# Numerical dimension and locally ample curves

## TL;DR

This paper investigates the properties of curves with ample normal bundles and their cycle classes, establishing bounds on the numerical dimension of divisors and positioning of curve classes within cones of curves using q-ample divisors.

## Contribution

It extends previous results by removing singularity restrictions and applying q-ample divisor theory to study cycle classes of positive curves with ample normal bundles.

## Key findings

- Cycle class of a curve with ample normal bundle lies in the interior of the cone of curves.
- Cycle class of an ample curve lies in the interior of the cone of movable curves.
- Bound on the numerical dimension of divisors restricted to subvarieties with ample normal bundle.

## 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.