  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 and
ternary, 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.
