# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

211 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 $p_P(t)$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
66 views

116 views

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 : ... 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 ... 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 ... 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}? 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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},$$where the volume of a d-dimensional polytope P\... 0answers 79 views ### Effective classes in toric Kähler manifolds In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold (M,\omega) as a Kähler manifold with Kähler form \omega, and ... 0answers 32 views ### condition on rational polyhedral cone to guarantee dual cone is homogeneous Let \sigma\subseteq \Bbb R^d be a full-dimensional rational polyhedral cone which is strongly convex (i.e. \sigma\cap-\sigma=0). Definition. The cone \sigma is homogeneous if there are ... 0answers 27 views ### A regular sequence in a quotient by a “half lattice” defined by a toric manifold I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient$$P^{-1}(p) / \mathbb{K}$$by the ... 1answer 228 views ### Isomorphic equivariant sheaves are equivariantly isomorphic on a toric variety Let X be a toric variety containing the n-torus T\overset{i}{\hookrightarrow} X. The action of T extends naturally to an action on the sheaf i_*\mathcal{O}_T by$$(\alpha\cdot f)(x):=f(\...
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})...