Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
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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 $\sigma\subseteq V$...
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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 ...
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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 ...
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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 $k$ ...
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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.
...
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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 ...
user120812
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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?
user120812
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How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?
Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. ...
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Holomorphic line bundles on arbitrary simplicial toric varieties as restrictions
In the question titled "Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$" it was explained how any holomorphic line bundle on $\mathbb P^1 \times \mathbb P^1$ is of the form $\...
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2-faces of reflexive Delzant polytopes
Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?
Motivation. I would like more generally to get an answer to the following question:
...
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Blow-ups of toric varieties
I'm interseted in blow-ups of toric varieties, unfortunately, I don't understand the construction of a blow-up built by a refinement of a fan, if to be more specific, I didn't find any constructions.
...
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Connection on line bundle over general simplicial toric variety
In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...
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Can any simplicial toric variety be embedded in a product of projective spaces?
In this question - On a Hirzebruch surface. , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$.
My question is, does such an isomorphism exist ...
- 1,135
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
- 1,135
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Number of boundary divisors and colors of a Spherical variety
Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-...
user75933
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Locally toric resolutions of compactifications
Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
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Volume of $-K_X$ for a weighted projective variety
Let $X:=\mathbb P(a_0,a_1, \ldots, a_n)$ be a well formed weighted projective variety. Let $-K_X$ be its anticanonical divisor, then how to express its volume ${\rm vol}(-K_X)=(-K_X)^n$ in terms of $...
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When are these definitions of "toric variety" equivalent?
Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, ...
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Toric Desingularization Algorithms
There are certainly many algorithms to desingularize toric varieties (e.g https://arxiv.org/pdf/math/0411340.pdf). I would imagine in analogy with desingularizing surfaces these all involve blowing up ...
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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 ...
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Is the Chow ring of a wonderful model for a hyperplane arrangement isomorphic to the singular cohomology ring?
In the article "Hodge theory for combinatorial geometries" by Adiprasito, Huh and Katz, it it claimed in the proof of theorem 5.12 that there is a Chow equivalence between the de Concini-Processi ...
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When is the blow-up of a closed sub-variety of a toric variety a toric variety?
If $ X $ is a toric variety and one has a closed sub-variety $ Y \subseteq X $, is the blow-up $ \operatorname{Bl}_{Y}(X) $ a toric variety as well? I suspect not, but wanted to check with other ...
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Are rational varieties symplectically rationally connected?
Was it proven already that smooth rational complex projecitve varieties are symplectically rationally connected? I.e. some GW invariant with two point insertions is non zero. What about smooth toric ...
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When does a discrepant toric resolution induce a crepant resolution of a subvariety?
Let $Y$ be a complete intersection in a complete simplicial toric variety $X_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X_\Sigma)$. Suppose that $\phi:X_{\widehat{\Sigma}}\...
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Resolving $\mathbb Z_n$ action on $\mathbb C^2$
Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...
- 14k
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Relative canonical divisor associated to toric morphism induced by refinement of fan
Let $\phi:X'\to X$ be a morphism between toric varieties $X=X(\Delta), X'=X'(\Delta')$, induced by a refinement $\Delta'$ of $\Delta$. This refinement is obtained from a sequence of stellar ...
- 2,768
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Understanding projective space as fibrations of tori over spaces with boundaries
The toric manifold $\mathbb{CP}^1$ can be understood as a circle fibration over an interval $I$, with the circles having zero radius at the boundaries of the interval. How does one generalize this ...
- 1,135
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Why are the toric fibers of a toric manifold Lagrangian submanifolds?
How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general ...
- 1,135
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toroidal compactifications of modulis spaces of ppav's
Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...
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Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety
It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X_\Delta$.
My question is whether there is a relationship between the number of maximal cones ...
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Toric variety defined by the Weyl orbit of a minuscule weight
Let $\Phi$ be a (reduced, crystallographic) root system with Weyl group $\mathcal{W}$, and $p$ a (nonzero) minuscule weight for $\Phi$: its orbit $\mathcal{W}p$ is the set of vertices of a convex ...
