Let $R$ be a $k$-algebra of finite type over a perfect field $k$ of characteristic $p>0$. Let $X$ be an integral scheme of dimension $n$ which is projective over the spectrum of $R$ and smooth over $k$. Let $D$, $E$, and $H$ be divisors on $X$ and $\lambda\in\mathbb{Q}_{\geq 0}$ such that

1. (i)

   ${\mathcal{O}}_{X}(H)$ is an ample, globally generated line bundle,

2. (ii)

   $h^{0}(X,{\mathcal{O}}_{X}(mD))>0$ for some $m>0$, and

3. (iii)

   the $\mathbb{Q}$-divisor $E-\lambda D$ is nef.

Then the sheaf $\mathcal{O}_{X}(K_{X/k}+E+dH)\otimes_{\mathcal{O}_{X}}\tau(\lambda\cdot\|D\|)$ is globally generated for all $d\geq n+1$.