If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil generated by $s_i$ and $s_j$ does not have a curve in its base locus?

I know that for generic choices for the basis it is rather trivial, but I have to work with a fixed basis from which I don't have much information.

A simple example would be a basis of monomials for the global sections of $\mathcal{O}_{\mathbb{P}^2}(n)$. There are many monomials sharing a common factor, but one can easily choose two relatively prime.