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We study the jumps in the archimedean height of the Ceresa cycle, asintroduced by R. Hain in his work on normal functions on moduli spaces ofcurves, and as further analyzed by P. Brosnan and G. Pearlstein in terms ofasymptotic Hodge theory. Our work is based on a study of the asymptoticbehavior of the Hain-Reed beta-invariant in degenerating families of curves. Weshow that the height jump of the Ceresa cycle at a given stable curve is equalto the so-called "slope" of the dual graph of the curve, and we characterizethose stable curves for which the height jump vanishes. We also obtain ananalytic formula for the height of the Ceresa cycle for a curve over a functionfield over the complex numbers, and characterize in analytic terms when theheight of the Ceresa cycle vanishes.

Jumps in the height of the Ceresa cycle