Questions tagged [toric-varieties]
The toric-varieties tag has no usage guidance.
70
questions
0
votes
0
answers
10
views
The action of the torus $T_N$ on the semigroup algebra $C[M]$
Toric Varieties written by David Cox, John Little and Hal Schenck
p.18 Equivalence of constructions
Before starting our main result, we need to study the action of the torus $T_N$ on the semigroup ...
- 1
0
votes
0
answers
17
views
existence of affine open torus-invariant subspaces
I am looking for a reference text which contains a proof of the following fact.
If $\phi
:X\to Y$ is a morphism of toric varieties and $X$ is affine, then $\phi$ factors through a torus-invariant ...
- 3,116
0
votes
0
answers
42
views
Torus action on a prime divisor of Toric Varieties
Let △ be a fan, X(△) be a toric variety and D be a prime divisor of X(△).
Let Tn be a torus and t∈Tn. Then why is t*D a prime divisor of X(△)?
- 1
2
votes
1
answer
104
views
Intersection of (Toric varieties) and (Flag varieties)
For me, a variety $X$ is assumed to be irreducible and normal over some algebraically closed field $k$ of characteristic zero.
I call $X$ a toric variety if it has an algebraic torus $T$ which embeds $...
- 13.3k
1
vote
0
answers
32
views
Reference request: Toric Varieties in reverse order
I am looking for a book/text which introduces toric varieties in reverse order. Instead of starting with lattices, cones, fans and convex geometry, it should start with a toric variety $X$ and define ...
- 3,116
7
votes
0
answers
104
views
For Hirzebruch Surfaces, does the tangent exact sequence split?
Consider the projection $\pi:\mathbb{F}_n \rightarrow \mathbb{P}^1$. Is it true that the following exact sequence $$0 \rightarrow T_{\pi} \rightarrow T_{\mathbb{F}_n} \rightarrow \pi^*T_{\mathbb{P}^1} ...
- 91
4
votes
0
answers
74
views
Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[I cross-posted this question as https://mathoverflow.net/questions/431454. (That version is also slightly updated.)]
Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \...
- 41
0
votes
0
answers
101
views
Is $V(y) \subset \mathbb C^2$ not an affine toric variety?
From Toric Varieties by Cox, Little, Schenck:
An affine toric variety is an irreducible affine variety $V$ containing a torus $T_N\cong (\mathbb C^*)^n$ as a Zariski open subset such that the action ...
- 11.4k
1
vote
0
answers
17
views
Is the ideal of a subscheme of a toric variety B-saturated?
Let $X$ be a normal toric variety over $\mathbb{C}$ with no torus factors. Let $S$ be the Cox ring of $X$, and $B \subset S$ be the irrelevant ideal. Given an ideal sheaf $\mathcal{I} \subseteq \...
2
votes
0
answers
102
views
Transition Functions of the Weighted Projective Space $\mathbb{P}^{2}_{\{3,2,1\}}$.
I am trying to find the transition functions of the weighted projective space $\mathbb{P}^{2}_{\{3,2,1\}}$. Bear with, I am very new to algebraic geometry.
To my understanding, $\mathbb{P}^{2}_{\{3,2,...
- 1,513
0
votes
0
answers
48
views
Definition of affine toric variety
Sorry for my bad English.
I'm trouble about a definition of an affine toric varieties.
I often see a definition of affine toric varieties as follow;
An affine toric variety is an irreducible affine ...
- 1,770
1
vote
0
answers
120
views
Introductory papers in Mirror Symmetry for algebraic geometry students
Assuming that a student knows algebraic geometry at the level of Cox's
(1) Ideals, Varieties and Algorithms
and
(2) Toric Varieties
What are some good papers (meaning, suitable in light of (1) and (2))...
- 713
3
votes
0
answers
173
views
How to learn math after your PhD is finished [closed]
Question: How does someone go about learning advanced topics in Math after they're done with their PhD?
Specific example:
You've done your undergrad and masters degrees in math and learned from ...
