Explicit Constructions of Halphen Pencils

We construct rational elliptic surfaces of index two by explicitly constructing their associated Halphen pencils in the projective plane $\mathbb{P}^2$. For each of the types of singular fibers that occur we construct at least one example having that type of fiber and in fact, for some, we construct all possible examples. We establish a precise dictionary between the fibers in a rational elliptic surface and the corresponding plane curves and, in particular, we study the singularities of the curves appearing in a Halphen pencil.