[Paper Review] On the numerical dimension of pseudo-effective divisors in positive characteristic

Paolo Cascini, Christopher D. Hacon+2 more|arXiv (Cornell University)|Jun 27, 2012

Algebraic Geometry and Number Theory13 references5 citations

TL;DR

This paper establishes that in positive characteristic, a pseudo-effective R-divisor D on a smooth projective variety has positive numerical dimension if it is not numerically equivalent to the negative part in its divisorial Zariski decomposition. The proof relies on the Frobenius morphism and a new vanishing result, recovering a key result from characteristic zero that previously depended on Kawamata-Viehweg vanishing.

ABSTRACT

Let X be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if D is a pseudo-effective R-divisor on X which is not numerically equivalent to the negative part in its divisorial Zariski decomposition, then the numerical dimension of D is positive. In characteristic zero, this was proved by Nakayama using vanishing theorems.

Motivation & Objective

• To extend Nakayama's characteristic zero result on numerical dimension to positive characteristic, where standard vanishing theorems fail.
• To show that the numerical dimension of a pseudo-effective R-divisor is positive if it is not numerically equivalent to the negative part in its divisorial Zariski decomposition.
• To establish a new vanishing result using the Frobenius morphism, enabling the lifting of sections from curves in positive characteristic.
• To prove that for nef divisors, the numerical dimension defined via asymptotic growth of sections equals the intersection-theoretic dimension.
• To demonstrate that key results in birational geometry can be recovered in positive characteristic through systematic use of Frobenius techniques.

Proposed method

• Use the Frobenius morphism to derive a general vanishing result valid in arbitrary characteristic, relying on asymptotic Serre vanishing.
• Construct a divisorial Zariski decomposition for pseudo-effective R-divisors, defining the negative part $N_{\sigma}(D)$ via numerical invariants $\sigma_{\Gamma}(D)$.
• Apply Theorem 1.2 to show that sections of $\mathcal{O}_X(\lfloor mD\rfloor + A)$ can be lifted from intersections of general ample divisors $W = H_1 \cap \cdots \cap H_r$ when $W$ avoids the non-nef locus $\mathbf{B}_-(D)$.
• Use the non-vanishing of the intersection number $(D^j \cdot H^{n-j})$ to ensure positivity of the restriction to $W$, enabling lower bounds on $h^0$.
• Employ induction on dimension and restriction to general hyperplane sections to compare $\kappa_\sigma(D)$ and $\nu(D)$ for nef divisors.
• Leverage alterations to extend results from smooth to normal projective varieties, preserving numerical invariants under pullback.

Experimental results

Research questions

• RQ1 Can Nakayama's result on the positivity of the numerical dimension of pseudo-effective divisors be extended to positive characteristic?
• RQ2 Can the failure of Kawamata-Viehweg vanishing in positive characteristic be overcome using the Frobenius morphism to recover key birational geometry results?
• RQ3 Is the numerical dimension $\kappa_\sigma(D)$ of a nef divisor equal to the intersection-theoretic dimension $\nu(D)$ in positive characteristic?
• RQ4 Under what conditions can sections of $\mathcal{O}_X(\lfloor mD\rfloor + A)$ be lifted from subvarieties to the whole variety in positive characteristic?
• RQ5 How does the non-nef locus $\mathbf{B}_-(D)$ relate to the divisorial Zariski decomposition in positive characteristic?

Key findings

• If $D$ is a pseudo-effective $\mathbf{R}$-divisor on a smooth projective variety over an algebraically closed field of positive characteristic and $D$ is not numerically equivalent to $N_\sigma(D)$, then $\kappa_\sigma(D) \geq 1$.
• There exists an ample divisor $A$ and a constant $C > 0$ such that $h^0(X, \mathcal{O}_X(\lfloor mD\rfloor + A)) \geq Cm$ for all $m \gg 0$.
• Theorem 1.2 establishes that if $W = H_1 \cap \cdots \cap H_r$ avoids $\mathbf{B}_-(D)$ with $r < \dim X$, then the restriction map $H^0(X, \mathcal{O}_X(\lfloor mD\rfloor + A)) \to H^0(W, \mathcal{O}_X(\lfloor mD\rfloor + A)|_W)$ is surjective for all $m \geq 1$.
• For a nef $\mathbf{R}$-divisor $D$, the numerical dimension $\kappa_\sigma(D)$ equals the intersection-theoretic dimension $\nu(D)$, defined as the largest $j$ such that $(D^j \cdot H^{n-j}) \neq 0$.
• The equality $\kappa_\sigma(D) = \nu(D)$ holds not only on smooth varieties but also on normal projective varieties via alterations and pullback invariance.
• The Frobenius morphism enables a new vanishing result that replaces the role of Kawamata-Viehweg vanishing in characteristic zero, allowing the recovery of key positivity results in positive characteristic.