TITLE: Why is the Hodge class of \bar{M_g} big and nef?
QUESTION [8 upvotes]: Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first chern class of the pushforward \pi_*(ω). Among other things the hodge class, together with the boundary divisors, freely generates the Picard group of \bar{M_g}.

Question: Why is lambda big and nef?

REPLY [5 votes]: Some multiple of lambda is defined on the coarse moduli space and this is the pullback of an ample bundle on \bar{A_g}, the Satake-Baily-Borel compactification of A_g. Since \bar{M_g} maps birationally onto its image in \bar{A_g}, it follows that lambda is nef and big, in fact also semi-ample (some multiple is base point free) on the coarse moduli space. 
(The map to \bar{A_g } contracts the boundary divisor corresponding to irreducible nodal curves so lambda is not ample.)