# Questions tagged [toric-varieties]

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### 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 ...
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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 ...
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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 ...
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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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### 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 ...
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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?
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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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### 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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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 $\sum_i \... 0 votes 0 answers 65 views ### closed orbit of smooth toric variety is defined by regular sequence Let$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)$... 2 votes 0 answers 91 views ### Blow up of toric variety corresponds to subdivision of cone Look at the lattice$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)$. Then it ... 1 vote 0 answers 82 views ### Weighted projective space as toric variety I am trying to solve exercise 3.3.10 from Cox' book on toric varieties. I have no trouble with part (a), but it seems to me that for part (b) I have to wirte down the variety$X_{\Sigma}$explicitly, ... 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.... 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 ... 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 ... 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 ... 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: ... 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 ... 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\... 1 vote
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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 ... 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 $j$ ...
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?