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Construct schemes like toric varieties on manifolds

I'm reading a book about toric varieties, and some thoughts occured to me. Toric varieties are constructed by fans which are the union of rational cones in $\mathbb{R}^n$. Is it possible to construct ...
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Blowup of a simplicial affine toric variety at the fixed point of the torus action

In this question, all cones are strongly convex, rational, polyhedral cones. We shall adopt the convention that, if a lowercase Greek letter $\sigma$ denotes a simplicial cone, then the uppercase ...
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Torus action/limit approach to show that open embedding $U_\tau \subseteq U_\sigma$ implies $\tau$ is face of $\sigma$

Context Both the Fulton and Cox/Little/Schenck books on Toric Varieties include an exercise to show that if an inclusion $\tau\subset \sigma$ of cones induces an open embedding $U_\tau \hookrightarrow ...
Joseph's user avatar
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What Toric Variety does this fan correspond to?

Let $\Sigma$ be the fan defined by $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\star\}$, where $\sigma_1=\operatorname{cone}(e_1)$, $\sigma_2=\operatorname{cone}(e_2)$, $\sigma_3=-\sigma_1$, $\sigma_4=-\...
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Redundancy in the definition of a Toric Variety

So as I have it, a toric variety is a complex variety $X$ with an open embedding of a torus $T^n$ with dense image, and morphism: $$a:X\times_{\mathbb C}T^n\longrightarrow X$$ which extends the ...
Chris's user avatar
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Free Commutative Monoid Quotient by Relations?

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with ...
Chris's user avatar
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1 answer
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Is the weighted projective space $\mathbb{P}(1,2,2)$ smooth?

In the course of learning about the reolution of singularities in toric geometry, I have come across something which doesn't make much sense to me. Consider the weighted projective space $$ \mathbb{P}(...
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Klyachko's classification of toric vector bundles - an ongoing discussion

I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra) I am ...
sagirot's user avatar
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Clarifying an example for quotient stacks: Whether diagonal is closed substack

In trying to remember an example about quotient stacks, I think I've got something turned around. I am trying to determine whether the diagonal is a closed substack of the (product of) quotient stack(...
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Toric vector bundle - Klyachko's classification

I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra). I am ...
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When does a smooth projective toric variety admit a symplectic structure?

I'm trying to understand the relationship between toric varieties and their associated polytopes. There is a very crisp result in the case of symplectic toric varieties, namely that there is a 1:1 ...
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Reference request: Calabi-Yau toric manifold condition in algebraic geometry

I'm currently studying toric Calabi-Yau manifolds, and in particular, am looking at how we can construct them from fans. A fact keeps coming up in the papers I am reading, for instance in https://...
anna's user avatar
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Motivation for the study of affine semigroups

I have recently been studying affine semigroup (= Semigroups that is commutative, finitely generated, embeddable in a lattice) in the context of toric varieties (Cox, Little & Schenck). I was ...
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The action of the torus $T_N$ on the semigroup algebra $C[M]$

Toric Varieties written by David Cox, John Little and Hal Schenck p.18 Equivalence of constructions Before starting our main result, we need to study the action of the torus $T_N$ on the semigroup ...
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Torus action on a prime divisor of Toric Varieties

Let △ be a fan, X(△) be a toric variety and D be a prime divisor of X(△). Let Tn be a torus and t∈Tn. Then why is t*D a prime divisor of X(△)?
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Intersection of (Toric varieties) and (Flag varieties)

For me, a variety $X$ is assumed to be irreducible and normal over some algebraically closed field $k$ of characteristic zero. I call $X$ a toric variety if it has an algebraic torus $T$ which embeds $...
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Reference request: Toric Varieties in reverse order

I am looking for a book/text which introduces toric varieties in reverse order. Instead of starting with lattices, cones, fans and convex geometry, it should start with a toric variety $X$ and define ...
Nico's user avatar
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For Hirzebruch Surfaces, does the tangent exact sequence split?

Consider the projection $\pi:\mathbb{F}_n \rightarrow \mathbb{P}^1$. Is it true that the following exact sequence $$0 \rightarrow T_{\pi} \rightarrow T_{\mathbb{F}_n} \rightarrow \pi^*T_{\mathbb{P}^1} ...
Changho Han's user avatar
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?

[I cross-posted this question as https://mathoverflow.net/questions/431454. (That version is also slightly updated.)] Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \...
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Is $V(y) \subset \mathbb C^2$ not an affine toric variety?

From Toric Varieties by Cox, Little, Schenck: An affine toric variety is an irreducible affine variety $V$ containing a torus $T_N\cong (\mathbb C^*)^n$ as a Zariski open subset such that the action ...
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Is the ideal of a subscheme of a toric variety B-saturated?

