Example of nef and big, not ample What would be a common, simple example of a nef and big divisor that is not ample? Are there any common, less simple examples? Are there any common strategies for finding examples?

The easiest way to get examples is to observe that nefness and bigness are preserved under pullbacks via birational morphisms, but ampleness isn't.

So let $Y$ be normal and let $f: X \to Y$ be any birational morphism that is not an isomorphism. If $A$ is an ample divisor on $Y$, then $f^*A$ is nef and big but not ample.

The simplest such example would be $f: X \to \mathbf P^2$ the blowup of a point, and $H$ a line in $\mathbf P^2$. Then $f^*H$ is nef and big, but it has degree 0 on the exceptional divisor, so it isn't ample.

Examples obtained in this way all have a special property: the nef and big divisor $f^*A$ we obtain is always _semi-ample_ , meaning that some multiple of it is basepoint-free (a.k.a. globally generated). But in general a nef and big divisor need not be semi-ample. Examples of this kind are more subtle: see Section 2.3 of _Positivity in Algebraic Geometry I_ by Lazarsfeld for a nice exposition.