Given a Lagrangian sphere in a symplectic 4–manifold $(M, \omega)$ with $b^{+}=1$, we
find embedded symplectic surfaces intersecting it minimally. When the Kodaira
dimension $\kappa$ of $(M,\omega)$ is $-\infty$, this minimal intersection property turns out to be
very powerful for both the uniqueness and existence problems of Lagrangian spheres.
On the uniqueness side, for a symplectic rational manifold and any class which is not
characteristic, we show that homologous Lagrangian spheres are smoothly isotopic,
and when the Euler number is less than 8, we generalize Hind and Evans’ Hamiltonian
uniqueness in the monotone case. On the existence side, when $\kappa=-\infty$, we give a
characterization of classes represented by Lagrangian spheres, which enables us to
describe the non-Torelli part of the symplectic mapping class group.