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

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 $V$.Comment: 28 pages, typos added. To appear on Trans.Amer. Math. So

P^r-scrolls arising from Brill-Noether theory and K3-surfaces

In this paper we study examples of P^r-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from classical Lazarsfeld's results in. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of the moduli space of such K3's and of the moduli space of semistable torsion-free sheaves of a given Mukai-vector on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a general K3 as well as Brill-Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces whose base curve has general moduli.Comment: published in Manuscripta Mathematic

Equivalence of families of singular schemes on threefolds and on ruled fourfolds

The main purpose of this paper is twofold. We first want to analyze in details the meaningful geometric aspect of the method introduced in the previous paper [12], concerning regularity of families of irreducible, nodal "curves" on a smooth, projective threefold $X$. This analysis highlights several fascinating connections with families of other singular geometric "objects" related to $X$ and to other varieties. Then, we generalize this method to study similar problems for families of singular divisors on ruled fourfolds suitably related to $X$.Comment: 22 pages, Latex 2e, submitted preprin

Brill--Noether loci of stable rank--two vector bundles on a general curve

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls.Comment: 9 pages, submitted preprin

Extensions of line bundles and Brill--Noether loci of rank-two vector bundles on a general curve

In this paper we study Brill-Noether loci for rank-two vector bundles and describe the general member of some components as suitable extensions of line bundles.Comment: 31 pages; revised version after referees' comments; to appear in Revue Roumaine de Math\'ematiques Pures et Appliqu\'ee

Hilbert schemes of some threefold scrolls over F_e

Hilbert schemes of suitable smooth, projective 3-fold scrolls over the Hirzebruch surface F_e, with e > 1, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described. This article generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=1.Comment: 25 pages; revised version after referees' comments; to appear in Advances in Geometry; this article deals with the case of scrolls over F_e, e>1, and thus generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=

Gaps for geometric genera

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach.Comment: 9 pages, submitted preprin

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e=0,1 the scrolls fill up the entire component of the Hilbert scheme, while for e=2 the scrolls exhaust a subvariety of codimension 1.Comment: 12 pages; submitted pre-print; previous papers arXiv:1110.5464, for e=1, and arXiv:1406.0956, for any e, dealt with vector bundles having first Chern class which ensures that they are always uniform. This is not the case for the present pape