Basic classes and embedded spheres

For a given smooth four-manifold we study the relation between the simple type condition for Donaldson invariants and the intersections of basic classes with smoothly embedded spheres with self-intersection −2 and −3. ng the structure of Donaldson invariants imposed by the existence of embedded spheres we also recover one criterion for a manifold being of simple type (previously proven by Kronheimer and Mrówka using singular connections).