Let $k$ be a fixed algebraically closed field and $X/k$ an irreducible scheme smooth and proper over $k$. Can there exist a line bundles $\mathcal{L}, \mathcal{M}$ and an integer $m > 0$ so that

1.) $\dim_k \Gamma(\mathcal{L}) = 0$

2.) $\dim_k \Gamma(\mathcal{M}) > 0$

With $\mathcal{L}^m \cong \mathcal{M}^m$. If not, does an example exist if you drop smoothness (then replace $\mathcal{M}$ with the line bundle of an effective Weil divisor) or the condition that $k$ is algebraically closed?