The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
11 views

Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
4
votes
1answer
55 views

Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
1
vote
0answers
51 views

Calabi-Yau Toric Varieties

This is a rather naive question, but, from what I understand, we begin with a some reflexive polytope $P$. From the basic theory of toric varieties, we can construct a toric variety corresponding to ...
7
votes
0answers
82 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
4
votes
1answer
65 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
2
votes
0answers
24 views

How to show that the inverse image of a face of a simple polytope is a connected manifold?

If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
3
votes
2answers
105 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
4
votes
0answers
55 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
1
vote
1answer
54 views

Distinguished points of a cone

Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ ...
4
votes
1answer
65 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
1
vote
0answers
24 views

Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
0
votes
0answers
13 views

Uniqueness of supporting hyperplane for a face of a cone

In William Fulton's 'Introduction to Toric varieties' he says - " When $\sigma$ spans $V$ and $\tau$ is a facet of $\sigma$ then there is a $u \in \sigma ^{\vee}$ unique upto multiplication by a ...
2
votes
0answers
19 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
1
vote
1answer
29 views

Are the orbits of a symplectic toric manifold the fibers of its moment map?

A symplectic toric manifold, by definition, carries an effective torus action generated by a moment map. The orbits of the torus action are of course contained in the fibers of the moment map, but ...
2
votes
0answers
18 views

Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
2
votes
1answer
45 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
3
votes
0answers
52 views

Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with ...
2
votes
0answers
31 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
2
votes
1answer
88 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
0
votes
1answer
28 views

CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?

In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to ...
1
vote
0answers
27 views

Are the following Hamiltonian actions?

Let $(x,y)$ be the coordinate of $\mathbb{C}^2 \subset \mathbb{P}^1\times \mathbb{P}^1$. Is the $S^1$ action on $\mathbb{P}^1\times \mathbb{P}^1$ given by $$ t\cdot(x,y)=(tx,t^{-1}y) $$ Hamiltonian? ...
1
vote
0answers
51 views

Fibers of toric morpisms

Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber ...
0
votes
1answer
40 views

Question about toric ideal

In the proposition 1.2 contained in the following http://www.math.harvard.edu/~jbland/ma232b2_notes.pdf, I can't understand why a monomial satisfying (1.7) exists. Can you help me? Thanks.
3
votes
0answers
79 views

Good confusion-avoiding notation for gluing toric varieties from fans?

$ \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \DeclareMathOperator\Cone{Cone} $I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A ...
2
votes
1answer
97 views

Faces of cones give localizations of affine toric varieties

I'm trying to establish an explicit description of the open subsets of affine toric varieties given by faces of the underlying cone. Background For a rational convex polyhedral cone $\sigma\subseteq ...
6
votes
1answer
285 views

Dual of a rational convex polyhedral cone

A rational convex polyehedral cone $\sigma\subseteq\mathbb R^n$ is a set of the form $$ \sigma=\operatorname{Cone}(u_1,\dots,u_k) := \left\{ \sum_{i=1}^k r_i u_i \,\Bigg|\, r_i\ge ...
4
votes
2answers
116 views

Showing a quotient $\mathbb{Z}$ module is free

In Fulton's "Introduction to Toric Varieties" he repeatedly uses the following fact. Let $\sigma$ be a strongly convex rational polyhedral cone in a lattice $N$ and let $N_{\sigma}$ be the subgroup ...
3
votes
2answers
220 views

Prove that $(C \cap D)^\vee = C^\vee + D^\vee$.

Definitions and notations: Let $M$ to be the $n$-dimensional Euclidean space $\mathbb{R}^n$ and $N$ its dual space $M^\ast = \mathrm{Hom}_\mathbb{R}(M, \mathbb{R}).$ A subset $C \subset N$ is said to ...
4
votes
1answer
208 views

Motivation and Applications for Toric Varieties

I'm a graduate student of mathematics starting to study algebraic geometry with a focus on toric varieties (along Cox, Little, Schenk). From what I learned so far, I can grasp that toric varieties ...
4
votes
3answers
324 views

