1
vote
1answer
191 views

Is the Kähler cone of a toric variety always simplicial?

I am working on a familly of toric varieties which seem to have the following property: the closure of the Kähler cone is a simplicial cone (and even a smooth cone with respect to the natural ...
0
votes
2answers
202 views

When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?

Let $p_1,...,p_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X_k$ is toric and hence a Mori Dream Space. ...
3
votes
0answers
116 views

What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
5
votes
1answer
219 views

Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...
0
votes
1answer
131 views

Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer. Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
1
vote
1answer
144 views

Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
1
vote
0answers
79 views

How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...
2
votes
1answer
94 views

Does any smooth hypersurface in (C^*)^n admit a smooth normal crossings compactifcation as a hypersurface in a toric variety?

Question 1. Given a Laurent polynomial $f(z_1,\cdots,z_n)$, such that the corresponding zero locus $Z$ of $f$ in $(\mathbb{C}^*)^n$ is smooth, can we find a smooth toric variety $\bar{X}$ (together ...
1
vote
1answer
152 views

Rational surface singularities as Toric varieties

I originally asked basically this question on MSE here. From an appendix in Steinberg's "Conjugacy classes in algebraic groups" I found a list of the rational surface singularities. Equations of types ...
3
votes
0answers
135 views

determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$. Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...
1
vote
0answers
268 views

Are Grassmannians toric varieties? [closed]

Are Grassmannians $G(k,n)$ toric varieties for all possible $k,n$? If they are toric varieties, are there any descriptions for the fans?
2
votes
0answers
120 views

Intuition about Toroidal Embeddings

I have been trying to understand the very basics of toroidal embeddings, and the definitions on the face of them are not terribly daunting. I've been going with the "locally analytically looks like a ...
0
votes
1answer
134 views

Why the blowup of a toric variety corresponding to a subdivison of fan?

I heard the statement "the blowup of a toric variety corresponding to a subdivison of fan" many times, but could not find reference in the literature. What is the precised statement (blowup a point? ...
0
votes
0answers
175 views

Connectedeness of toric varieties

Hi, I would like to understand when a toric variety is connected. Given $\Delta$ a fan (possibly with infinitely many cones) in $\mathbb{R}^n$, $n\geq 2$ denote with $X_{\Delta}$ the associated toric ...
0
votes
0answers
167 views

Variation the definition of toric varieties

Let us stick to affine toric variety. By definition, a toric variety is a variety containing a torus $T \cong (\mathbb{C}^*)^n$, with the torus action on $T$ extend to the whole variety. The torus ...
6
votes
1answer
238 views

Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example). One ...
4
votes
0answers
129 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 ...
2
votes
0answers
156 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
243 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 ...
7
votes
0answers
345 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 ...
1
vote
0answers
138 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 ...
3
votes
3answers
365 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
119 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
211 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 ...
0
votes
0answers
167 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 ...
4
votes
1answer
638 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 ...
14
votes
0answers
564 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
673 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
52 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?
9
votes
2answers
459 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, ...
2
votes
1answer
348 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
223 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
255 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 $(-K_X)^n$ ...
4
votes
2answers
669 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
197 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 ...
3
votes
3answers
668 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 ...
1
vote
1answer
527 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
314 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
422 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 ...
5
votes
0answers
234 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
296 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
405 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
428 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
540 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. ...
5
votes
2answers
571 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 ...
4
votes
1answer
421 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
220 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 ...
1
vote
2answers
454 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
3answers
715 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?