Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
315
questions
26
votes
3
answers
2k
views
Is there always a toric isomorphism between isomorphic toric varieties?
Suppose two toric varieties are isomorphic as abstract varieties. Does it follow that there exists a toric isomorphism between them?
Edit: the comments below lead me to believe that I'm using the ...
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 ...
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 $v$ ...
18
votes
2
answers
8k
views
The canonical line bundle of a normal variety
I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
17
votes
2
answers
3k
views
What are some open problems in toric varieties?
In light of the nice responses to this question, I wonder what are some open problems in
the area of toric geometry? In particular,
What are some open problems relating to the algebraic ...
16
votes
1
answer
663
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 ...
16
votes
0
answers
1k
views
Homology classes of subvarieties of toric varieties
Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety.
Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero?
Background
If $X$ is a Kaehler variety, this is of ...
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 ...
14
votes
0
answers
436
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 ...
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
$$
A^*(...
12
votes
2
answers
1k
views
Is an affine "G-variety" with reductive stabilizers a toric variety?
Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose
$x\in X$ is a $G$-...
12
votes
0
answers
397
views
Rational points of weighted projective spaces
[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.]
Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
12
votes
0
answers
524
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
11
votes
2
answers
645
views
Detecting tilings by toric geometry
This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
11
votes
1
answer
843
views
Proving that a variety is not (isomorphic to) a toric variety
Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the ...
11
votes
0
answers
665
views
Big tangent bundle
Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
10
votes
2
answers
2k
views
Counting/constructing Toric Varieties
Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these ...
10
votes
3
answers
1k
views
Hamiltonian $S^1$ actions with isolated fixed points
I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $...
10
votes
1
answer
797
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 ...
10
votes
1
answer
1k
views
Cox rings of toric varieties over arbitrary fields
The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous ...
10
votes
0
answers
372
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 ...
9
votes
2
answers
884
views
Derived categories of toric varieties and convex geometry
Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).
One ...
9
votes
1
answer
874
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 ...
9
votes
0
answers
346
views
Computing Ext for toric divisors
Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...
8
votes
1
answer
1k
views
Are quadric hypersurfaces toric varieties?
A quadric hypersurface (over an algebraically closed field of characteristic zero) in $\mathbb{P}^n$ for $1\leq n\leq 3$ is a toric variety. (Namely, it's isomorphic to $\mathbb{P}^1\times\mathbb{P}^...
8
votes
2
answers
1k
views
Learning Quantum (Co)Homology and Landau Ginzburg Superpotential
I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer theory",...
8
votes
1
answer
1k
views
Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
8
votes
3
answers
1k
views
relation between toric geometry and log geometry
Hello,
I'm trying to understand the relation between the points of view of
log geometry (monoids) and toric geometry (fans).
Suppose that $k$ is a field and $P$ is a finitely generated monoid.
Then $...
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 ...
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 ...
7
votes
3
answers
1k
views
Has anyone researched additive analogues of toric geometry in characteristic zero?
One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...
7
votes
2
answers
2k
views
nef Cone of a Toric Variety
Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on ...
7
votes
1
answer
510
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
282
views
Separating a lattice simplex from a lattice polytope
Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...
7
votes
1
answer
818
views
Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?
It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...
7
votes
2
answers
2k
views
Deformations of Hirzebruch surfaces and toric action
Hi,
the Hirzebruch surface $F_n$ admits a deformation for $0\leq m\leq n$ defined by the equation
$$
\mathcal{M}=\{ ([x_0:x_1],[y_0:y_1:y_2],t) \in \mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{C}...
7
votes
1
answer
483
views
When are these definitions of "toric variety" equivalent?
Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, ...
7
votes
2
answers
587
views
Is $(x^2y,xy^2)$ log smooth?
Consider the map
$$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 $(\...
7
votes
1
answer
1k
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 ...
7
votes
0
answers
256
views
Cohomology of fibers of a morphism of a blowup of affine space
Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the ...
7
votes
0
answers
154
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=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
7
votes
0
answers
338
views
Derived category of toroidal varieties
This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem.
A ...
7
votes
0
answers
634
views
When should a moment polytope have "smooth" faces?
A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...
6
votes
2
answers
394
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 ...
6
votes
1
answer
3k
views
Clean introduction to toric varieties for an undergraduate audience
I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need ...
6
votes
2
answers
358
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 ...
6
votes
2
answers
526
views
Are the closures of the tori in the decomposition of a torified variety toric varieties?
In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed ...
6
votes
2
answers
296
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 $...
6
votes
1
answer
346
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 $...
6
votes
1
answer
359
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(\...