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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 ...
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1answer
35 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.
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64 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
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1answer
87 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 ...
5
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1answer
178 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 ...
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2answers
109 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
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2answers
211 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
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1answer
153 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
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3answers
240 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
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0answers
51 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 ...
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0answers
81 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
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2answers
191 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
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0answers
39 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
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1answer
156 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
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0answers
80 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 ...
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0answers
103 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 ...
6
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1answer
249 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= ...
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1answer
103 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
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1answer
127 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 ...
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1answer
133 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 ...
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1answer
67 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
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0answers
70 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
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1answer
112 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
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1answer
292 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 ...
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0answers
59 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
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0answers
75 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
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2answers
186 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
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0answers
131 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
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1answer
126 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. ...
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0answers
128 views

Exercise in David Cox “Toric Varieties”

I want to do an exercise in the book Toric Varieties (by David Cox) Exercise 3.3.5. Let $\overline{\phi}:N \rightarrow N'$ be a surjective $\mathbb{Z}$-linear mapping and let $\widehat{\sigma}$ ...
1
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1answer
231 views

How to find the canonical divisor on a nonsingular toric variety?

I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then $$ ...
3
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1answer
184 views

Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
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1answer
229 views

Hodge theory for toric varieties

Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various ...
2
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1answer
91 views

differential with logarithmic poles

where can I find the computation of the groups $H^i(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^j)$? Moreover, if $D$ is a divisor with normal crossing in $\mathbb{P}^n$, how can I compute the hypercohomology ...
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1answer
88 views

figure out a simple toric variety

consider the plane cones $s=\langle(1,0),(1,n)\rangle$ and $t=\langle(1,0),(1,-n)\rangle$. This produce a toric variety obtained glueing $k[x,xy^n]$ and $k[x,xy^{-n}]$ along $k[x]$. There is a more ...
4
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1answer
272 views

Cohomology of $\mathcal O_X$ for toric varieties

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases} $$ as for $\mathbb ...
5
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1answer
320 views

Equations of a projective toric variety

Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective ...