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

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 $\mathcal {E}$ on $X$, and integers $k\geq 0$, $\delta >0$, denote by ${\mathcal {V}}_{\delta } ({\mathcal {E}} (k))$ the subscheme of ${\mathbb {P}}(H^0({\mathcal {E}}(k)))$ parametrizing global sections of ${\mathcal {E}}(k)$ whose zero-loci are irreducible $\delta$-nodal curves on $X$. We present a new cohomological description of the tangent space $T_{[s]}({\mathcal {V}}_{\delta } ({\mathcal {E}} (k)))$ at a point $[s]\in {\mathcal {V}}_{\delta } ({\mathcal {E}} (k))$. This description enables us to determine effective and uniform upper bounds for $\delta$, which are linear polynomials in $k$, such that the family ${\mathcal {V}}_{\delta } ({\mathcal {E}} (k))$ 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 $[s] \in {\mathcal {V}}_{\delta } ({\mathcal {E}} (k))$ related to the postulation of the nodes of its zero-locus $C = V(s) \subset X$. 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 $\delta$ which are, respectively, cubic, quadratic and linear polynomials in $k$ ensuring the regularity of ${\mathcal {V}}_{\delta } ({\mathcal {E}} (k))$ at $[s]$. Finally, when $X= \mathbb {P}^3$, we also discuss some interesting geometric properties of the curves given by sections parametrized by ${\mathcal {V}}_{\delta } ({\mathcal {E}} \otimes \mathcal {O}_X(k))$.