Questions tagged [toric-varieties]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
0 answers
70 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,...
user avatar
  • 1,443
0 votes
0 answers
28 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 ...
user avatar
  • 936
0 votes
0 answers
53 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))...
user avatar
3 votes
0 answers
122 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 ...
user avatar
1 vote
0 answers
46 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 ...
user avatar
0 votes
1 answer
51 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 $\mathbb{P}^1$ ...
user avatar
  • 1,271
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 ...
user avatar
1 vote
0 answers
53 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 ...
user avatar
0 votes
1 answer
66 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 $$ \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}. $$ ...
user avatar
  • 964
1 vote
0 answers
25 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 $\Delta$, ...
user avatar
  • 2,239
0 votes
0 answers
29 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 \...
user avatar
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 ...
user avatar
1 vote
0 answers
32 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 ...
user avatar
7 votes
0 answers
65 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 ...
user avatar
  • 1,814
1 vote
0 answers
37 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. ...
user avatar
2 votes
0 answers
45 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 ...
user avatar
0 votes
0 answers
47 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?
user avatar
4 votes
2 answers
202 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 ...
user avatar
2 votes
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 ...
user avatar
1 vote
0 answers
27 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 \...
user avatar
0 votes
0 answers
65 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 $\Delta$. $V(\sigma)$ ...
user avatar
2 votes
0 answers
91 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 ...
user avatar
  • 1,133
1 vote
0 answers
82 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, ...
user avatar
  • 1,133
5 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....
user avatar
2 votes
1 answer
229 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 ...
user avatar
1 vote
0 answers
81 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 ...
user avatar
  • 1,067
2 votes
1 answer
84 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 ...
user avatar
1 vote
1 answer
126 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: ...
user avatar
0 votes
1 answer
214 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 ...
user avatar
2 votes
2 answers
244 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\...
user avatar
1 vote
0 answers
34 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 $\...
user avatar
  • 11
1 vote
0 answers
139 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 ...
user avatar
1 vote
1 answer
121 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_\...
user avatar
  • 2,659
2 votes
1 answer
50 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 ...
user avatar
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 $j$ ...
user avatar
  • 19.6k
0 votes
0 answers
46 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?
user avatar
  • 6,072
1 vote
1 answer
85 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 ^...
user avatar
  • 17.1k
5 votes
0 answers
414 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 ...
user avatar
  • 709
0 votes
1 answer
115 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.
user avatar
  • 128
1 vote
0 answers
21 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 $...
user avatar
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 ...
user avatar
2 votes
0 answers
240 views

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 ...
user avatar
  • 18.2k
1 vote
0 answers
139 views

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 $\Bbb P^3$)...
user avatar
  • 103
3 votes
1 answer
275 views

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 $...
user avatar
2 votes
1 answer
296 views

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 ...
user avatar
2 votes
1 answer
746 views

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 ...
user avatar
  • 103
1 vote
0 answers
57 views

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.) ...
user avatar
2 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 ...
user avatar
  • 6,962
1 vote
0 answers
110 views

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 - $0=\{(0,0)\}$ $\rho_1=\langle e_1\rangle$ $\rho_2=\langle-e_1+2e_2\rangle$ $\...
user avatar
  • 6,962
3 votes
1 answer
194 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\...
user avatar
  • 6,962