Isomorphism of lattices

This is an exercise that appears in some sheet of exercises in toric geometry. Take two lattices, $N=\mathbb{Z}^{3}$ and $N'=\operatorname{Span}_\mathbb{Z}\{(-1,-1,1), (-1,2,1), (2,-1,1)\}$. The ...
2
votes
0answers
58 views

Realizing a list of varieties as the toric variety of a fan

The locus of the following curves define affine varieties. $xy-z^m=0.$ $x^m+xy^2+z^2=0.$ $x^4+y^2+z^2=0.$ $x^2+y^3+z^2=0.$ $x^5+y^3+z^2=0.$ In Fulton's "Toric Varieties" book he constructs toric ...
2
votes
0answers
131 views

Procedure to find generators of the dual cone

Page 11 of Fulton's "Toric Varieties" gives the following procedure for finding generators of the dual of a convex polyhedral cone in $\mathbb{R}^d$: For each set of $n-1$ independent vectors ...
3
votes
2answers
285 views

Reference request: toric geometry

What is a good book on algebraic geometry, with focus on toric varieties, similar both in the philosophy and in the prestige of the autors to Modern Geometric Structures and Fields by Novikov and ...
4
votes
0answers
42 views

$H^{1}(O_{F})$ of a surface in a toric variety

I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five ...
3
votes
1answer
185 views

Example of a variety that is not toric

My question is simple, but I haven't seen it to be addressed anywhere: What would be a simple example of an affine variety that is not a toric variety? Toric varieties (the ones I have ...
2
votes
0answers
90 views

Injectivity of a rational map

I have a variety $C_{0}\subset\mathrm{Spec}\mathbb{C}[M_{Y}]$ living in some $Y$ (which is toric) and I am guessing, that it is $\mathbb{P}^{1}$. So I have to find a birational function ...
1
vote
0answers
132 views

Classification of Cones

I am attempting to classify (convex rational) cones in $\Bbb{R}^2$. We say here that $\sigma\subset\Bbb{R}^2$ is a cone if there exist $u,v\in\Bbb{Z}^2$ which $\Bbb{R}$-span the whole plane ...
7
votes
1answer
338 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
3
votes
1answer
120 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
3
votes
1answer
158 views

Finding a toric variety of a cone

I'm trying to find the toric variety associated to the cone $\sigma_0$ which is the region in the real plane with $x\geq 0$ and $y-x\geq 0.$ I found that it's dual cone is $\check{\sigma_0}$ the ...
6
votes
1answer
147 views

If a group scheme $G$ operates on another scheme $X$, how do you define orbits?

In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even ...
1
vote
1answer
72 views

How to recognize a glueing

This is an exercise in chapter 1 of Fulton's book "Introduction to toric varieties". Let $\Delta$ be the fan consisting of the cones $\sigma_1=\langle e_1, e_2\rangle$ and $\sigma_2=\langle ...
2
votes
0answers
75 views

Toric Variety from a fan with “identical” edges

I have this fan that I've been trying to construct a toric variety from. The problem is, it contains certain edges twice. These are the edges: $$(1,0,1)$$ $$(0,1,1)$$ $$(-1,-1,1)$$ $$(0,0,1)$$ ...
6
votes
1answer
119 views

Degree of Toric Divisors

Is it possible to calculate the degree of a toric divisor directly from the fan of the toric variety? If so, how is this done? Or is there some alternative way to calculate the degree of these ...
3
votes
1answer
427 views

Is locally free sheaf of finite rank coherent?

Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent? By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that ...
2
votes
0answers
65 views

Restriction of locally free sheaf associated projective modules

My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in ...
3
votes
0answers
82 views

Transition functions of toric projective bundle (Proposition in [Cox, Toric Varieties])

My reference: David Cox's "Toric Varieties" My question is the proof of Proposition 7.3.3. Proposition 7.3.3. The cones {$\sigma_i$ | $\sigma \in \Sigma$, $i = 0,\dots,r$} and their faces form ...
5
votes
2answers
202 views

If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's On ...
2
votes
0answers
141 views

GIT quotient for a certain torus action on an affine space

I'm reading various books and some notes and here is my question. Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by $$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, ...
2
votes
1answer
137 views

Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...