  Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been
conjectured by Viehweg that a quasi-projective manifold that admits a
generically finite morphism to the moduli stack of canonically polarized
varieties is necessarily of log general type. Given a quasi-projective surface
that maps to the moduli stack, we employ extension properties of logarithmic
pluri-forms to establish a strong relationship between the moduli map and the
minimal model program of the surface. As a result, we can describe the
fibration induced by the moduli map quite explicitly. A refined affirmative
answer to Viehweg's conjecture for families over surfaces follows as a
corollary.
