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?



Roots of line bundles
Yes. Take $\mathcal L$ the trivial line bundle, with a one-dimensional space of global sections, and $\mathcal M$ a nontrivial torsion line bundle, so $\mathcal M^k=\mathcal L$.


Then $\Gamma(\mathcal M)$ is certainly zero-dimensional, since otherwise $\mathcal M$ would have a nonvanishing section and be trivial or a somewhere vanishing section and then $\mathcal L$ would have a somewhere vanishing section and be nontrivial.


This provides a counterexample. Such examples exist on any smooth proper variety with a nontrivial Picard variety, meaning $\operatorname{dim}\_kH^1(X,\mathcal O\_X)>0$, such as curves of positive genus.


