Regularity Results for Families of Nodal Curves on Smooth Projective Threefolds and Postulation of Nodes

Given X a smooth projective threefold, E a rank-two vector bundle on X and k, δ two positive integers, we determine effective and uniform upper-bounds for δ, which are linear polynomials in k, such that the family Vδ(E(k)), parametrizing irreducible and δ-nodal curves which are zero-loci of global sections of the vector bundle E(k) on X, is smooth and of the expected dimension (regular, for short). We discuss some examples showing that our result is almost-sharp. Furthermore, when X is assumed to be a Fano or a Calaby-Yau threefold, we study in detail the regularity problem related to the postulation of nodes. Precisely, given [C] ∈ Vδ(E(k)), when the nodes of C are assumed to be in general position on X, on an irreducible divisor of X with at worst log-terminal singularities or on a l.c.i. and subcanonical curve in X, we find upperbounds on δ which are, respectively, cubic, quadratic and linear polynomials in k ensuring the regularity of Vδ(E(k)) at the point [C]. When X = P, we also discuss some interesting geometric properties of the curves parametrized by Vδ(E(k)).