Example of nef and big, not ample

Question

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?

Answer 1

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.

Answer 2

Using the implications for any rank vector bundles \begin{equation} \text{ample}\iff\text{positive}\Rightarrow\text{semi-positive}; \end{equation} in D.P.S. example 1.7, the authors constructed a nef rank $2$ vector bundle $E$ over an elliptic curve $\Gamma$ which is not semi-positive, so it is not an ample vector bundle.

To be exact, $\mathcal{O}_{\mathbb{P}(E)}(1)=L$ is a nef line bundle over the ruled surface $\mathbb{P}(E)=X$; by D.P. theorem 4.3 $\displaystyle\int_Xc_1(L)^2>0$, by D. J.-P. corollary 6.19 $L$ is also a big line bundle; but $L$ is not a semi-positive line bundle, that is $L$ is not an ample line bundle.

For a recap of these results, see D. J.-P. sections 6 and 18.

Post Scriptum

1. D.P.S. means J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Alg. Geom. 3 (1994) 295-345
2. D.P. means J.-P. Demailly, M. Păun - Numerical characterization of the Kähler cone of a compact Kähler manifold, Annals of Math. 159 (2004) 1247-1274
3. D.J.-P. means J.-P. Demailly - Analytic Methods in Algebraic Geometry (2010) Higher Education Press, Surveys of Modern Mathematics, Vol. 1