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We prove a sharp relative Clifford inequality for relatively special divisorson varieties fibered by curves. It generalizes the classical Cliffordinequality about a single curve to a family of curves. It yields a geographicalinequality for varieties of general type and Albanese-fibered by curves,extending the work of Horikawa, Persson, and Xiao in dimension two to arbitrarydimensions. We also apply it to deduce a slope inequality for some arbitrarydimensional families of curves. It sheds light on the existence of a mostgeneral Cornalba-Harris-Xiao type inequality for families of curves.  The whole proof is built on a new tree-like filtration for nef divisors onvarieties fibered by curves. One key ingredient of the proof is to estimate thesum of all admissible products of nef thresholds with respect to thisfiltration.

Relative Clifford inequality for varieties fibered by curves