The smooth classification of elliptic surfaces with \(b^ +>1\) (Q1341288)

A compact complex surface \(S\) is elliptic if there is a holomorphic map \(\pi : S \to C\), where \(C\) is a complex curve and the generic fiber is a smooth elliptic curve. In this interesting paper the authors study the case \(C = \mathbb{C} \mathbb{P}^1\) and construct a family \({\mathcal I}\), such that every minimal elliptic surface (fibered over \(\mathbb{C} \mathbb{P}^1)\) is equivalent up to a deformation to an element of \({\mathcal I}\). The family \({\mathcal I}\) is built using a basic example (blow up the nine base points of a generic pencil of cubic curves in \(\mathbb{C} \mathbb{P}^2)\) and two operations. The first one is an appropriate fiber sum and the second is the logarithmic transformation. The main result depends on the computation of some nontrivial coefficients of the Donaldson polynomial of an elliptic surface.