moduli spaces - Why is the Hodge class of bar{M_g} big and nef?

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.)