- 713
1
vote
0
answers
56
views
Gluing of $\operatorname{Spec} k[x,y], \operatorname{Spec} k[x^{-1},x^{-1}y],$ and $\operatorname{Spec} k[y^{-1},xy^{-1}]$
In Fulton's book on toric varieties he (on page 7) goes through the example of $\mathbb{P}^2$.
He considers the fan given by $((1,0),(0,1),(-1,-1))$ and shows via its dual that the variety ...
- 462
0
votes
1
answer
62
views
Homology of toric varieties
In the proof of theorem 5.1.1 of this paper by Dotsenko, Shadrin, and Vallette, they define this (smooth, normal) toric variety $B(n)$, note that there is a surjective toric morphism to $\mathbb{P}^1$ ...
- 1,315
0
votes
1
answer
47
views
Question regarding Proper Mapping
I'm reading toric variety from Cox et. al. and there is a theorem about proper mapping that I cant solve.
Let $f : X \rightarrow Y$ be a continuous mapping of locally compact first
countable Hausdorff ...
- 390
1
vote
0
answers
60
views
Action of $\mathbb C^*$ as torus of toric variety $\mathbb C$
I am reading Toric Variety now and as a well documented example we know $\mathbb{C^s}$ is toric variety with torus $\mathbb(C^*)^s$. I am wondering how will the action of $\mathbb(C^*)^s$ will look ...
- 390
0
votes
1
answer
83
views
Question on the Color of the Horospherical Variety $\mathrm{SL}_2(\mathbb{C})/U \hookrightarrow \mathrm{Bl}_0(\mathbb{C^2})$
I am now studying on the horospherical variety. For example, I am observing $\mathrm{SL}_2(\mathbb{C})/U$ where $U$ is a unipotent subgroup
$$
\begin{pmatrix}
1 & * \\ 0 & 1
\end{pmatrix}.
$$
...
- 994
1
vote
0
answers
47
views
Intersection multiplicity on toric varieties using simplicial cones
In the book "Introduction to toric varieties" by William Fulton (p. 100) there is a formula for the intersection product of two subvarieties. Let $\tau,\sigma$ be cones in the fan $\Delta$, ...
- 2,562
0
votes
0
answers
32
views
points in affine toric variety
I am reading Toric Variety from Cox, little and In proposition 1.3.1, I saw this proposition about which I have one question
As you can see, they have written that the map $\gamma: S \rightarrow \...
- 390
2
votes
1
answer
103
views
Torus $T=\text{Spec}(K[M])$ for the character lattice $M=\{(a_0,....,a_n)\in\mathbb{Z}^{n+1}\mid \sum_{i=0}^na_i=0\}$
For $N=M=\mathbb{Z}^n$ and $e_0,...,e_n$ standard basis of $\mathbb{Z}^n$ the torus $T=\text{Spec}(K[M])=\text{Spec}(K[x_1^{\pm 1},...,x_n^{\pm 1}])=\mathbb{G}_m^n$, where $x_i=\chi^{e_i}$ with ...
- 91
1
vote
0
answers
35
views
A question regarding the definition of an affine toric variety
I am reading about affine toric varieties from Toric Varieties by Cox, Little, and Shenck, and the definition of an affine toric variety is given as follows:
An affine toric variety is an irreducible ...
- 390
8
votes
0
answers
82
views
Where do toric varieties appear naturally?
I'm reading Fulton's book. There's an awesome theorem that classifies all smooth toric surfaces as blowups at points starting from either $P^2$ or some Hirzebruch surface.
I want to be more excited ...
- 1,844
1
vote
0
answers
54
views
initial degenerations
I am trying to understand the article "Realization spaces for tropical fans" https://arxiv.org/pdf/0909.4582.pdf by Eric Katz and Sam Payne.
Let $X\subset T$ be a subvariety of the torus.
...
