Big $q$-ample Line Bundles

TL;DR

This paper characterizes q-ample line bundles on complex projective schemes by their restrictions to augmented base loci, providing a criterion for q-ampleness based on subvarieties.

Contribution

It establishes an equivalence between q-ampleness of a line bundle and its restriction to the augmented base locus, extending the understanding of q-ampleness in algebraic geometry.

Findings

q-ampleness is characterized by restrictions to augmented base loci

Failure of q-ampleness on a variety implies non-q-ampleness on some divisor

Provides a criterion for q-ampleness in terms of subvariety restrictions

Abstract

A recent paper of Totaro develops a theory of $q$-ample bundles in characteristic 0. Specifically, a line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\geq m_0$ implies $H^i(X,\mathcal{F}\otimes \mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension 1 subscheme $D$ of $X$ such that the restriction of $L$ to $D$ is not $q$-ample.