Symplectic rational blow-ups on rational 4-manifolds

Abstract

We prove that if a symplectic 4-manifold X X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4 4 -manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition.