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. 



Properties of divisors when moving from char 0 to char p.
No, that is not true: the property of being nef, ample, etc. is not stable under specialization (although it is stable under generization). For instance, begin with $Y=\mathbb{P}^2\_{\mathbb{Z}}$, i.e., $\text{Proj} \ \mathbb{Z}[s,t,u]$ together with its natural projection $$\pi:Y\to \text{Spec}\ \mathbb{Z}.$$ Consider the homogeneous ideal $$I = \langle st(t-s),s(u-pt),t(u-ps),u(t-s),u(u-ps) \rangle.$$ The corresponding zero scheme is the union of three disjoint sections of $\pi$, namely $[1,0,0]$, $[0,1,0]$ and $[1,1,p]$. Let $\nu:X\to Y$ be the blowing up of $Y$ along $I$. 


There is a canonically defined pullback map of invertible sheaves, $$f:\nu^\*\omega\_{Y/\mathbb{Z}} \to \omega\_{X/\mathbb{Z}},$$
which identifies $\nu^\*\omega\_{Y/\mathbb{Z}}$ with $\omega\_{X/\mathbb{Z}}(-\underline{E})$ for a unique Cartier divisor $\underline{E}$ on $X$, the exceptional divisor of $\nu$. Now consider the invertible sheaf $L = (\nu^\*\mathcal{O}\_Y(2))(-\underline{E})$. 


Over $\mathbb{Z}[1/p]$, the invertible sheaf $L$ is nef, and even globally generated. Essentially this is because the associated ideal $I[1/p]$ is generated by the homogeneous generators of degree $2$; indeed, the one cubic generator $st(t-s)$ is already in the ideal generated by the quadratic generators, $$st(t-s) = \frac{1}{p}\left(s[t(u-ps)]-t[s(u-pt)]\right).$$ However, in characteristic $p$ this is not nef: the strict transform of the line $Z(u)$ has intersection number $-1$ with $L$. 


**Edit.** Replace "openness" claim by "stable under generization", see comments below.


