Blow-up formulas for (-2)-spheres

TL;DR

This paper extends existing blow-up formulas for (-1)-spheres in 4-manifolds to include (-2)-spheres, providing explicit formulas for the Donaldson invariants involving these spheres.

Contribution

It introduces a method to derive blow-up formulas for (-2)-spheres, expanding the understanding of Donaldson invariants in 4-manifold topology.

Findings

Derived explicit blow-up formulas for (-2)-spheres in 4-manifolds.

Extended the method used for (-1)-spheres to include (-2)-spheres.

Provided formulas involving universal series for Donaldson invariants.

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 $$ D_c(\exp(t\tau)) = D_c(B^2(t) + S^2(t)/2 \tau^2) on \tau^perp if c\cdot \tau is even. $$