Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
318
questions
3
votes
1
answer
179
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 $\sigma\subseteq V$...
9
votes
1
answer
889
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 ...
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 ...
5
votes
0
answers
161
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
0
answers
84
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
0
answers
274
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
1
answer
818
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
vote
0
answers
84
views
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 $\mathbb{F}_3$. ...
2
votes
0
answers
112
views
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 $\...
4
votes
1
answer
181
views
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:
...
4
votes
1
answer
1k
views
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.
...
1
vote
0
answers
52
views
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)/(\...
0
votes
1
answer
402
views
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 ...
8
votes
1
answer
2k
views
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 ...
4
votes
1
answer
190
views
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$-...
2
votes
1
answer
174
views
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 ...
7
votes
1
answer
518
views
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 $...
7
votes
1
answer
487
views
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, ...
3
votes
1
answer
164
views
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 ...
5
votes
0
answers
128
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 ...
16
votes
1
answer
668
views
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 ...
2
votes
1
answer
356
views
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 ...
2
votes
0
answers
191
views
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 ...
4
votes
1
answer
98
views
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}}\...
5
votes
1
answer
167
views
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 ...
3
votes
1
answer
246
views
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
votes
0
answers
191
views
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 ...
3
votes
1
answer
194
views
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 ...
2
votes
0
answers
150
views
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 ...
3
votes
1
answer
244
views
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 ...
4
votes
1
answer
318
views
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 ...
15
votes
4
answers
1k
views
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 ...
2
votes
0
answers
273
views
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 ...
4
votes
1
answer
606
views
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 ...
2
votes
2
answers
604
views
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, ...
1
vote
0
answers
220
views
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).$$ Here $\mathbf{a}_1,\...
8
votes
0
answers
675
views
How to calculate the top Chern class of a "functorial" vector bundle on a moduli space of sheaves?
Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...
2
votes
0
answers
103
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
3
votes
1
answer
151
views
Taking powers of polytopes
I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known.
Imagine this situation where I want to think of such a thing : say ...
6
votes
1
answer
919
views
Is there any structure theorem for piecewise linear functions?
I was wondering if such statements are known like "any piecewise linear function from $\mathbb{R}^d \rightarrow \mathbb{R}$ can be written as $\sum_{i=1}^k \alpha_i (\text{ some $2$ piece linear ...
13
votes
1
answer
1k
views
Chow rings of smooth toric varieties
In his 1987 article The geometry of toric varieties Danilov gives a combinatorial presentation of Chow rings of complete smooth toric varietes. Given a complete unimodular fan $\Sigma$ we have
$$
A^*(...
2
votes
1
answer
163
views
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 ...
2
votes
1
answer
215
views
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 ...
3
votes
0
answers
149
views
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 ...
2
votes
0
answers
115
views
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?...
3
votes
1
answer
320
views
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 ...
3
votes
1
answer
402
views
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 ...
5
votes
2
answers
323
views
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\ |...
4
votes
1
answer
240
views
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 ...
4
votes
2
answers
3k
views
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 ...