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

Algebraic Geometry