One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, then $V$ and $W$ intersect.) However, the same does not seem to hold for most other easy examples of surfaces. For instance, any ruled surface $S \to C$ has non-intersecting curves (take the fibers over any two distinct points of $C$). Furthermore, any surface $S$ obtained from $\mathbb{P}^2_k$ by blowing up points $p_1, \ldots, p_n$ has two non-intersecting curves: take two lines that intersect transversely at $p_1$ and avoid $p_2, \ldots, p_n$, and lift them to curves on $S$.

Thus, my question:

Is there any nonsingular algebraic surface other than the projective plane such that any two curves on the surface have nontrivial intersection?

(Note: assume the base field is algebraically closed.)