# Questions tagged [toric-geometry]

For questions related to toric geometry. The objects of study in toric geometry are toric varieties. Toric varieties are called ‘toric’ because they are equipped with a ‘torus action’. By a torus we mean the linear algebraic group $C^∗ ×\dots× C^∗$, not the torus from topology. A toric variety contains a torus as an open subset and this defines the torus action.

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### Principal bundle (or torsor) for a diagonalizable group over a torus

Let $T$ be an algebraic torus and $G$ a diagonalizable group; both are over an algebraically closed field $k$ of characteristic $0$ (take $k=\mathbb C$, if you like). I am trying to understand ...
1 vote
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### A semigroup $\mathbb{N}\mathcal{A}$ is saturated in $M$ if and only if $\mathbb{N}\mathcal{A}=\text{Cone}(\mathcal{A})\cap M$

I'll first recall some definitions here for convenience. Given a finite set $S$ in a real vector space $V$, $$\text{Cone}(S)=\{\sum\limits_{u\in S} \lambda_u u\mid \lambda_u\geq 0\}.$$ An affine ...
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### How to see this assertion about coordinate rings and a localisations

I am studying 'The Homogeneous Coordinate Ring of a Toric Variety' by David Cox, and in the proof of his theorem 2.1 he defines $U_\sigma := \{x\in \mathbb{C}^{\Sigma(1)} : x^{\hat{\sigma}}\neq0\}$ ...
1 vote
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### Conditions for evaluation morphisms $H^{0}(X, -K_X) \rightarrow k^n$ to be surjective.

Suppose I have a projective toric variety $X = X(N)$ (over a field $k$) associated to a polytope $N$, and I have $p_1,...,p_n \in D$ generically chosen points in the toric boundary $D$ of $X$. I am ...
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1 vote
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### toroidal compactifications of moduli spaces of ppav

Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...