On the Arakelov Geometry of Moduli Spaces of Curves
  In this paper we compute the asymptotics of the metric on the line bundle over the moduli space of curves that arises when attempting to compute the archimedean height of the algebraic cycle $C-C^-$ in the jacobian of a smooth projective curve of genus $g$. One way to express the results is to say that the metric extends (more or less) to the line bundle found by Moriwaki that has non-negative degree on every complete curve in $\Mbar_g$ not contained in the boundary. 
Richard Hain and David Reed