Globally generated, nef and big line bundles which are not ample on a K3 surface

Question

Let $X$ be a K3 surface over $\mathbb{C}$. On a $K3$ surface we know that $Pic(X)\cong Num(X)\cong NS(X)$. A class $L\in Num(X)$ is called movable if $L.C\geq 0$ for every curve $C$ in $X$. It just means that $L$ is a movable class if it is nef.

The interior of this cone is the ample cone, by Nakai's criterion. So we could have big and nef bundles which are not ample isn't it?

Also can we have globally generated, big and nef line bundles on a K3 surface which are not ample? I suppose on the Kummer surface, we could find such examples, is that right?

Answer 1

Let $S_0\subset P^3$ be a quartic surface with a node and let $\pi:S\to S_0$ be the minimal desingularization. Then, since $\pi$ is crepant, $S$ is a K3 surface, $L=\pi^*(O_{P^3}(1))$ is globally generated and big, but not ample since the corresponding morphism contracts a $(-2)$-curve. In particular, when $S$ is a Kummer surface (and $S_0$ has sixteen nodes), you will find examples. I think the presence of such $(-2)$-curves is essentially the only obstruction for a nef and big line bundle to be ample.

Answer 2

Any $(-2)$-curve on a K3 surface can be contracted and the corresponding divisor is globally generated (hence nef) and big: Let $C$ be the $(-2)$-curve and $A$ an arbitrary ample divisor. Then $$D=2A+(A\cdot C)C$$ is a nef and big divisor. It's big, because it is (ample)+(effective) and it is nef, because it could potentially be not nef only on $C$ (any irreducible curve with a different support will intersect $A$ positively and $C$ non-negatively) and $$ D\cdot C= 2A\cdot C + (A\cdot C)(-2) =0 .$$

This also shows that an appropriate multiple of $D$ defines a birational morphism that contracts $C$ and nothing else.

On the other hand, if the K3 surface does not contain any $(-2)$-curves, then every effective curve has non-negative self-intersection. Those with self-intersection $0$ define an elliptic fibration and those with positive self-intersection are ample. This implies that any nef and big divisor is ample. So we obtain:

Claim A K3 surface admits a nef and big but not ample divisor if and only if it contains a smooth rational curve.