  The degeneration of the Quillen metric for a one-parameter family of Riemann
surfaces has been studied by Bismut-Bost and Yoshikawa. In this article we
propose a more geometric point of view using Deligne's Riemann-Roch theorem. We
obtain an interpretation of the singular part of the metric as a discriminant
and the continuous part as a degeneration of the metric on Deligne products,
which gives an asymptotic development involving the monodromy eigenvalues. This
generalizes the results of Bismut-Bost and is a version of Yoshikawa's results
on the degeneration of the Quillen metric for general degenerations with
isolated singularities in the central fiber.
