Family Switching Formula and the $-n$ Exceptional Rational Curves

Abstract

In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial generalization of Fintushel-Stern's argument for four-manifolds. We also derive an algebraic analogue of the family switching formula, allowing the P^1 fibration to carry singular fibers. At the end, we apply the idea of switching to compare the canonical algebraic Kuranishi models of different classes on the so-called "universal family".