Equivalence of families of singular schemes on threefolds and on ruled fourfolds

The main purpose of this paper is twofold. We first analyze in detail the meaningful geometric aspect of the method introduced in [12], concerning families of irreducible, nodal “curves” on a smooth, projective threefold $X$. This analysis gives some geometric interpretations not investigated in [12] and highlights several interesting connections with families of other singular geometric “objects” related to $X$ and to other varieties. Then, we use this method to study analogous problems for families of singular divisors on ruled fourfolds suitably related to $X$. This enables us to showthat Severi varieties of vector bundles on $X$ can be rephrased in terms of “classical” Severi varieties of divisors on such fourfolds.