On the numerical dimension of pseudo-effective 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.

v.3 (17 pages): typos corrected and other minor changes. Additional remarks on the singular case. To appear in the American Journal of Mathematics

References in corpus (2)

• The diminished base locus is not always closed
• The non-nef locus in positive characteristic

Cited by in corpus (3)

• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
• F-singularities in families
• Volume and Hilbert function of R-divisors