- 91
2
votes
0
answers
49
views
How many tropical polynomials give rise to the same variety? (reference request)
An n-variable polynomial $p(x)=\bigoplus_{i=1}^n \beta_i \textbf{x}^{\alpha_i}$ (here $\textbf{x}$ is the tuple $(x_1,x_2,...,x_n)\in \mathbb{R}^n$) in the tropical (max-plus) semiring has an ...
- 363
0
votes
0
answers
49
views
Dense torus in $\mathbb{P}^n$
What is the dense torus in $\mathbb{P}^n$?
Is this the complement of all coordinate hyperplanes?
Is there only one possibility?
- 91
4
votes
2
answers
263
views
Normalization of an affine toric variety is toric
In the book "Toric Varieties" by Cox-Little-Schenck, Proposition 1.3.8 is left as an exercise. It essentially characterizes what the normalization of an affine toric variety is. The ...
- 693
2
votes
0
answers
92
views
Are Riemann surfaces toric?
For a smooth toric projective variety $X$, I'm able to use the result
$H^0(X,\mathcal{O}(D)) \cong \bigoplus_{m \in \Delta_D \cap M} \mathbb{C}\cdot \chi^m$
to understand global sections of a given ...
- 414
1
vote
0
answers
50
views
Why a polynomial factors in the coordinate ring of an affine chart of the blow up of $\mathbb{A}^{n+1}$ (as a toric variety) at the origin?
I am studying a theorem by Ishi in which she proves there is a resolution of the singularity of a hypersurface defined on de affine space considered as the toric variety given by the fan $\sum_i \...
- 11
0
votes
0
answers
95
views
closed orbit of smooth toric variety is defined by regular sequence
Let $X_\Sigma$ be a smooth toric variety corresponding to a fan $\Sigma$, $V(\sigma)\subseteq X_\Sigma$ the closure of the orbit $O(\sigma)$ corresponding to a cone $\sigma$ of $\Delta$. $V(\sigma)$ ...
- 91
2
votes
0
answers
117
views
Blow up of toric variety corresponds to subdivision of cone
Look at the lattice $N= \mathbb{Z}^3/\mathbb{Z}(1,1,2)$, let $u_0,u_1,u_2$ be the images of the standard basis elements of $\mathbb{Z}^3$ and consider the cone $\sigma = \text{Cone}(u_0,u_1)$. Then it ...
- 1,175
1
vote
0
answers
118
views
Weighted projective space as toric variety
I am trying to solve exercise 3.3.10 from Cox' book on toric varieties. I have no trouble with part (a), but it seems to me that for part (b) I have to wirte down the variety $X_{\Sigma}$ explicitly, ...
- 1,175
5
votes
0
answers
61
views
Is a toric variety still a toric variety after change of coordiantes?
I am reading Cox, Little, Shenck Toric Varieties, and I had a question about the definition of a toric variety. One of the examples (on page 12) uses
$$x^2 - y^3 = 0$$
as an example of a toric variety....
- 107
2
votes
1
answer
389
views
Isomorphic Lattices, complex Tori and their relation to Jacobians
Let $g >1$ a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
...
1
vote
0
answers
102
views
Confusion about Affine Toric Varieties/ Toric Ideals
I have just started reading about Toric varieties from David Cox and I have a couple of very basic questions about toric ideals and aff. toric varieties.
1) Eventually, a toric ideal can be defined ...
- 1,067
2
votes
1
answer
131
views
Q about torus automorphism example
I am working through the book "introduction to tropical algebra" and page 70 and 71 example 2.2.10 states
"An automorphism of the torus is an invertible map specified by
Laurent monomials. Thus the ...
- 107
1
vote
1
answer
162
views
Why does every toric ideal correspond to an affine toric variety?
I'm reading the book Toric Varieties by Cox, Little and Schenk, and have a small question about the proof of their Proposition 1.1.11 (on page 16). The key part of the proposition is this:
Claim: ...
