The toric-varieties tag has no usage guidance.

**5**

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884 views

### relation between toric geometry and log geometry

Hello,
I'm trying to understand the relation between the points of view of
log geometry (monoids) and toric geometry (fans).
Suppose that $k$ is a field and $P$ is a finitely generated monoid.
Then ...

**1**

vote

**2**answers

558 views

### Do subsets of generators of a toric ideal generate a toric ideal?

Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite
generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give ...

**14**

votes

**2**answers

2k views

### What are some open problems in toric varieties?

In light of the nice responses to this question, I wonder what are some open problems in
the area of toric geometry? In particular,
What are some open problems relating to the algebraic ...

**4**

votes

**1**answer

470 views

### Secondary fans and Stanley Reisner ideals

Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$.
I have heard that the ...

**2**

votes

**1**answer

358 views

### Is a (quasi)projective toric variety (Q)Proj of its homogeneous coordinate ring?

This is really two questions. First, consider a normal toric variety $X_\Sigma$. Its homogeneous coordinate ring
$$R=\mathbb C[x_1,...,x_{|\Sigma(1)|}]$$
is graded by $A_{n-1}(X)$. In analogy ...

**7**

votes

**2**answers

3k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...

**5**

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**0**answers

483 views

### When should a moment polytope have “smooth” faces?

A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...

**6**

votes

**2**answers

911 views

### Counting/constructing Toric Varieties

Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these ...

**4**

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**2**answers

521 views

### Relationship between topological cohomology and $\ell$-adic cohomology

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope ...

**9**

votes

**1**answer

795 views

### Cox rings of toric varieties over arbitrary fields

The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous ...

**12**

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**0**answers

447 views

### A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
...

**4**

votes

**1**answer

800 views

### Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope?

In this link, Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.
I want to see ...

**5**

votes

**2**answers

696 views

### Software for computing multi-graded Hilbert series

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1)
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$\frac{1 - ...

**3**

votes

**2**answers

785 views

### Moment map for toric actions — online references?

Consider a toric variety, defined as a (normal?) complex projective variety $X$ together with an algebraic action of $(\mathbb C^*)^n$ with finitely many orbits. Now we have two "real symplectic" ...

**8**

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**2**answers

606 views

### Hamiltonian S^1 actions with isolated fixed points

I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian S^1 action with isolated fixed points, that does not admit a compatible S^1 ...