  We study the jumps in the archimedean height of the Ceresa cycle, as
introduced by R. Hain in his work on normal functions on moduli spaces of
curves, and as further analyzed by P. Brosnan and G. Pearlstein in terms of
asymptotic Hodge theory. Our work is based on a study of the asymptotic
behavior of the Hain-Reed beta-invariant in degenerating families of curves. We
show that the height jump of the Ceresa cycle at a given stable curve is equal
to the so-called "slope" of the dual graph of the curve, and we characterize
those stable curves for which the height jump vanishes. We also obtain an
analytic formula for the height of the Ceresa cycle for a curve over a function
field over the complex numbers, and characterize in analytic terms when the
height of the Ceresa cycle vanishes.