Let $X$ be a normal toric variety over $\mathbb{C}$ with no torus factors. Let $S$ be the Cox ring of $X$, and $B \subset S$ be the irrelevant ideal. Given an ideal sheaf $\mathcal{I} \subseteq \...
Maciej Gałązka's user avatar
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1 answer
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Transition Functions of the Weighted Projective Space $\mathbb{P}^{2}_{\{3,2,1\}}$.

I am trying to find the transition functions of the weighted projective space $\mathbb{P}^{2}_{\{3,2,1\}}$. Bear with, I am very new to algebraic geometry. To my understanding, $\mathbb{P}^{2}_{\{3,2,...
CoffeeCrow's user avatar
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Definition of affine toric variety

Sorry for my bad English. I'm trouble about a definition of an affine toric varieties. I often see a definition of affine toric varieties as follow; An affine toric variety is an irreducible affine ...
Yos's user avatar
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Introductory papers in Mirror Symmetry for algebraic geometry students

Assuming that a student knows algebraic geometry at the level of Cox's (1) Ideals, Varieties and Algorithms and (2) Toric Varieties What are some good papers (meaning, suitable in light of (1) and (2))...
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How to learn math after your PhD is finished [closed]

Question: How does someone go about learning advanced topics in Math after they're done with their PhD? Specific example: You've done your undergrad and masters degrees in math and learned from ...
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Gluing of $\operatorname{Spec} k[x,y], \operatorname{Spec} k[x^{-1},x^{-1}y],$ and $\operatorname{Spec} k[y^{-1},xy^{-1}]$

In Fulton's book on toric varieties he (on page 7) goes through the example of $\mathbb{P}^2$. He considers the fan given by $((1,0),(0,1),(-1,-1))$ and shows via its dual that the variety ...
fish_monster's user avatar
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1 answer
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Homology of toric varieties

In the proof of theorem 5.1.1 of this paper by Dotsenko, Shadrin, and Vallette, they define this (smooth, normal) toric variety $B(n)$, note that there is a surjective toric morphism to $\mathbb{P}^1$ ...
Aidan's user avatar
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1 answer
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Question regarding Proper Mapping

I'm reading toric variety from Cox et. al. and there is a theorem about proper mapping that I cant solve. Let $f : X \rightarrow Y$ be a continuous mapping of locally compact first countable Hausdorff ...
user631697's user avatar
1 vote
0 answers
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Action of $\mathbb C^*$ as torus of toric variety $\mathbb C$

I am reading Toric Variety now and as a well documented example we know $\mathbb{C^s}$ is toric variety with torus $\mathbb(C^*)^s$. I am wondering how will the action of $\mathbb(C^*)^s$ will look ...
user631697's user avatar
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1 answer
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Question on the Color of the Horospherical Variety $\mathrm{SL}_2(\mathbb{C})/U \hookrightarrow \mathrm{Bl}_0(\mathbb{C^2})$

I am now studying on the horospherical variety. For example, I am observing $\mathrm{SL}_2(\mathbb{C})/U$ where $U$ is a unipotent subgroup $$ \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}. $$ ...
J1U's user avatar
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Intersection multiplicity on toric varieties using simplicial cones

In the book "Introduction to toric varieties" by William Fulton (p. 100) there is a formula for the intersection product of two subvarieties. Let $\tau,\sigma$ be cones in the fan $\Delta$, ...
LegNaiB's user avatar
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points in affine toric variety

I am reading Toric Variety from Cox, little and In proposition 1.3.1, I saw this proposition about which I have one question As you can see, they have written that the map $\gamma: S \rightarrow \...
user631697's user avatar
2 votes
1 answer
151 views

Torus $T=\text{Spec}(K[M])$ for the character lattice $M=\{(a_0,....,a_n)\in\mathbb{Z}^{n+1}\mid \sum_{i=0}^na_i=0\}$

For $N=M=\mathbb{Z}^n$ and $e_0,...,e_n$ standard basis of $\mathbb{Z}^n$ the torus $T=\text{Spec}(K[M])=\text{Spec}(K[x_1^{\pm 1},...,x_n^{\pm 1}])=\mathbb{G}_m^n$, where $x_i=\chi^{e_i}$ with ...
Mandarine's user avatar
1 vote
0 answers
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A question regarding the definition of an affine toric variety

I am reading about affine toric varieties from Toric Varieties by Cox, Little, and Shenck, and the definition of an affine toric variety is given as follows: An affine toric variety is an irreducible ...
user631697's user avatar
8 votes
0 answers
99 views

Where do toric varieties appear naturally?

