# Questions tagged [toric-varieties]

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### Construct schemes like toric varieties on manifolds

I'm reading a book about toric varieties, and some thoughts occured to me. Toric varieties are constructed by fans which are the union of rational cones in $\mathbb{R}^n$. Is it possible to construct ...
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### Blowup of a simplicial affine toric variety at the fixed point of the torus action

In this question, all cones are strongly convex, rational, polyhedral cones. We shall adopt the convention that, if a lowercase Greek letter $\sigma$ denotes a simplicial cone, then the uppercase ...
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### Redundancy in the definition of a Toric Variety

So as I have it, a toric variety is a complex variety $X$ with an open embedding of a torus $T^n$ with dense image, and morphism: $$a:X\times_{\mathbb C}T^n\longrightarrow X$$ which extends the ...
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### Free Commutative Monoid Quotient by Relations?

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with ...
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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 ...
1 vote
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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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1 vote
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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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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 ...