Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

Filter by
Sorted by
Tagged with
2
votes
0answers
39 views

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 ...
5
votes
2answers
242 views

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 ...
2
votes
1answer
59 views

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 $p_P(t)$ ...
0
votes
0answers
51 views

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 ...
2
votes
0answers
65 views

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 ...
1
vote
0answers
59 views

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 ...
2
votes
0answers
75 views

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 ...
1
vote
0answers
67 views

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 ...
2
votes
0answers
66 views

Birational contraction of toric vector bundle

Let $X$ be the toric vector bundle over $\mathbb{P}(1,1,1,2)$ with grading matrix $$ \left(\begin{array}{cccccc} 1 & 1 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 & ...
1
vote
0answers
60 views

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 ...
5
votes
1answer
228 views

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 $H^*_T(X,\mathbb{Z})$ ...
3
votes
0answers
62 views

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 $\mathbb{R}^{n*}$ ...
3
votes
0answers
84 views

(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 ...
3
votes
0answers
127 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 ...
2
votes
1answer
127 views

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 ...
1
vote
0answers
78 views

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 $R$ ...
1
vote
0answers
68 views

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 ...
2
votes
0answers
106 views

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 ...
4
votes
1answer
171 views

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 $\...
5
votes
1answer
139 views

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 neither ...
3
votes
1answer
112 views

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 = \...
3
votes
0answers
118 views

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 ...
1
vote
0answers
119 views

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 ...
5
votes
0answers
179 views

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 ...
3
votes
1answer
209 views

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 \...
2
votes
0answers
116 views

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 : ...
1
vote
0answers
77 views

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 ...
4
votes
0answers
96 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 ...
3
votes
0answers
58 views

Hypertoric varieties in dimension 4?

Are the only smooth hypertoric varieties in real dimension 4 obtained as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?
6
votes
1answer
404 views

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 ...
2
votes
0answers
94 views

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 ...
4
votes
0answers
157 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 ...
10
votes
0answers
199 views

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 ...
6
votes
1answer
187 views

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 $...
1
vote
1answer
82 views

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 ...
6
votes
1answer
162 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 ...
1
vote
0answers
37 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\...
1
vote
0answers
79 views

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 ...
2
votes
0answers
32 views

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 ...
1
vote
0answers
27 views

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 ...
6
votes
1answer
228 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(\...
2
votes
0answers
59 views

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})...
0
votes
0answers
77 views

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 $...
3
votes
1answer
63 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 $\sigma\subseteq V$...
7
votes
1answer
404 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 ...
5
votes
0answers
140 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 ...
5
votes
0answers
87 views

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 $k$ ...
1
vote
0answers
59 views

Fibration of a toric symplectic manifold from a fibration of the moment polytope

This question is regarding fiber bundles, both whose fibers and total space are toric symplectic manifolds. The structure group on the fiber is a subgroup of the structure group of the total space. ...
2
votes
0answers
198 views

Set theoretic complete intersections in toric varieties

Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety? Is there an example of a smooth ...
4
votes
1answer
460 views

Complete intersections in toric varieties

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?

1
2 3 4 5