2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/110139In 1995 D. Joyce explicitly constructed a series of self-dual metrics with torus action on the connected sums of complex projective planes. In this paper we explicitly construct the twistor spaces of some of Joyce's self-dual metrics. Starting from a fiber space whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the required twistor spaces. In the construction an important role is played by flops, a useful operation in algebraic geometry.22 pages, 9 figures; V2: errors and typos corrected, presentation improvedDifferential GeometryAlgebraic Geometry53C25; 14C05text