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 $$D_c(\exp (te))=D_c(B(t))on{e}^{\perp }ifc\cdot eiseven,$$ $$D_c(\exp (te))=D_{c-e}(S(t))on{e}^{\perp }ifc\cdot eisodd,$$ 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 $$D_c(\exp (t\tau ))=D_c(B^2(t)+S^2(t)/2{\tau }^2)on{\tau }^{p}erpifc\cdot \tau iseven.$$

6 pages, AMS-latex version 1.1