Blow-up formulas for (-2)-spheres

Abstract

Let $X$ be a simply connected 4-manifold containing a $(-1)$-sphere $e$. Fintushel and Stern prove that $$ D_c(\exp(te)) = D_c(B(t)) on e^\perp if c\cdot e is even, $$ $$ D_c(\exp(te)) = D_{c-e}(S(t)) on e^\perp if c \cdot e is odd, $$ for some universal series $B(t),S(t) \in \Q[x][[t]]$ with $x$ the class of a point. We show that their method can easily be extended to $(-2)$-spheres $\tau$ to give blow up formulas like $$ D_c(\exp(t\tau)) = D_c(B^2(t) + S^2(t)/2 \tau^2) on \tau^perp if c\cdot \tau is even. $$