2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73501We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with self-intersection -1, i.e. they are minimal. In addition, none these smooth structures admit an underlying symplectic structure. Shortly after the appearance of a preliminary version of this article, Park, Stipsicz, and Szabo used the techniques described herein to show that the complex projective plane blown up at 5 points has infinitely many distinct smooth structures. In the final section of this paper we give a somewhat different construction of such a family of examples.11 pages, More typos and minor errors correctedGeometric TopologySymplectic Geometry14J26, 53D05, 57R55, 57R57text