Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective variety.
Now let $L$ be a line bundle over $X_{\mathbb{Z}[1/N]}$ and $L_\mathbb{C}$ and $L_p$ be the corresponding line bundle over $X=X_{\mathbb{Z}[1/N]}\times\text{Spec}(\mathbb{C})$ and $X_p$ respectively.
How do the properties of $L_\mathbb{C}$ (ample, nef, big, effective) relate to the properties of $L_p$?
For example, if $L_\mathbb{C}$ is nef does this imply that $L_p$ is nef?
Any references or starting points would be greatly appreciated.