# Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

272
questions

1
vote

1
answer

50
views

### Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

2
votes

0
answers

76
views

### When is a smooth point of a projective, simplicial, toric variety $ X_{\Sigma} $ compatibly $ F $-split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

0
votes

0
answers

45
views

### How to compute the multiplicity of a strongly convex, rational, polyhedral cone $ \sigma $?

In David Cox, John Little and Hal Schenck's book Toric Varieties page 302, Chapter 6, Section 4, Proposition 6.4.4, the authors state the following proposition. If $ \Sigma $ is a simplicial fan of ...

3
votes

0
answers

131
views

### When is a wonderful compactification a toric variety?

Given a (projective) hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$, in which we assume the intersection of all hyperplanes is $0$, and a building set $G$, De Concini and Procesi define the ...

1
vote

0
answers

80
views

### Line bundles on toric varieties associated to Weyl chamber

I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...

2
votes

1
answer

72
views

### What is the subdivision corresponding to the blowup of a toric divisor of a singular toric variety?

Let $X$ be an affine toric variety corresponding to the cone $\sigma$. If $X$ is smooth, blowups of toric strata correspond to star subdivisions of $\sigma$. Suppose that $X$ is singular and let $D \...

2
votes

0
answers

67
views

### Does one only need to look at torus invariant curves to calculate the Seshadri constant for a point of a toric variety?

If $ X $ is an irreducible projective variety, $ L $ is a Nef divisor on $ X $, $ x $ is a point of $ X $, and $ \pi: \operatorname{Bl}_{x}(X) \to X $ is the natural projection morphism, then the ...

1
vote

1
answer

167
views

### Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety

Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...

1
vote

0
answers

95
views

### Doubt in the proof of Mcduff''s method of probes

I have been reading the paper "Displacing Lagrangian toric fibers by probes" by Dusa Mcduff, here is the arxiv link https://arxiv.org/pdf/0904.1686.pdf.
I have a doubt in the proof of lemma $...

2
votes

1
answer

79
views

### What is the minimal $ n $ such that the Cox ring of the blow up of a simplicial, $ r $-dimensional, toric variety at $ n $ points in g.p. is not f.g.?

In Shigeru Mukai's paper "Counterexample to Hilbert's 14th Problem for the 3-dimensional Additive Group," Mukai proved that if $ \frac{1}{r+1}+\frac{1}{n-r-1} \le \frac{1}{2} $, then the ...

5
votes

0
answers

202
views

### Do algebraic tori have no $H^1$?

If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...

4
votes

1
answer

83
views

### Almost toric mutations

I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,...

3
votes

0
answers

149
views

### Resolutions of configuration space of the projective line where the complement is of "Tate type"

I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...

5
votes

0
answers

85
views

### Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?

[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_m,f_n \ne 0$ ...

1
vote

0
answers

85
views

### Simplicial approximation theorem for toric varieties

Given abstract simplicial complexes $K$ and $L$, one constructs topological spaces $|K|$ and $|L|$. Simplicial approximation theorem says for any continuous map $f: |K|\to |L|$ that there exists ...

0
votes

1
answer

93
views

### Lattice polytope toric varieties under rescaling

Is there any interesting relation/map between the toric variety $X_P$ associated to a lattice polytope $P$ and the toric variety associated to the polytope rescaled by some integer factor?

2
votes

1
answer

229
views

### Quasismooth vs smooth in a smooth toric variety

Let $X$ be a toric variety, and let $\pi: \mathbb A^n-V(B) \to X = (\mathbb A^n-V(B))/(\mathbb C^*)^\rho$ be the quotient map defining $X$ in the Cox construction. A subvariety $Y\subset X$ is called ...

3
votes

1
answer

194
views

### Is there a Chevalley map for spherical varieties?

If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...

1
vote

0
answers

52
views

### How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$

I am given that $k$ is a field and $G$ is the monoid consisting of all monomials
$X^iY^j$, where $j$ is between $0$ and $3i$.
I am trying to compute the quotient of the monoid algebra $kG$ by the ...

3
votes

0
answers

167
views

### A question regarding Corollary 4.12 in Mumford's "Analytic construction of deg. Ab. Var."

Let $S=Spec A$ be the spectrum of an integrally closed, excellent and Noetherian ring. In his paper, David Mumford constructs an $S$-group scheme $G$. He shows that the torsion of $G_s$ is $p$-...

