Let $X$ be a smooth projective surface over $\mathbb{C}$. We know that nef line bundles form a closed convex cone whose interior is the ample cone. My doubt is the other direction, is it possible to have an ample class $C$ such that $C=C\_1+C\_2$ where neither $C\_i$ is nef? In fact I know that $C\in |L|$ where $L$ is very ample.


Edit:


I want $C\_i$ to be effective. I am interested in the case when $X$ is the Jacobian of a genus 2 curve.



Can an ample class be written as a sum of two classes which are not even nef?
Take an arbitrary surface which has lots of non-nef effective curves. In particular, take one that has two non-nef effective curves on "opposite" sides of the cone of effective curves. Then some linear combination of these two curves will lie in the interior of the cone and will be ample.


In particular, take a K3 surface with at least 2 smooth rational curves on it. This will satisfy the above condition.


For an explicit example take a general quartic surface in $\mathbb P^3$ that contains a (smooth) plane conic. It is relatively easy to see that the intersection of this quartic with the plane containing the given conic is the union of two smooth conics (the original and another one), so you have two $(-2)$-curves whose sum is the hyperplane class of this embedding to $\mathbb P^3$. Voilà.


