On the numerical dimension of pseudo-effective divisors in positive characteristic

TL;DR

This paper proves that on smooth projective varieties over algebraically closed fields of positive characteristic, pseudo-effective divisors not equivalent to their negative part have positive numerical dimension, extending results known in characteristic zero.

Contribution

It establishes the positivity of the numerical dimension for certain pseudo-effective divisors in positive characteristic, generalizing Nakayama's characteristic zero results.

Findings

Numerical dimension is positive for divisors not equivalent to their negative part.

Extends Nakayama's theorem from characteristic zero to positive characteristic.

Provides new insights into the geometry of divisors in positive characteristic.

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.