TITLE: Nef divisors with few global sections
QUESTION [6 upvotes]: Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$?
Edit: In fact I'm interested in nef line bundles D, not just divisors.

REPLY [2 votes]: Another extremely simple example:
Let $S = \mathrm{Bl}_p \mathbb{P}^2$ be the blowup of $\mathbb{P}^2$ at a point, and look at the divisor $H-E$ on $S$, where $E$ is the exceptional divisor and $H$ is the pullback of a hyperplane class on $\mathbb{P}^2$.  Notice $h^0(H-E)=2 = \dim S$, since sections correspond to lines in $\mathbb{P}^2$ passing through $p$ (up to scalars).  But $(H-E)^2=0$, so $H-E$ is nef.