Tag: toric

ag geometry algebraic: polynomial inlays of toric varieties of polytopes?

Background: Leave $ P $ be an integral polytope, and $ X_P $ The toric variety associated with the normal range.

$ X_P $ It is always projective, because the collection of characters corresponding to the points $ mathbb {Z} ^ n cap P $ together they give an inlay of $ X_P $ in projective space.

However, the dimension of this embedding is the number of integer points, which is generally exponentially large in a reasonable description of $ P $.

Questions:

  • Suppose that $ P $ is given by $ Ax leq b $ in $ mathbb {Q} ^ n $, with $ A in M_ {n times m} ( mathbb {Q}) $ Y $ b in mathbb {Q} ^ n $and that they promised us that $ P $ is integral is there a projective incorporation of $ X_P $ that only requires $ POLY (| A |, | b |) $ bits to specify?
  • There is a family $ A_n, b_n $ such that the minimum dimension of an inlay grows exponentially in $ | A |, | b | $?
  • Are there any polytope parameters (in the sense of parametrized complexity) that control the size of a minimum (and efficiently computable) insert?

I'm deliberately lazy about whether the coding of $ A $ is in binary or unary; Inlays of polynomial or pseudopolinomial size would be interesting for me.

Motivation: I am curious about whether there are polytope parameters that are evident through simple additions of the corresponding toric variety, and that could help with computational problems on the side of the polytope.

For example, if we know that $ X_P $ It is a smooth complete intersection and we have the equations that cut it, we can calculate its Euler characteristic using the formula on page 146 of "About Chern's classes and Euler's characteristic for complete non-singular intersections" by Vicente Navarro Aznar. This would count the number of vertices of $ P $, which is usually a $ # P $ Difficult problem Of course, most polytopes will not provide a smooth toric variety or a complete intersection, and it is very likely that calculating the scale is difficult, so this observation is of limited use.

Anyway, I'm curious to know if we can measure the complexity of the politoe by the complexity of the toric variety as a projective variety. The basic question is whether or not we can efficiently find small inlays in general, hence this question.

abstract algebra: toric variety associated with the cone on a polytope

Leave $ P = [0, m] subset M _ { mathbb {R} ^ 2} $ be the line segment and consider the cone about $ P $.

What is the toric variety of the cone about $ P $?

The point is that I'm not completely sure how to build a cone $ sigma $ of the polytope $ P $. Should it be just the cone generated by $ me_1 $? In which case we would not get the variety $ mathbb {C} times T ^ 1 $, where $ T $ denotes the bull?

What does the "affine variety of the cone of a polytope" really mean? I'm sorry if this question seems too trivial, but I couldn't find any reference that answers this question explicitly.

Reference: Calabi-Yau toric varieties

I am looking for a reference of the following fact:

Leave $ N cong mathbb {Z} ^ n $ a trellis and $ P subset N _ { mathbb {R}} = N otimes _ { mathbb {Z}} mathbb {R} $ a convex lattice polytope. Leave $ P_1, points, P_k $ be soft (all $ P_j $are elementary simplifications) subdivision of $ P $ which consists of convex lattice polytopes and takes the fan $ Sigma $ considering the cones on the faces of the $ P_j $& # 39; s. Then the toric variety $ V _ {Sigma} $ It is a soft Calabi-Yau.

Algebraic geometry ag: Picard groups of toric varieties in positive characteristics

For a toric variety $ X _ {Sigma} $ about the complex numbers associated with a fan $ Sigma $ There is a simple short and exact sequence that calculates the divisor class group. To each dimensional cone $ rho $ in the fan there is a divisor of Weil invariant of the bull $ D _ { rho} $ (which is the closure of the associated bull orbit in $ X _ {Sigma} $) The exact short sequence is
$ M rightarrow bigoplus _ { rho in Sigma (1)} mathbb {Z} cdot D _ { rho} rightarrow text {Cl} (X _ { Sigma}) rightarrow 0 $ (where $ M $ it is the network of characters and the sum in the middle is over all the rays associated with the fan).

This description is based on the correspondence of Orbit Cone which is tested in the book by Cox Little and Schenck (which deals only with toric varieties on $ mathbb {C} $) The proof they present does not directly generalize to the characteristic zero fields and, although I have seen that many theorems move to algebraically closed fields, it is not clear to me that this description does generalize. Leave $ lambda ^ n $ denote the cocaracter associated with $ n $ in the network of characters. For example, correspondence requires that the intersection $ U _ { sigma_1} cap U _ { sigma_2} = U _ { sigma_1 cap sigma_2} $ and this in turn is based on the Proposition that $ n $ is in a cone $ sigma $ If and only if $ lim_ {t rightarrow 0} lambda ^ n (t) $ converges on $ U _ {sigma} $. This seems to make use of the fact that $ mathbb {C} $ It has an appropriate topology.

