On the diffeomorphism groups of elliptic surfaces

In this paper we determine for relatively minimal elliptic surfaces with positive Euler number the image of the natural representation of the group of orientation preserving self-diffeomorphisms on $\Hbar$, the second homology group reduced modulo torsion. To this end we construct as many embedded spheres of square -2 such that an isometry not induced from any combination of reflections at such spheres or from 'complex conjugation' can be shown not to be induced from some diffeomorphism at all. This is done with the help of Seiberg-Witten-invariants.