1
vote

1
answer

191
views

### A connection between equivariant and non-equivariant cohomology of toric variety

Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$.
The $T$-equivariant cohomology $H^*_T(X)$ (with coefficients in a field, say) is an ...

2
votes

1
answer

161
views

### Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...

5
votes

0
answers

195
views

### Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...

3
votes

1
answer

168
views

### If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?

Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?

7
votes

2
answers

521
views

### Is $(x^2y,xy^2)$ log smooth?

Consider the map
$$f:\mathbb C^2\to\mathbb C^2$$
$$(x,y)\mapsto(x^2y,xy^2)$$
We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...

5
votes

2
answers

278
views

### From Delzant polytope to lattice polytope

By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But
is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the ...

2
votes

2
answers

354
views

### Embedding of a blow-up

In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H_1,H_2$ of type $(2,1)$ and set $Y = X\cap H_1\cap H_2$.
Let $Z$ be the blow-up of $X$ ...

3
votes

0
answers

139
views

### Smooth proper varieties over the integers that are not toric

Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...

0
votes

0
answers

111
views

### Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$

This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...

3
votes

0
answers

120
views

### Polytope algebra and toric vareties

Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.
For a simple polytope $P$ let us ...

2
votes

1
answer

199
views

### Equivalence of sequences of blowups of $\mathbb{P}^3$

Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...

4
votes

1
answer

367
views

### Cohomology of divisors on Hirzebruch surfaces

Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...

0
votes

1
answer

216
views

### What is a toric lattice? [closed]

What is a toric lattice? and how can I construct one in Macaulay2 and compute its basis? is there any alternative method to make one? Since I went through the whole ...

2
votes

0
answers

88
views

### Amoeba for a K3 surface in $\mathbb {CP}^3$

Let $X=X_\Delta$ be the toric variety associated to a reflexive polyhedron $\Delta$.
Consider a Calabi-Yau hypersurface $Y\subset X$, and the image of $Y$ under the moment map $\mu:X\to \Delta$ has ...

3
votes

1
answer

176
views

### How to create a toric variety whose Cox ring has a specific grading?

If one wanted to obtain a fan for a toric variety of dimension $ n>1 $ whose Cox ring is $ \mathbb{Z}^{2} $ graded with weights $ \{(a_{i},b_{i})\}_{i=1}^{n+2} $, then one could let $ B $ be the $ ...

2
votes

1
answer

273
views

### When does the Hirzebruch surface have a nef anticanonical divisor?

Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...

12
votes

0
answers

299
views

### Rational points of weighted projective spaces

Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive integers. Denote by $P_\underline{a}$ the corresponding weighted projective space over $k$, i.e. the quotient of $U:=\...

3
votes

0
answers

77
views

### Structure of fibers of (complex) moment map of hypertoric variety

I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an ...

3
votes

0
answers

132
views

### Smooth toric compactification of $\mathbb C^n$

By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...

1
vote

1
answer

205
views

### divisors in non-compact toric varieties

Let $X$ be a smooth quasi-projective toric variety of dimension $n$ over $\mathbb C$.
Take it to be non-compact, so its fan is not complete.
(A good example to keep in mind is a toric Calabi-Yau.)
If ...

3
votes

1
answer

199
views

### Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...

4
votes

0
answers

82
views

### Toric Bézier patches

Toric Bézier patches (as described in https://arxiv.org/abs/0706.2116) are maps from a lattice polytope $P$ to the positive part of its associated toric variety $X_P$. While they are not the inverse ...

2
votes

0
answers

128
views

### Deformation of toric varieties to complete intersections

I am looking for some systematic study/examples of families of projective complete intersection varieties degenerating to a projective toric variety. In particular, given a projective toric variety, ...

14
votes

0
answers

339
views

### How should we think about the algebraic moment map?

My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...

4
votes

0
answers

187
views

### When is a toric variety a Poincare duality space?

When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a ...

2
votes

1
answer

159
views

### Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...

3
votes

1
answer

204
views

### When an action on open dense subvariety by an algebraic group extends to variety

A toric variety $X$ over $k$ is a variety which contains an algebraic torus
($T= \mathbb{G}_k^s$)
as a dense open subset such that the action of the torus on itself extends to the whole of
$X$. Slogan:...

4
votes

1
answer

247
views

### the map on Picard groups induced by restriction to a toric subvariety

Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character ...

3
votes

0
answers

146
views

### Hodge structure on intersection cohomology of toric varieties

Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the ...

3
votes

1
answer

214
views

### Cohomology ring of a hypersurface in toric variety

Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$.
...