e. https://en.wikipedia.org/wiki/Du_Val_singularity. It is well known that they are classified by ADE, because the exceptional divisors arising in the ...
Federico Carta's user avatar
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Hyperplane sections of non-singular projective toric varieties

Let $X^n\subset \mathbb{P}^N$ be an embedding of a non-singular projective toric variety (where variety stands for a reduced irreducible scheme over $\mathbb{C}$, and toric means normal variety ...
Grigory Solomadin's user avatar
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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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Is the minimal Chern number of a toric manifold at least 2?

I would like to show that the minimal Chern number $N_M$ of a toric manifold $M$ is at least $2$, where $$ N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \...
BrianT's user avatar
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Inferring properties of toric manifolds through Delzant's description

Let $(M,\omega, \mathbb{T})$ be a symplectic toric manifold. It is well-known that the properties of $M$ can be retrieved by looking at the moment polytope $\Delta$ image of the momentum map $$ \mu : ...
BrianT's user avatar
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1 vote
1 answer
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Secondary fan and KN strata

Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...
Jesse Huang's user avatar
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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
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Hypertoric varieties in dimension 4?

Are the only smooth hypertoric varieties in real dimension 4 obtained as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?
Filip's user avatar
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7 votes
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Why only some del Pezzo are toric?

Let us define smooth del Pezzo surfaces $dP_r$ as the blowup of $r$ generic points in $\mathbb{CP}_2$. One can show that if we request $dP_r$ to be Fano, then $r=0,...,8$. In theoretical physics ...
Federico Carta's user avatar
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A generalization of toric varieties

Let $M$ be a monoid with cancelation whose groupification is $\mathbb Z^d$ ($d$ finite). Even without assuming a finite generation of $M$, it seems to me that (a) $X=Spec\, \mathbb C M$ contains the ...
Adam's user avatar
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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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10 votes
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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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6 votes
2 answers
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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 $...
Igor Makhlin's user avatar
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1 vote
1 answer
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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 ...
Avi Steiner's user avatar
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6 votes
1 answer
222 views

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 ...
Kovalevskaia's user avatar
1 vote
0 answers
41 views

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\...
Avi Steiner's user avatar
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1 vote
0 answers
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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 ...
BrianT's user avatar
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3 votes
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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 ...
Avi Steiner's user avatar
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1 vote
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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 ...
BrianT's user avatar
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6 votes
1 answer
361 views

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(\...
Walter Simon's user avatar
2 votes
0 answers
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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})...
BrianT's user avatar
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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 $...
BrianT's user avatar
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3 votes
1 answer
177 views

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 $\sigma\subseteq V$...
Fred Rohrer's user avatar
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9 votes
1 answer
882 views

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 ...
Igor Makhlin's user avatar
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6 votes
0 answers
172 views

"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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