Stability and singularities of relative hypersurfaces

TL;DR

This paper investigates the stability and singularities of relative hypersurfaces over curves, establishing an instability criterion for fibers and relating it to the log canonical threshold and Nakayama's Zariski decomposition.

Contribution

It introduces a new instability condition for fibers of relative hypersurfaces and connects it with bounds on the log canonical threshold and divisor decompositions.

Findings

Provides an upper bound on the log canonical threshold

Establishes an instability condition for fibers

Connects stability analysis with Zariski decomposition

Abstract

We study relative hypersurfaces over curves, and prove an instability condition for the fibres. 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.