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

Flaminio Flamini

• 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.