  In 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.
