Questions tagged [toric-varieties]

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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 ...
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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 \...
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53 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 $\Delta$. $V(\sigma)$ ...
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Reference request for Non-toric varieties

I am working on a problem that possibly needs to consider non-toric varieties. I would like to know if there are books or papers which systematically introduce basic/advanced properties of non-toric ...
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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 ...
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36 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, ...
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35 views

When an Action on open dense subvariety by an Algebraic Group extends to Variety

A toric variety $X$ over $k$ is a variety which contains an algebraic torus ($T= \mathbb{G}_k^s$) as a dense open subset such that the action of the torus on itself extends to the whole of $X$. Slogan:...
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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....
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83 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 ...
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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 ...
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35 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 ...
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1answer
92 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: ...
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85 views

Torus embedding, torus action, toric variety

How are torus embeddings, torus actions and toric varieties related? Specifically, if I have a toric variety how can I recover the torus action and if I have a torus action how can I form the toric ...
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1answer
93 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 ...
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166 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\...
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28 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 $\...
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80 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 ...
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1answer
77 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_\...
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1answer
47 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 ...
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1answer
75 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 $j$ ...
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43 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?
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81 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 ^...
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257 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 ...
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1answer
91 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.
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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 $...
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1answer
84 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 ...
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136 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 ...
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122 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 $\Bbb P^3$)...
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1answer
208 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 $...
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1answer
218 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 ...
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1answer
581 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 ...
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54 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.) ...
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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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82 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 - $0=\{(0,0)\}$ $\rho_1=\langle e_1\rangle$ $\rho_2=\langle-e_1+2e_2\rangle$ $\...
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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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108 views

Alternate proof that the intersection of half spaces associated to facets of a cone, is that cone

Let $\sigma$ be a convex polyhedral cone and $V$ be span ($\sigma$). Then associated to any facet $\tau$ of $\sigma$ there is a unique $u_\tau$ up to scalar multiplication such that $Ann \, u_\tau \...
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Where does the linear map whose minors determine the rational normal cone, come from?

$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\im}{\operatorname{Im}}$ Let the rational normal cone be given by the image of $\Phi: \C^2 \to \C^{d+1}$, parametrized by $(s,t) \to (s^d, s^{d-1}t,...t^d)$. ...
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294 views

Is this toric variety the blowup of $\mathbb C^2$ at some point?

Let $u_1=e_1,\quad u_0=e_1+2e_2,\quad u_2=e_2$. Consider the fan consisting of the following cones $\sigma_1= \langle u_1,u_0\rangle$, $\sigma_2=\langle u_0,u_2\rangle$ and their faces. Then the toric ...
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Does this theorem hold for real part of a toric variety? + Reference request - Real toric varieties.

Let $X$ be a complex toric variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$. I would like to learn a little more about the ...
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108 views

On two-dimensional non-singular complete toric varieties

Let $v_0,v_1,\dots,v_d=v_0$ be a sequence of lattice points in $\mathbb Z^2$, in counterclockwise order (see figure below), such that successive pairs generate $\mathbb Z^2$ as a $\mathbb Z$-module. ...
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31 views

Realizing $\mathbf{C}$ as 1-dimensional normal toric variety

Take $N = \mathbf{Z}$, $N_\mathbf{R} = \mathbf{R}$. Then I know that the only cones are the intervals $\sigma_+ = [0,\infty)$, $\sigma_- =(-\infty,0]$, and $\tau = \{0\}$. Consider the fans $\{\...
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53 views

Toric variety corresponding to coordinate axes in $\mathbb{R}^2$

I have just learned how to construct a toric variety from a fan and I am a bit confused. Let $\Sigma$ be the fan that consists of the coordinate axes in $\mathbb{R}^2$, i.e. $\Sigma = \{ \sigma_0, \...
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126 views

What is this action of $\mathbb Z_2$ on $\mathbb C^2$ that gives the following affine toric variety?

Let $\sigma$ be the cone in $\mathbb R^2$ given by $\langle e_1, e_1+2e_2\rangle$. The corresponding affine variety $U_\sigma=\mathcal{Z}(x^2-yz)\subseteq \mathbb C^3$ I am trying to understand the ...
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on the real part of a toric variety

Corresponding to a strongly convex rational polyhedral cone in the lattice $\sigma$ in the lattice $N$ we have the affine toric variety $U_\sigma=\operatorname{Hom_{semi group}}(\sigma^\vee\cap M,\...
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37 views

Where do we use that $\sigma$ is a maximal dimension cone?

Let $N$ be a lattice (free $\mathbb Z$ - module of rank $n$) with dual $M=\operatorname{Hom}_{\mathbb Z - \text{mod}}(N,\mathbb Z)$ and let $\sigma$ be a cone in $N\otimes\mathbb R\cong\mathbb R^n$. ...
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646 views

When are vector bundles on toric varieties also toric varieties?

Let $X$ be a toric variety, and $\pi:E\to X$ a vector bundle, say of rank $2$. You can think of $X=\mathbb P^1$. When is the total space of $E$, or of $P(E)$, a toric variety? What do I need in ...