On The Moduli of Surfaces Admitting Genus Two Fibrations Over Elliptic Curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes ${\cal H}(1,X(d),n)$ of morphisms of degree $n$ from elliptic curves to the modular curve $X(d)$, $d\geq 3$. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line ${\mathbb A}^1$ with fibers determined by the components of ${\cal H}(1,X(d),n)$.