  The goal of this work is to study positivity of subvarieties with nef normal
bundle in the sense of intersection theory. After Ottem's work on ample
subschemes, we introduce the notion of a nef subscheme, which generalizes the
notion of a subvariety with nef normal bundle. We show that restriction of a
pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective
(resp. big). We also show that ampleness and nefness are transitive properties.
We define the weakly movable cone as the cone generated by the pushforward of
cycle classes of nef subvarieties via proper surjective maps. This cone
contains the movable cone and shares similar intersection-theoretic properties
with it, thanks to the aforementioned properties of nef subvarieties. On the
other hand, we prove that if $Y\subset X$ is an ample subscheme of codimension
$r$ and $D|_Y$ is $q$-ample, then $D$ is $(q+r)$-ample. This is analogous to a
result proved by Demailly-Peternell-Schneider. We use the theory of $q$-ample
divisors, as developed by Totaro, throughout the paper.
