# 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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### Is an extension of an algebraic group by a multiplicative group a semidirect product?

This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample. Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$...
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• 7,235
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 ...
• 231
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### toric variety associated to schur polynomial?

In a Schur polynomial, which is homogeneous, monomials are added with fixed coefficients. Collecting the indices of monomials seems to define a toric data for a projective variety. But the ...
1 vote
128 views

### Induced map on cohomology from inclusion of toric varieties

Suppose you have an equivariant closed immersion of toric varieties $Y \subset X$, and suppose further that they are both smooth (meaning that the torus of $X$ restricts to the torus on $Y$). Suppose ...
• 20.2k
1 vote
151 views

### Reference for a result in toric geometry

I want to know where I can find the proof of the following theorem : if $X$ is a smooth toric variety, then $H^{\bullet}(X) \cong CH^{\bullet}(X)$. Thanks in advance !
324 views

### First Chern class of toric manifolds

I have been reading a Mirror Symmetry monograph, and its physical arguments seem to imply that all toric manifolds have semipositive-definite first Chern class. Is this true, and if yes, how does ...
61 views

### What do we know about the set of all fans?

I know the question is not very rigorous, but I have been trying to prove some facts about toric varieties and I think giving this set some structure would be very helpful. So, suppose you fix a ...
• 740
1 vote