Let X be a smooth projective variety over bC, let A=operatornameSpecR where R is a finitely generated bZ-algebra containing bZ, and let X_A→ A be a spreading out of X over A, so that X_A×_A bCcong X. Thus X_A is a family of characteristic p>0 models of X. Weak ordinarity conjecture. There exists a Zariski-dense set of closed points U⊆ A such that, for every point p∈ U, writing X_p=X_A×_A k(p), the Frobenius morphism induces a bijection H^i(X_p,operatorname O_X_p)longrightarrow H^i(X_p,F^e_operatorname O_X_p) for every i and every e. This conjecture predicts a dense set of reductions with ordinary Frobenius action on the cohomology of the structure sheaf; it is motivated by work of Mustaţă and Srinivas, but the supplied text does not state whether it has been resolved.

Weak ordinarity conjecture

Let $X$ be a smooth projective variety over $\bC$ , let $A=\operatorname{Spec}R$ where $R$ is a finitely generated $\bZ$ -algebra containing $\bZ$ , and let $X_A\to A$ be a spreading out of $X$ over $A$ , so that $X_A\times_A\bC\cong X$ . Thus $X_A$ is a family of characteristic $p>0$ models of $X$ . Weak ordinarity conjecture. There exists a Zariski-dense set of closed points $U\subseteq A$ such that, for every point $p\in U$ , writing $X_p=X_A\times_A k(p)$ , the Frobenius morphism induces a bijection

H^i(X_p,\operatorname{\mathcal O}_{X_p})\longrightarrow H^i(X_p,F^e_*\operatorname{\mathcal O}_{X_p})

for every $i$ and every $e$ . This conjecture predicts a dense set of reductions with ordinary Frobenius action on the cohomology of the structure sheaf; it is motivated by work of Mustaţă and Srinivas, but the supplied text does not state whether it has been resolved.

Equivalent formulations 3

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The weak ordinarity conjecture
Let $V$ be an $n$ -dimensional smooth projective variety over a field $k$ of characteristic zero. Given a model of $V$ over a finitely generated $b$ -subalgebra $A$ of $k$ , and writing $V_s$ for the fiber over a closed point $sb$ of $boperatorname{Spec} A$ , the weak ordinarity conjecture. There exists a Zariski-dense set of closed points $S \subseteq \operatorname{Spec} A$ such that the action induced by Frobenius on $H^n(V_s, \mathcal{O}_{V_s})$ is bijective for every $s \in S$ . The conjecture is a weak form of ordinarity: ordinarity implies the stated Frobenius-bijectivity condition, but the converse need not hold. Its status here is refuted.
source: Bhargav Bhatt, Karl Schwede and Shunsuke Takagi, “The weak ordinarity conjecture and F-singularities”, arXiv:1307.3763 (2016).
The Weak Ordinarity Conjecture
Let $Y$ be a smooth, connected projective variety over an algebraically closed field $k$ of characteristic $0$ . Given a model $Y_A$ over a finitely generated $\mathbb{Z}$ -algebra $A$ , consider the closed points $s\in\operatorname{Spec}A$ for which the action of Frobenius on
$\operatorname{H}^{\dim Y_s}(Y_s,\mathcal{O}_{Y_s})$
is bijective. Weak Ordinarity Conjecture. This set of points is dense in $\operatorname{Spec}A$ . The conjecture concerns reduction modulo primes and the relationship between Frobenius action and characteristic-zero singularities; the source records its role in implications for dense F-pure type, but gives no resolution status.
source: Eloísa Grifo and Craig Huneke, “Symbolic powers of ideals defining F-pure and strongly F-regular rings”, arXiv:1702.06876 (2017).
The weak ordinarity conjecture
Let $X$ be an $n$ -dimensional smooth projective variety over a field $k$ of characteristic zero. Given a model of $X$ over a finitely generated $b\mathbb{Z}b$ -subalgebra $A$ of $k$ , there exists a Zariski-dense set of closed points $S \subseteq \operatorname{Spec} A$ such that the action of Frobenius on $H^n(X_s, \mathcal{O}_{X_s})$ is bijective for all $s \in S$ .

Weak ordinarity conjecture. The assertion above holds for every such $X$ and every model.

This conjecture concerns Frobenius actions on the top coherent cohomology of reductions of smooth projective varieties. It is stated as strictly weaker than ordinarity in the sense of Bloch and Kato and is used to relate characteristic-zero multiplier ideals to characteristic- $p$ test ideals.
source: Axel Stäbler, “Reductions of non-lc ideals and non F-pure ideals assuming weak ordinarity”, arXiv:1804.02922 (2019).

Sources & referencesView supporting material

Primary source

Zsolt Patakfalvi, Karl Schwede and Kevin Tucker, “Positive characteristic algebraic geometry”, arXiv:1412.2203 (2017).

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Weak ordinarity conjecture

Equivalent formulations 3

The weak ordinarity conjecture

The Weak Ordinarity Conjecture

The weak ordinarity conjecture

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