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Application of toric varieties for problems that do not mention them
I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?
To give ...
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Is there a "fundamental theorem of toric geometry"?
I have some questions about toric geometry.
1) Can any toric variety (I mean not necessarily smooth or projective, ...) be constructed from a fan ?
2) Suppose $T_1$ and $T_2$ are toric varieties ...
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Moment map for complete flags variety
Let $M:=U(n)/T^n$ be a complete flag variety, where $U(n)$ is an unitary group and $T^n \simeq (S^1)^n$ consists of its diagonal matrices. I have heard the following construction of a symplectic ...
- 1,990
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2
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The boundary of toric varieties
Let $\mathcal{X}$ be a toric variety, with $T$ a torus embedded as an open set
in $\mathcal{X}$ (and where the algebraic action of $T$ extends to $\mathcal{X}$). As I am not a toric specialist at all, ...
- 4,353
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vote
0
answers
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Equivariant derived category versus graded derived category
Everything here has the Zariski topology.
Let $T=(\Bbb{C}^*)^d$, and define an action of $T$ on $\Bbb{C}^n$ by $$t\cdot x=(t^{\mathbf{a}_1}x_1,\ldots, t^{\mathbf{a}_n}x_n).$$ Here $\mathbf{a}_1,\...
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How to calculate the top Chern class of a "functorial" vector bundle on a moduli space of sheaves?
Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...
- 1,990
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Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
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Taking powers of polytopes
I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known.
Imagine this situation where I want to think of such a thing : say ...
- 2,146
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Is there any structure theorem for piecewise linear functions?
I was wondering if such statements are known like "any piecewise linear function from $\mathbb{R}^d \rightarrow \mathbb{R}$ can be written as $\sum_{i=1}^k \alpha_i (\text{ some $2$ piece linear ...
- 2,146
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Chow rings of smooth toric varieties
In his 1987 article The geometry of toric varieties Danilov gives a combinatorial presentation of Chow rings of complete smooth toric varietes. Given a complete unimodular fan $\Sigma$ we have
$$
A^*(...
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In a nonsingular complex toric variety, is an algebraic cycle over a facet of the quotient polytope integral?
Let $X$ be a nonsingular complex toric variety with moment map $\mu : X \to P$ over a convex polytope $P$. Given a facet $F$ of $P$, its preimage $\mu^{-1}(F)$ is a complex codimension 1 subvariety of ...
user94803
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Projectively equivalent toric varieties
Say I have two normal lattice polytopes $P$ and $Q$ in $\mathbb{R}^n$ (lattice $\mathbb{Z}^n$) with the same number of lattice points $N+1$. Then they define two toric varieties $X_P$ and $X_Q$ which ...
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A question about smooth convex lattice polygons
Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...
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Constructive Resolution of Toric Singularities via Model Theory
Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
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Toric structures on projective space
Consider the symplectic manifold $\mathbb P^n$ equipped with the Fubini-Study symplectic form $\omega$. Given $n+1$ "generic" points $z_0,\dots,z_n$ on $\mathbb P^n$, is there an effective Hamiltonian ...
- 768
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Cohen-Macaulay non-normal toric variety
Given a quasi-smooth toric variety $X$ in the sense of Gelfand-Kapranov-Zelevinsky,
i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$-factorial toric variety and ...
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When is $\mathbb C^d\setminus\mathcal Z$ simply connected?
Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho_1,\cdots,\rho_d\}$. Consider the co-ordinate ring $\mathbb C[x_1,\cdots,x_n]$. Let $\mathcal Z=\bigcup_C\mathcal V(x_i\ |...
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Is there a relation between the singularities and the divisor class group of a simplicial toric variety
Let $\Delta$ be a simplicial fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges and $\{u_1,\cdots,u_d\}$ are the primitive vectors along the edges. Let $A$ be the divisor class group of the ...
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2
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Clarification on the definition of a quotient singularity
I am working on the quotient construction of a simplicial toric variety as described in chapter 5 of this book. I have tried the following two examples -
The fan $\Delta$ in $\mathbb R^2$ consists of ...
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