Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
321
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Almost toric mutations
I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,...
- 755
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votes
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Resolutions of configuration space of the projective line where the complement is of "Tate type"
I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...
- 7,716
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
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Simplicial approximation theorem for toric varieties
Given abstract simplicial complexes $K$ and $L$, one constructs topological spaces $|K|$ and $|L|$. Simplicial approximation theorem says for any continuous map $f: |K|\to |L|$ that there exists ...
- 11
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votes
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Lattice polytope toric varieties under rescaling
Is there any interesting relation/map between the toric variety $X_P$ associated to a lattice polytope $P$ and the toric variety associated to the polytope rescaled by some integer factor?
- 1,425
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answer
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Quasismooth vs smooth in a smooth toric variety
Let $X$ be a toric variety, and let $\pi: \mathbb A^n-V(B) \to X = (\mathbb A^n-V(B))/(\mathbb C^*)^\rho$ be the quotient map defining $X$ in the Cox construction. A subvariety $Y\subset X$ is called ...
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Is there a Chevalley map for spherical varieties?
If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...
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How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$
I am given that $k$ is a field and $G$ is the monoid consisting of all monomials
$X^iY^j$, where $j$ is between $0$ and $3i$.
I am trying to compute the quotient of the monoid algebra $kG$ by the ...
- 569
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A question regarding Corollary 4.12 in Mumford's "Analytic construction of deg. Ab. Var."
Let $S=Spec A$ be the spectrum of an integrally closed, excellent and Noetherian ring. In his paper, David Mumford constructs an $S$-group scheme $G$. He shows that the torsion of $G_s$ is $p$-...
- 1,745
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A connection between equivariant and non-equivariant cohomology of toric variety
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$.
The $T$-equivariant cohomology $H^*_T(X)$ (with coefficients in a field, say) is an ...
- 21.3k
2
votes
1
answer
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Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$
I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
- 1,037
5
votes
0
answers
283
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Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$
A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
- 103
4
votes
1
answer
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If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?
Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
- 143
7
votes
2
answers
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Is $(x^2y,xy^2)$ log smooth?
Consider the map
$$f:\mathbb C^2\to\mathbb C^2$$
$$(x,y)\mapsto(x^2y,xy^2)$$
We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
- 18.5k
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votes
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From Delzant polytope to lattice polytope
By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But
is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the ...
- 143