Explicit constructions of Halphen pencils of index two

Abstract

We describe many explicit constructions of Halphen pencils of index two and their associated rational elliptic surfaces. For each of the types of singular fiber that occur we construct at least one Halphen pencil yielding that type of fiber and in fact, for some, we construct all possible examples. More generally, we also study the singularities of a curve ${\mathcal{𝒞}}_{3m}$ appearing in a Halphen pencil of index $m$ in terms of the log canonical threshold of the pair $({\mathbb{P}}^2,{\mathcal{𝒞}}_{3m})$.