I've read the following in a paper and am not sure why it's true (I'm very beginner at this stuff). Let $X$ be a normal, projective variety (over $\mathbb{C}$ and irreducible if it makes a difference) and $\mathcal{L}$ an ample line bundle on $X$ with $h^0(X,\mathcal{L})=2$. If $\dim X\geq 2$, then $\mathcal{L}$ must have a nonempty base locus?
All I know is that if there's no base locus, then you can define a morphism $X\to\mathbb{P}(H^0(X,\mathcal{L})^*)\cong\mathbb{P}^1$ (and I think I know that the image at least is not a point, else $\mathcal{L}$ will have a nonvanishing global section, which implies it's trivial -- is this right?).
If true, would it be true in general that if $\dim X\geq h^0(X,\mathcal{L})$ (with the same assumptions on $X,\mathcal{L}$), then $\mathcal{L}$ has a nonempty base locus? If not, is it at least true when $h^0=3$? Thanks for your help.