I'm reading Fulton's book. There's an awesome theorem that classifies all smooth toric surfaces as blowups at points starting from either $P^2$ or some Hirzebruch surface. I want to be more excited ...
Andy's user avatar
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1 vote
1 answer
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initial degenerations

I am trying to understand the article "Realization spaces for tropical fans" https://arxiv.org/pdf/0909.4582.pdf by Eric Katz and Sam Payne. Let $X\subset T$ be a subvariety of the torus. ...
Mandarine's user avatar
2 votes
0 answers
57 views

How many tropical polynomials give rise to the same variety? (reference request)

An n-variable polynomial $p(x)=\bigoplus_{i=1}^n \beta_i \textbf{x}^{\alpha_i}$ (here $\textbf{x}$ is the tuple $(x_1,x_2,...,x_n)\in \mathbb{R}^n$) in the tropical (max-plus) semiring has an ...
Brendan Mallery's user avatar
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Dense torus in $\mathbb{P}^n$

What is the dense torus in $\mathbb{P}^n$? Is this the complement of all coordinate hyperplanes? Is there only one possibility?
Mandarine's user avatar
4 votes
2 answers
394 views

Normalization of an affine toric variety is toric

In the book "Toric Varieties" by Cox-Little-Schenck, Proposition 1.3.8 is left as an exercise. It essentially characterizes what the normalization of an affine toric variety is. The ...
Luis Ferroni's user avatar
2 votes
0 answers
102 views

Are Riemann surfaces toric?

For a smooth toric projective variety $X$, I'm able to use the result $H^0(X,\mathcal{O}(D)) \cong \bigoplus_{m \in \Delta_D \cap M} \mathbb{C}\cdot \chi^m$ to understand global sections of a given ...
locally trivial's user avatar
1 vote
0 answers
77 views

Why a polynomial factors in the coordinate ring of an affine chart of the blow up of $\mathbb{A}^{n+1}$ (as a toric variety) at the origin?

I am studying a theorem by Ishi in which she proves there is a resolution of the singularity of a hypersurface defined on de affine space considered as the toric variety given by the fan $\sum_i \...
LeoMontoya's user avatar
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0 answers
120 views

closed orbit of smooth toric variety is defined by regular sequence

Let $X_\Sigma$ be a smooth toric variety corresponding to a fan $\Sigma$, $V(\sigma)\subseteq X_\Sigma$ the closure of the orbit $O(\sigma)$ corresponding to a cone $\sigma$ of $\Delta$. $V(\sigma)$ ...
Mandarine's user avatar
2 votes
0 answers
186 views

Blow up of toric variety corresponds to subdivision of cone

Look at the lattice $N= \mathbb{Z}^3/\mathbb{Z}(1,1,2)$, let $u_0,u_1,u_2$ be the images of the standard basis elements of $\mathbb{Z}^3$ and consider the cone $\sigma = \text{Cone}(u_0,u_1)$. Then it ...
S.Farr's user avatar
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1 vote
0 answers
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Weighted projective space as toric variety

I am trying to solve exercise 3.3.10 from Cox' book on toric varieties. I have no trouble with part (a), but it seems to me that for part (b) I have to wirte down the variety $X_{\Sigma}$ explicitly, ...
S.Farr's user avatar
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5 votes
0 answers
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Is a toric variety still a toric variety after change of coordiantes?

I am reading Cox, Little, Shenck Toric Varieties, and I had a question about the definition of a toric variety. One of the examples (on page 12) uses $$x^2 - y^3 = 0$$ as an example of a toric variety....
bottledcaps's user avatar
2 votes
1 answer
627 views

Isomorphic Lattices, complex Tori and their relation to Jacobians

Let $g >1$ a natural number and $\mathbb{C}^g$ complex vector space which is isomorphic to $\mathbb{R}^{2g}$ is real vector space. An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a ...
user267839's user avatar
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2 votes
0 answers
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Confusion about Affine Toric Varieties/ Toric Ideals

I have just started reading about Toric varieties from David Cox and I have a couple of very basic questions about toric ideals and aff. toric varieties. 1) Eventually, a toric ideal can be defined ...
fhn's user avatar
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2 votes
1 answer
185 views

Q about torus automorphism example

I am working through the book "introduction to tropical algebra" and page 70 and 71 example 2.2.10 states "An automorphism of the torus is an invertible map specified by Laurent monomials. Thus the ...
robotsheepboy's user avatar
2 votes
1 answer
220 views

Why does every toric ideal correspond to an affine toric variety?

I'm reading the book Toric Varieties by Cox, Little and Schenk, and have a small question about the proof of their Proposition 1.1.11 (on page 16). The key part of the proposition is this: Claim: ...
Oskar Henriksson's user avatar
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1 answer
417 views

A basis for the Picard group of the blow up at $\mathbb{P}^2$ at two points

(Toric) Intro: (Superflous, just to consider the initial data) From a toric point of view, the blow up of $\mathbb{P}^2$ at two points (which we can assume are $[0:1:0]$ and $[0:0:1]$ is given by a ...
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