On families of complex curves over ℙ¹ with two singular fibers

Let f: S → ℙ¹ be a family of genus g ≥ 2 curves with two singular fibers F₁ and F₂. We show that F₁ = F₂* and F₂ = F₁* are dual to each other, S is a ruled surface, the geometric genera of the singular fibers are equal to the irregularity of the surface, and the virtual Mordell–Weil rank of f is zero. We prove also that c₁²(S) ≤ -2 if g = 2, and c₁²(S) ≤ -4 if g > 2. As an application, we will classify all such fibrations of genus g = 2.