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}.



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> **Question**: Why is lambda big and nef? 
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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.)


