Stability and Singularities of Relative Hypersurfaces

This article has been accepted for publication in International mathematics research notices, Published by Oxford University Press. We study relative hypersurfaces over curves, and prove an instability condition for the fibers. This gives an upper bound on the log canonical threshold of the relative hypersurface. We compare these results with the information that can be derived from Nakayama's Zariski decomposition of effective divisors on relative projective bundles.