- 1,335
0
votes
1
answer
279
views
A basis for the Picard group of the blow up at $\mathbb{P}^2$ at two points
(Toric) Intro: (Superflous, just to consider the initial data)
From a toric point of view, the blow up of $\mathbb{P}^2$ at two points (which we can assume are $[0:1:0]$ and $[0:0:1]$ is given by a ...
user618650
2
votes
2
answers
291
views
Couple of questions about Picard group of $\mathbb{C}^*$
I'd like to compute the Picard group of $\mathbb{P}^n\times \mathbb{C}^*$.
So using toric geometry I've easily found $\text{Cl}(\mathbb{P}^n\times \mathbb{C}^*)\simeq \text{Cl}(\mathbb{P}^n)\oplus\...
user618650
1
vote
0
answers
37
views
Critical values of the evaluation map for rational curves in toric surface
I'm looking at rational curves in a toric surface. Such curves have a parametrization of the form
$$t\dashrightarrow \chi \prod_{j=1}^m (t-\alpha_j)^{n_j} \in (\mathbb{C}^*)^2 ,$$
for some scalars $\...
- 11
1
vote
0
answers
174
views
Rational Normal Curve of degree d.
Let A consist of the columns of the $2\times (d+1)$ matrix
$$A=\begin{pmatrix}
d & d-1 & \cdots & 1&0\\
0 & 1 & \cdots & d-1 &d
\end{pmatrix}$$
Then consider the map
...
1
vote
1
answer
147
views
Pullback of divisors in toric varieties
Let $X=X_\Sigma$ and $X'=X_{\Sigma'}$ be toric varieties such that the fan $\Sigma'$ is a refinement of the fan $\Sigma$, and denote by $f:X'\to X$ the corresponding toric morphism.
Let $D=\Sigma a_\...
- 2,689
2
votes
1
answer
58
views
Irreducible varieties with easy computable class group
In Hartshorne, the author mentions that computing the class group and Picard group of an irreducible variety is in general a very hard problem. I would like to know some easy examples where these ...
user437748
5
votes
1
answer
93
views
Is smoothness required in order for this $\mathbb{F}_p$ point counting formula to hold?
Let $\Delta$ be a fan, and $X(\Delta)$ the toric variety.
The formula in section 7 here
asserts that $|X(\mathbb{F}_q)| = \sum_{k= 0}^n (q - 1)^k d_{n - k}$, where $d_j$ is the numberof $j$ ...
- 20.1k
0
votes
0
answers
49
views
Are proper toric varieties necessarily projective?
Let $X$ be a proper complex algebraic variety with an open subset isomorphic to $T=(\mathbb{C}^*)^n$, and the $T$ action on itself extends to $X$. Is $X$ necessarily projective?
user93417
1
vote
1
answer
86
views
Questions related to a map tensored with $\mathbb R$
Context
I'm studying Audin's book - Torus Action on Symplectic manifolds, chapter VII where she builds Toric varieties from fans.
It is defined on page 231 (2 ed) a linear map
$$\pi : \mathbb Z ^...
- 17.3k
6
votes
0
answers
535
views
Why is the algebraic torus an affine variety?
I am reading Toric Varieties by Cox, Little, and Schenck. I am stuck on the definition an algebraic torus, given in Part 1.1, page 10, which states:
The affine variety $(\mathbb{C}^*)^n$ is a group ...
- 731
0
votes
1
answer
136
views
When toric variety is affine?
Let $\Delta$ be a fan and let $X$ be a toric variety associated to $\Delta$. Is there a quick way to tell when $X$ is affine by looking at the fan?
Thanks.
- 138
1
vote
0
answers
23
views
(symmetric) generators for cohomology group of a del pezzo surface of degree 6
I'm working on the surface $X$ which is birational to the blowing-up of $\mathbb{P}^1\times \mathbb{P}^1$ at two points. When I consider its cohomology group $H^2(X,\mathbb{Z})$, I can use a basis as $...
0
votes
1
answer
128
views
When is a projective space smooth from the toric perspective?
I am studying the lecture notes on toric geometry (for physicists) by Bouchard (arxiv:hep-th/0702063) and Skarke (arXiv:hep-th/9806059). In these lecture notes, the authors give the following ...
- 623