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

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

2002-02-04

arxiv.org/abs/math/0202022v3

Mathematics

Algebraic Geometry

28 pages, typos added. To appear on Trans.Amer. Math. Soc

Scientific paper

The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given $X$ a smooth projective threefold, $\E$ a rank-two vector bundle on $X$, $L$ a very ample line bundle on $X$ and $k \geq 0$, $\delta >0 $ integers and denoted by $V= {\V}_{\delta} ({\E} \otimes L^{\otimes k})$ the subscheme of ${\Pp}(H^0({\E} \otimes L^{\otimes k}))$ parametrizing global sections of ${\E} \otimes L^{\otimes k}$ whose zero-loci are irreducible and $\delta$-nodal curves on $X$, we present a new cohomological description of the tangent space $T_{[s]}({\V}_{\delta} ({\E} \otimes L^{\otimes k}))$ at a point $[s]\in {\V}_{\delta} ({\E} \otimes L^{\otimes k})$. This description enable us to determine effective and uniform upper-bounds for $\delta$, which are linear polynomials in $k$, such that the family $V$ is smooth and of the expected dimension ({\em 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 Calaby-Yau threefold, we study in detail the regularity property of a point $[s] \in V$ related to the postulation of the nodes of its zero-locus $C_s =C \subset X$. 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 $V$ at $[s]$. Finally, when $X= \Pt$, we also discuss some interesting geometric properties of the curves given by sections parametrized by $V$.

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