Assume I have a line bundle $\mathcal{L}$ on a projective, nonsingular variety $X$ over, say, an algebraically closed field (in fact, you may assume $\mathbb{C}$). Given global sections $s_0,\ldots,s_k\in\mathcal{L}(X)$, how to prove that they do not generate $\mathcal{L}$ globally?
I'd think that this is best done by contradiction, but I can't think of any theorems that strongly rely on the assumption that a line bundle is globally generated by certain particular sections. Note that $\mathcal{L}$ may very well be globally generated (in fact, in my case, it most certainly is) - but I want to prove that certain sections do not generate it globally. So, what I am basically asking for, are results that include a base-point free linear system in their assumptions, so I can use that to produce a contradiction. My first impule was to consider the induced morphism $\phi:X\to\mathbb{P}^k$, but that's already where I get stuck.
If it helps, the case $X=\mathbb{P}^n$ and $\mathcal{L}=\mathcal{O}_X(1)$ is already interesting to me, but it'd (of course) be better if the strategy works even in the more general setting.