Stability and singularities of relative hypersurfaces

We study relative hypersurfaces, and prove an instability condition for the fibres. This is the starting point for an investigation of the geometry of effective divisors on relative projective bundles. 1 Introduction and discussion of the results We work over the complex field. Let E be a vector bundle of rank r ≥ 3 and degree d on a smooth projective curve B of genus b. Consider the relative projective bundle P: = PB(E) with its projection pi: PB(E) − → B. Let OP(1) be the tautological sheaf. Let us consider a relative hypersurface X ⊂ P; this means for us an element of a linea