Is there any example of a soft fan? $ Sigma $ and a finite field $ mathbb {F} $ such that the associated toric variety $ X _ { Sigma, mathbb {F}} $ has a Picard group that is not isomorphic to the Picard group of $ X _ { Sigma, mathbb {C}} $, or it is true that the Picard group in this case only depends on the fan and not on the definition field. Also, what about more than $ bar { mathbb {F}} $?

geometry ag.algebraica: When is the vector beam on the toric variety a toric variety?

Is it true that a set of vectors on a toric variety is also a toric variety if and only if it is divided? If so, how do we show it?

This seems to be the content of a commentary in the Oda Tata lectures on bull inlays, although the language is slightly different, and there is no evidence.

sg.symplectic geometry – Is the minimum Chern number of a toric variety at least 2?

I would like to show that the minimum number of Chern $ N_M $ of a toric manifold $ M $ At least $ 2 $, where
$$
N_M: = underset {l> 0} { min} lbrace existence A in H_2 (M; mathbb {Z}) : langle c, A rangle = l rbrace,
$$
$ c $ denotes the first Chern class $ (M, omega) $ (for any choice of $ omega $– compatible complex structure), and $ langle.,. rangle $ It is the natural pair between cohomology and homology groups.

I do not know how to prove this, but the following interpretation of Chern's first class could help.

Leave $ (M2d, omega, mathbb {T}) $ be a toric colic, where $ omega $ It is the symplectic form and $ mathbb {T} $ is a $ d $three-dimensional bull that acts effectively and in a Hamiltonian manner in $ (M, omega) $. Visit $ M $ as a symplectic reduction of $ mathbb {C} ^ n $ by the action of a $ k $three-dimensional subtorus $ mathbb {K} subset (S ^ 1) ^ n $ (therefore identifying $ mathbb {T} simeq (S ^ 1) ^ n / mathbb {K} $), it can be shown that there is a natural isomorphism
$$
H_2 (M; mathbb {Z}) simeq text {Lie} ( mathbb {K}) _ { mathbb {Z}},
$$
where the integral network $ text {Lie} ( mathbb {K}) _ { mathbb {Z}} $ It is the core of the exponential map. $ exp: text {Lie} ( mathbb {K}) to mathbb {K} $. For any choice of $ omega $-compatible almost complex structure in $ M $, the first class of chern $ c in H ^ 2 (M; mathbb {Z}) simeq text {Lie} ( mathbb {K}) _ { mathbb {Z}} ^ * $ writes:
$$
c (m) = underset {j = 1} { overset {n} sum} m_j m in text {Lie} ( mathbb {K}) _ { mathbb {Z}} quad iota (m) = (m_1, …, m_n),
$$
where $ iota: text {Lie} ( mathbb {K}) hookrightarrow mathbb {R} ^ n $ It is the inclusion of Lie algebras induced by inclusion. $ mathbb {K} subset (S ^ 1) ^ n $.

Of course, in general (when $ M $ it is not toric), $ N_M $ it can be equal to $ 1 $, and one may even have that $ langle c, H_2 (M; mathbb {Z}) = 0 $ (in which case it is often written $ N_M = infty $). However, since any toric collector has a decomposition in complex cells, it seems that $ N_M $ it should be at least $ 2 $.

Any help would be appreciated. Thanks in advance.

Geometría ag.algebraica – Stabilizers of the toric battery

Leave $ mathcal {X} $ be an O-ring on a field $ k $. This can be defined by a stacked fan, that is, a pair $ ( Sigma, beta) $, where $ Sigma $ is a fan in a lattice $ L $Y $ beta: L to N $ It is a lattice homomorphism with finite cokernel.

Assume that $ mathcal {X} $ It's soft and DM. The automorphism groups of points of $ mathcal {X} (k) $ They are of the form $ mu_n $, $ n geq 1 $. I would like to know if it is possible to bind. $ n $ (or cousins ​​that are divided $ n $) in terms of $ | coker (?) | $, or in some other way. I'm sure this is a stupid question, but this topic is very new to me.