14
votes
Has anyone researched additive analogues of toric geometry in characteristic zero?
The theory of Luna and Vust (Plongements d'espaces homogènes. Comment. Math. Helv. 58 (1983), 186–245.) on equivariant compactifications of homogeneous varieties works actually for any connected ...
- 15.5k
12
votes
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Complete intersections in toric varieties
Any smooth projective toric variety is rational, in particular simply connected.
Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete ...
- 66.3k
11
votes
Has anyone researched additive analogues of toric geometry in characteristic zero?
Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map.
But there is a foundational paper on the ...
- 21.1k
9
votes
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Volume of $-K_X$ for a weighted projective variety
I am just rewriting my comment above as an answer.
Let $S=k[x_0,\dots,x_n]$ be the $\mathbb{Z}_{\geq 0}$-graded $k$-algebra with every $x_i$ homogeneous of degree $a_i.$ Denote by $X$ the associated ...
9
votes
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Relationship between fans and root data
(1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the based root datum ${\rm BRD}(G)$.
(2) A spherical homogeneous space $Y=...
- 14.1k
8
votes
Application of toric varieties for problems that do not mention them
Since you did not mention you wanted applications to mathematical problems, I allow myself to take examples from other sciences, for instance biology.
The following problem : "On a given position in ...
- 7,300
8
votes
Application of toric varieties for problems that do not mention them
To any matroid $M$ on ground set $E$ we associate the characteristic polynomial
$$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$
where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is ...
- 7,875
8
votes
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Application of toric varieties for problems that do not mention them
There are lots of applications of toric varieties to singularities, e.g., the proof of the weak factorization theorem in characteristic zero. (Indeed, the name of the linked paper is "Torification and ...
- 5,958
8
votes
Derived categories of toric varieties and convex geometry
That's a great question!
In the last few years there is plenty of research on the subject - and a few (interesting) open questions.
Before Kawamta's result - note that toric manifolds satisfy $rk(...
- 441
8
votes
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Separating a lattice simplex from a lattice polytope
This is possible and here is how to do this. We will use an inductive argument, assume that the statement holds for polytops of dimension $<n$ and prove it for dimension $n$.
Take any vertex $v$ of ...
- 28.9k
8
votes
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Three-dimensional analogues of Hirzebruch surfaces
The 3-dimensional analogues should be $\mathbb{P}(O(a)\oplus O(b)\oplus O(c))$, yes. These are exactly the $\mathbb{P}^2$-bundles over $\mathbb{P}^1$.
In general, any projectivization of a vector ...
- 549
7
votes
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Resolving $\mathbb Z_n$ action on $\mathbb C^2$
I think the answer is no. In your notation, take $n=5$, $p=1$ and $q=2$. If you consider the blowup at the origin $X\to \mathbb C^2$, then $\mathbb Z_5$ acts on $X$ with two isolated fixed points: at ...
- 6,253
7
votes
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Is the Chow ring of a wonderful model for a hyperplane arrangement isomorphic to the singular cohomology ring?
The isomorphism $H^\cdot(Y)\cong Ch^\cdot(X)$ is shown by
Eva Feichtner and Sergey Yuzvinsky in Feichtner, E. & Yuzvinsky, S. Invent. Math. (2004) 155: 515. https://doi.org/10.1007/s00222-003-...
- 594
7
votes
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Number of boundary divisors and colors of a Spherical variety
There are relations coming from computing Picard groups in different ways. For simplicity let $G$ be semisimple. Let $x\in X$ be in the open $B$-orbit. Then $Bx=B/B_x$ and $Gx=G/G_x$ are open in $X$.
...
- 15.5k
7
votes
Closures of torus orbits in flag varieties
A point in the Grassmannian $ x \in G(k, n) $ defines a matroid $ M = M(x)$. Associated to this matroid is a matroid polytope $P(M)$. The torus orbit closure through $ x $ is the toric variety ...
- 4,612
7
votes
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Isomorphic equivariant sheaves are equivariantly isomorphic on a toric variety
Edit. The short answer is that this fails for every toric variety except when $T$ equals all of $X$. I edited the original answer to prove this, and also to prove the positive result below by @S. ...
7
votes
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Cohomology of divisors on Hirzebruch surfaces
Sure. To compute $H^0$ one can first pushforward to the base $\mathbb{P}^1$. If $a \ge 0$ one obtains
$$
p_*\mathcal{O}(a\Gamma + bF) \cong
p_*\mathcal{O}(a\Gamma) \otimes \mathcal{O}(b) \cong
S^a(\...
- 39.3k
7
votes
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Is there a Chevalley map for spherical varieties?
Edit: The answer to question 1 is yes if $G/H$ is a symmetric variety as the OP pointed out.
For arbitrary spherical varieties the answer is no in general. If my memory serves me right, the spherical ...
- 15.5k
6
votes
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Proving that a variety is not (isomorphic to) a toric variety
The question of algorithmically deciding if an ideal is binomial after a (suitable, e.g. linear) automorphism of affine space is decidable and various algorithms are discussed in "When is a polynomial ...
- 1,961
6
votes
Is there any structure theorem for piecewise linear functions?
If you weren't assuming the piecewise linear function $f: \mathbb R^d \to \mathbb R$ is continuous, then I think this is true: we can just choose functions
$$ f_i = \begin{cases}
\text{(some linear ...
- 2,175
6
votes
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Picard group of toric varieties
Edit: I have elaborated on this approach to the Picard group in Section 2 of my preprint.
The question was answered in the comments above, but only for the case of torsion-free Picard group. However, ...
6
votes
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Can any simplicial toric variety be embedded in a product of projective spaces?
There are well known examples of smooth (hence simplicial) complete toric varieties which are not projective. See for example p. 71 of Fulton's book Introduction to Toric Varieties. Any such variety ...
- 76
6
votes
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From Delzant polytope to lattice polytope
There is a way to turn a Delzant polytope into a lattice polytope, but there is no natural or canonical way. Note that the Delzant condition implies that the normal vector to each facet (codimension 1 ...
- 1,398
6
votes
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On intersection theory on toric varieties
No, let $P_\Delta= (\mathbb P^1)^3$ and let $C$ be a curve in $(\mathbb P^1)^2$. Then the Chow group of curves in $\mathbb P_\Delta$ is freely generated by copies of $\mathbb P^1$ in the three ...
- 148k
5
votes
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When are these definitions of "toric variety" equivalent?
As noticed by Dave Anderson, (4) does not imply (2) because the generic stabilizer might be finite but non-trivial. But still, let $E$ be the stabilzer subgroup scheme of $x$ as in (5). Because $T$ is ...
- 15.5k
5
votes
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2-faces of reflexive Delzant polytopes
Haase and Melnikov have proved here that every lattice polytope can be realized as a face of some reflexive polytope. They do this by an iterative procedure that increases the dimension by one until ...
- 85.5k
5
votes
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Irreducibility of Gelfand-Serganova strata
The strata need not be irreducible.
Quoting page 2 of Knutson, Lam, Speyer (https://arxiv.org/abs/0903.3694v1): "the strata can have essentially any singularity [Mn88]. In particular, the nonempty ...
- 24.2k
5
votes
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When does the Hirzebruch surface have a nef anticanonical divisor?
If $r \ge 3$ the exceptional section has negative intersection with anticanonical divisors), so the answer is $r \le 2$.
- 39.3k
5
votes
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Almost toric mutations
Mutation doesn't even change the integral affine base, which is why it doesn't change the symplectic manifold. All you're doing is changing the way the integral affine base is drawn. If you're given ...
- 7,005
5
votes
If $ Z $ is an $ n $-dimensional, projective variety, containing $ \mathbb{G}_{m}^{n} $, what is the obstruction to $ Z $ being toric?
Even in the case when the image of the morphism is open, there are plenty of non-toric examples. Consider the standard toric action on $\mathbb{CP}^3$ and consider one of the torus invariant divisors $...
- 6,995
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