We compute the Dongaldson SU(2)-invariants of the double cover of IP~2 branched over a sinooth aigebraic curve of degree eight. From this we deduce a formula for the relative invariants of the blow-up of the Gompf nucleus N_2, and we show how this gives a blow-up formula for a class of 4-manifolds which includes essentially all the sinply connected 4 manifolds known to hve big jiffcomorphism group. We apply the result on the nucleus also to prove a formula for the invariants of minimal simply connected elliptic surfaces which reduces the computation to the case of geometric genus one. In particular, we compute all the Donaldson invariants of minimal simply connected elliptic surfaces without multiple fibers. Our main tool is Donaldson-Floer theory.
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