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

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

A formal proof of the optimal frame setting for Dynamic-Frame Aloha with known population size

In Dynamic-Frame Aloha subsequent frame lengths must be optimally chosen to maximize throughput. When the initial population size ${\cal N}$ is known, numerical evaluations show that the maximum efficiency is achieved by setting the frame length equal to the backlog size at each subsequent frame; however, at best of our knowledge, a formal proof of this result is still missing, and is provided here. As byproduct, we also prove that the asymptotical efficiency in the optimal case is $e^{-1}$, provide upper and lower bounds for the length of the entire transmission period and show that its asymptotical behaviour is $\sim ne-\zeta \ln (n)$, with $\zeta=0.5/\ln(1-e^{-1})$.Comment: 22 pages, submitted to IEEE Trans. on Information Theor

Does Empirical Embeddedness Matter? Methodological Issues on Agent-Based Models for Analytical Social Science

The paper deals with the use of empirical data in social science agent-based models. Agent-based models are too often viewed just as highly abstract thought experiments conducted in artificial worlds, in which the purpose is to generate and not to test theoretical hypotheses in an empirical way. On the contrary, they should be viewed as models that need to be embedded into empirical data both to allow the calibration and the validation of their findings. As a consequence, the search for strategies to find and extract data from reality, and integrate agent-based models with other traditional empirical social science methods, such as qualitative, quantitative, experimental and participatory methods, becomes a fundamental step of the modelling process. The paper argues that the characteristics of the empirical target matter. According to characteristics of the target, ABMs can be differentiated into case-based models, typifications and theoretical abstractions. These differences pose different challenges for empirical data gathering, and imply the use of different validation strategies.Agent-Based Models, Empirical Calibration and Validation, Taxanomy of Models

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