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Abstract

We introduce the $2$-nodal spherical deformation of certain singular fibers
of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to
${\mathbb{CP}}^{2}\#6({\overline{\mathbb{CP}}^{2}})$,
${\mathbb{CP}}^{2}\#7({\overline{\mathbb{CP}}^{2}})$, and
$3{\mathbb{CP}}^{2}\#k({\overline{\mathbb{CP}}^{2}})$ for $k=16, 17, 18, 19$
using combinations of such deformations, symplectic blowups, and (generalized)
rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$-nodal spherical deformations of certain singular fibers of genus $g \geq 3$ fibrations.