Questions tagged [toric-varieties]

Toric variety is embedding of algebraic tori.

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3 votes
0 answers
102 views

What sort of spaces show up as intersection complexes of toric degenerations of Calabi-Yau Varieties?

Roughly, a toric degeneration is a proper flat family $f:\mathcal{X}\to D$ of relative dimension $n$ with the properties that $\mathcal{X}_t$ is an irreducible normal Calabi-Yau and $\mathcal{X}_0$ is ...
1 vote
0 answers
99 views

Seeing $\mathbb{CP}^2 \mathbin\# \overline{\mathbb{CP}^2}$ as a symplectic reduction of different manifolds

I have been reading the paper "Remarks on Lagrangian intersections on toric manifolds" by Abreu and Macarini, which gives several non-displaceability results by avoiding the use of ...
1 vote
1 answer
64 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
87 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
72 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
210 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 ...
4 votes
3 answers
2k views

Are projective toric varieties, locally complete intersection?

Let $X^n \subset \mathbb{P}^N$ to be a toric projective variety. Is $X$ a local complete intersection? Is being a local complete intersection an intrinsic property, independent of embedding?
2 votes
0 answers
139 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}...
3 votes
1 answer
110 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 \...
1 vote
1 answer
202 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 ...
2 votes
0 answers
110 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
0 answers
121 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 $...
10 votes
1 answer
799 views

Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...
2 votes
1 answer
95 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
301 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
180 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
115 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
177 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 $...
19 votes
2 answers
1k views

About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
6 votes
2 answers
359 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 ...
1 vote
0 answers
100 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 ...
5 votes
1 answer
446 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 $\...
0 votes
1 answer
97 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
341 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 ...
6 votes
1 answer
349 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
77 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 ...
2 votes
1 answer
243 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
0 answers
171 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
249 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
244 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
264 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 ...
7 votes
2 answers
589 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 $(\...
3 votes
1 answer
182 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?
20 votes
5 answers
1k views

From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
1 vote
2 answers
410 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
162 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 ...
2 votes
1 answer
221 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 ...
0 votes
0 answers
132 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
167 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 ...
4 votes
1 answer
545 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 ...
4 votes
1 answer
217 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 $ ...
0 votes
1 answer
259 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
98 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 ...
2 votes
0 answers
198 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 $R$ ...
3 votes
1 answer
437 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 $(-1,0)$,...
3 votes
0 answers
90 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 ...
4 votes
1 answer
312 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
175 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
236 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 ...
6 votes
2 answers
683 views

What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...

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