On the structure of Pedersen-Poon twistor spaces

We study algebro-geometric properties of certain twistor spaces over $n\boldsymbol{CP}^2$ with two dimensional torus actions, whose existence was proved by Pedersen and Poon. We show that they have a pencil whose general members are non-singular toric surface, and completely determine the structure of the reducible members of the pencil, which are also toric surfaces. In the course of our proof, we describe behaviors of the above pencil under equivariant smoothing. Relation between the weighted dual graphs of the toric surfaces in the pencil and similar invariant of the above torus action on $n\boldsymbol{CP}^2$ is also determined