Families of nodal curves on projective threefolds and their regularity via postulation of nodes

Abstract

Abstract. The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold X, a rank-two vector bundle E on X, and integers k ≥ 0, δ > 0, denote by V δ (E(k)) the subscheme of P(H 0 (E(k))) parametrizing global sections of E(k) whose zero-loci are irreducible δ-nodal curves on X. We present a new cohomological description of the tangent space. This description enables us to determine effective and uniform upper bounds for δ, which are linear polynomials in k, such that the family V δ (E(k)) is smooth and of the expected dimension (regular, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when X is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point [s] ∈ V δ (E(k)) related to the postulation of the nodes of its zero-locus C = V (s) ⊂ X. Roughly speaking, when the nodes of C are assumed to be in general position either on X, or on an irreducible divisor of X having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in X, we find upper bounds on δ which are, respectively, cubic, quadratic and linear polynomials in k ensuring the regularity of V δ (E(k)) at [s]. Finally, when X = P 3 , we also discuss some interesting geometric properties of the curves given by sections parametrized by V δ (E ⊗ O X (k)).