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.