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


Author: Flaminio Flamini
Journal: Trans. Amer. Math. Soc. 355 (2003), 4901-4932
MSC (2000): Primary 14H10, 14J60; Secondary 14J30, 14J32, 14J45
DOI: https://doi.org/10.1090/S0002-9947-03-03199-4
Published electronically: July 28, 2003
MathSciNet review: 1997590
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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 ${\mathcal{V}}_{\delta} ({\mathcal{E}} \otimes \mathcal{O}_X(k))$.


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Additional Information

Flaminio Flamini
Affiliation: Dipartimento di Matematica, Universitá degli Studi “Roma Tre", Largo San Leonardo Murialdo, 1 - 00146 Roma, Italy
Address at time of publication: Dipartimento di Matematica, Universitá degli Studi di L’Aquila, Via Vetoio-Loc. Coppito, 67010 L’Aquila, Italy
Email: flamini@matrm3.mat.uniroma3.it

DOI: https://doi.org/10.1090/S0002-9947-03-03199-4
Keywords: Families of nodal curves, postulation of nodes, projective threefolds
Received by editor(s): June 25, 2002
Published electronically: July 28, 2003
Additional Notes: The author is a member of Cofin GVA, EAGER and GNSAGA-INdAM
Article copyright: © Copyright 2003 American Mathematical Society

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