Title: Numerical dimension and locally ample curves,
Abstract: In the paper \cite{Lau16}, it was shown that the restriction of a
pseudoeffective divisor $D$ to a subvariety $Y$ with nef normal bundle is
pseudoeffective. Assuming the normal bundle is ample and that $D|_Y$ is not
big, we prove that the numerical dimension of $D$ is bounded above by that of
its restriction, i.e. $\kappa_{\sigma}(D)\leq \kappa_{\sigma}(D|_Y)$. The main
motivation is to study the cycle classes of "positive" curves: we show that the
cycle class of a curve with ample normal bundle lies in the interior of the
cone of curves, and the cycle class of an ample curve lies in the interior of
the cone of movable curves. We do not impose any condition on the singularities
on the curve or the ambient variety. For locally complete intersection curves
in a smooth projective variety, this is the main result of Ottem \cite{Ott16}.
The main tool in this paper is the theory of $q$-ample divisors.