Questions tagged [toric-varieties]
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61
questions
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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,...
- 1,443
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0
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28
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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 ...
- 936
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0
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53
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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))...
- 497
3
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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 ...
- 497
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0
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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 ...
- 140
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votes
1
answer
51
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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 $\mathbb{P}^1$ ...
- 1,271
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1
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47
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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 ...
- 371
1
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0
answers
53
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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 ...
- 371
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1
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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
$$
\begin{pmatrix}
1 & * \\ 0 & 1
\end{pmatrix}.
$$
...
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0
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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 $\Delta$, ...
- 2,239
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0
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29
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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 \...
- 371
2
votes
1
answer
72
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
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0
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32
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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 ...
- 371
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votes
0
answers
65
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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 ...
- 1,814
1
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0
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37
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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.
...
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0
answers
45
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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 ...
- 321
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0
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47
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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
202
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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 ...
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0
answers
76
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 ...
- 386
1
vote
0
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27
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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
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65
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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
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0
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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 ...
- 1,133
1
vote
0
answers
82
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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, ...
- 1,133
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votes
0
answers
53
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....
- 97
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votes
1
answer
229
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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
...
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0
answers
81
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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
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1
answer
84
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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 ...
1
vote
1
answer
126
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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: ...
- 1,227
0
votes
1
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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
244
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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
34
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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
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0
answers
139
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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
121
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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_\...
- 2,659
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votes
1
answer
50
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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
83
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 $j$ ...
- 19.6k
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0
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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?
- 6,072
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1
answer
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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 ^...
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0
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414
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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 ...
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votes
1
answer
115
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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.
- 128
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vote
0
answers
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(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
105
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 ...
- 609
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Why do we require that a toric variety have an action extending the torus action?
An affine toric variety is defined as a variety $X$ that contains a torus as an open dense subset such that the natural action of torus on itself extends to an action on $X$.
I would imagine that the ...
- 18.2k
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0
answers
139
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Blow Up of an arbitrary subvariety on $\Bbb P^3$.
Suppose the coordinates of $\Bbb P^3$ are $\{x_1,x_2,x_3,x_4\}$, and in the affine patch $x_1=1$ an ideal $I=\langle\, f_1 , f_2,f_3\rangle$ is given ($f_i$'s are homogeneous polynomials on $\Bbb P^3$)...
- 103
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votes
1
answer
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Criterion on nefness of a divisor on algebraic surfaces
Question 1: Let $X$ be a smooth rational surface with anti-canonical cycle, i.e, $-K_X$ is effective and its irreducible components form a polygon. Say, assume that $-K_X=\sum_\limits{i=1}^N D_i$ and $...
2
votes
1
answer
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Why take the dual cone when constructing toric variety?
I am reading an introduction to toric varieties. I don't understand why we are taking the monoid associated to the dual cone instead of simply taking the monoid associated to the cone. Is there any ...
- 23
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votes
1
answer
746
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Mori Cone of a surface
Suppose we have generic surface in a Toric variety (say an elliptically fibered 3-fold), it can be branched over a divisor too, and suppose we actually know the complete set of divisors over this ...
- 103
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vote
0
answers
57
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How to describe projective normality of a toric variety in terms of a fan?
Correct me if I am wrong, but my understanding of projective normality is that if you embed the toric variety in projective space, then it is normal (i.e. the ring of functions is integrally closed.) ...
- 11
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votes
0
answers
37
views
Is $\overline {G^\circ\cdot p}$ a toric variety?
Consider the algebraic torus $(\mathbb C^*)^n$. Let $G$ be a subgroup of $(\mathbb C^*)^n$ that is also a reductive group. Let $G^\circ$ be the connected component of $G$ containing the identity ...
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0
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How to find the distinguished points of this cone?
Consider the cone - $\langle e_1,-e_1+2e_2\rangle$ in the lattice $N=\mathbb Z^2$. Then it has the following faces -
$0=\{(0,0)\}$
$\rho_1=\langle e_1\rangle$
$\rho_2=\langle-e_1+2e_2\rangle$
$\...
- 6,962
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votes
1
answer
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views
How does one determine the singular points of a toric variety?
Consider the toric surface corresponding to the fan $\Delta$ consisting of $$\sigma_1=\langle e_1,-e_1+2e_2\rangle$$ $$\sigma_2=\langle-e_1+2e_2,-e_1-2e_2\rangle$$ $$\sigma_3=\langle -e_1-2_2,e_1\...
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