Symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines

Abstract

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg–Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces.