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4
votes
0answers
164 views

Derived category of toroidal varieties

This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem. A ...
5
votes
1answer
351 views

global sections of structure sheaf on the toric Calabi-Yau

Let P be a lattice polytope and lying in $ N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an ...
3
votes
0answers
305 views

“Step-by-Step” toric resolution process?

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial). The classical toric ...
8
votes
0answers
399 views

Big tangent bundle

Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
2
votes
3answers
599 views

Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...
2
votes
0answers
174 views

About Alexeev and Nakamura's paper “on Mumford's construction of degenerating abelian varieites”

Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow ...
0
votes
0answers
138 views

Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.

I am trying to figure out something concerning the index of lattices. The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...
3
votes
3answers
447 views

Toric Fano manifolds with Picard number 1

As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb ...
1
vote
1answer
148 views

Recommendations for binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ ...
4
votes
1answer
229 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
1
vote
2answers
242 views

Computing rational cohomology of smooth (not necessarily compact) toric varieties

The title pretty much says it all. I am looking for references (lecture notes, books, readable articles, suggestions), preferably example laden, that explain how to compute the rational cohomology of ...
1
vote
0answers
226 views

tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero. Let $\overline{G}$ be its wonderful compactification. I denote by $\overline{T}$ the closure of the torus $T$ in ...
3
votes
1answer
297 views

How to recover toric invariants tropically?

My excuses in advance in case my question is too vague (which is mainly due to the fact that I'm not really familiar with tropical/toric geometry, but at least I still believe that the content of the ...
5
votes
1answer
1k views

Clean introduction to toric varieties for an undergraduate audience

I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need ...
16
votes
0answers
653 views

Homology classes of subvarieties of toric varieties

Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety. Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero? Background If $X$ is a Kaehler variety, this is ...
17
votes
4answers
768 views

The Constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
2
votes
0answers
55 views

Associating an ideal to a subdivision

Given a coherent subdivison of a fan, how does one find a torus invariant ideal sheaf whose (normalized) blowup is the given subdivision?
8
votes
1answer
520 views

Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...
9
votes
2answers
490 views

Detecting tilings by toric geometry

This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask. Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
8
votes
0answers
445 views

Proving that a variety is not (isomorphic to) a toric variety

Is there an algorithmic (or other) way to prove that a (projective) variety is not isomorphic to a toric variety? I'd be happy with an algebraic answer (for affine or projective varieties), using the ...
3
votes
1answer
397 views

Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper

Hello, in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR COMPLEX ALGEBRAIC HYPERSURFACES": http://arxiv.org/pdf/math/0205011.pdf There is a lemma about the relation between intersection ...
3
votes
1answer
289 views

When a quotient singularity is toric?

Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order, Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?
2
votes
2answers
304 views

Bound on the (anticanonical) degree of toric Fano varieties

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ ...
5
votes
2answers
916 views

nef Cone of a Toric Variety

Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on ...
1
vote
0answers
207 views

Non-toric affine variety containing a torus

Part of the definition of an affine toric variety is that the action of the torus sitting as an open dense subset of the variety extends algebraically to the whole variety. Is there an easy example ...
4
votes
3answers
709 views

Do projective hypersurfaces contain projective toric varieties?

Is there an example of a smooth projective hypersurface in $\mathbb{P}^n_k$ ($k=\overline{k}$) that does not contain any projective toric varieties (edit: of positive dimension)? Or is it the case ...
3
votes
0answers
422 views

Intersection Theory on a toric variety

Hi All, I'm having some trouble understanding a result about calculating $D.C$ on a toric variety. The proposition I am trying to follow is from Cox, Little, Schenck. Either there is a mistake with ...
1
vote
1answer
585 views

Description of a birational map, Fulton's “Introduction to toric varieties”

This is an exercise of "Introduction to toric varieties" by Fulton, page 71, section 3.4, (the last one in this page.) The problem is initiated by constructing a complete (toric) variety which is not ...
2
votes
2answers
354 views

Toric automorphism of P1 times P1 blown up at four pts

A toric morphism between toric varieties is a morphism that is equivariant w.r.t. to the toric action, see e.g. section 3.2, Notes by H.Verrill and D.Joyner for definitions. In particular, any toric ...
5
votes
2answers
442 views

Are the closures of the tori in the decomposition of a torified variety toric varieties?

In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed ...
6
votes
0answers
263 views

Do simplicial toric varieties have “lots” of base point free linear systems?

Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system? If $X$ is not ...
8
votes
0answers
308 views

Computing Ext for toric divisors

Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute $$ ...
4
votes
3answers
479 views

Number of $(-1)$ curves on toric surfaces

Hello. My question is: Is it possible that a smooth complete toric surface has infinitely many $(-1)$-curves. I know that there is a blow-up of $\mathbb P^2$ in 9 points containing infintely many ...
2
votes
4answers
532 views

What is the fan of the toric blow-up of $\mathbb{P}^3$ along the union of two intersecting lines?

Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines? Everything I find in the literature is for blow-ups along smooth ...
1
vote
1answer
564 views

Smooth complete toric surfaces

Hi, I'm trying to find all the smooth complete toric surfaces, following section 2.5 of Fulton's book. There is one exercise given that I'm hung up on, hopefully someone can help me out a bit. ...
6
votes
2answers
670 views

Deformations of Hirzebruch surfaces and toric action

Hi, the Hirzebruch surface $F_n$ admits a deformation for $0\leq m\leq n$ defined by the equation $$ \mathcal{M}=\{ ([x_0:x_1],[y_0:y_1:y_2],t) \in \mathbb{P}^1 \times \mathbb{P}^2 \times ...
20
votes
3answers
1k views

Is there always a toric isomorphism between isomorphic toric varieties?

Suppose two toric varieties are isomorphic as abstract varieties. Does it follow that there exists a toric isomorphism between them? Edit: the comments below lead me to believe that I'm using the ...
1
vote
0answers
423 views

Toric Kahler Manifolds

One definition of Toric Kahler manifolds that I know is the following - $(M,J,\omega)$ is a Kahler manifold of dimension n with an $\textit{effective}$ $\textbf{T}^n$ action which preserves the ...
4
votes
1answer
449 views

Intersection of curves on projective toric surface and some enumerative questions

Reading on the tropical approach to enumerative geometry I have come across the claim: given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / ...
1
vote
0answers
241 views

What is known about the Picard scheme of a complete toric variety over C?

Let $X$ be a complete toric variety over a field $k$. Its Picard scheme is defined to be the scheme representing the functor $Pic_{(X/k)(\text{fppf})}$, where $Pic_{X/k}: Sch_k^{op}\to Set$ sends a ...
3
votes
2answers
510 views

On homology of Toric varieties

Lets $X$ be a simply connected projective toric variety of dimension $n$. Lets $\tau_1,\cdots,\tau_k$ be the set of $(n-1)$-dimensional cones of corresponding fan which is in one-to-one ...
3
votes
1answer
228 views

Can cones (toric monoids) be built as colimits of their faces?

Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...
3
votes
3answers
861 views

Are projective toric varieties, locally complete intersection?

Let $X^n \subset \mathbb{P}^N$ to be a toric projective variety. Is $X$ a local complete intersection? Is being a local complete intersection an intrinsic property, independent of embedding?
12
votes
2answers
790 views

Is an affine “G-variety” with reductive stabilizers a toric variety?

Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose $x\in X$ is a ...
17
votes
2answers
778 views

About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
1
vote
1answer
147 views

stabilizer of convex cones in a linear space

Let $V$ be a finite dimensional vector space over $\mathbb{R}$, and $C\subset V$ a convex cone of the form $C=\mathbb{R}_{\geq0}v_i$ for finitely many $v_i$'s in $V$. How can one describe the ...
2
votes
1answer
242 views

About toric varieties---properties of stabilizers

Let P be a normal variety over an algebraically closed field k, G a torus over k acting on P, assume that the stabilizer of the generic point of P is reduced (resp. connected or both), is it ture then ...
5
votes
3answers
871 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
2answers
549 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
2answers
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 ...