Abstract:
For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, \delta}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with only $\delta$ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper-bounds on $\delta$ ensuring that the corresponding Severi variety is everywhere smooth of codimension $\delta$ in $|C|$ (regular, for short). In particular, we focus on surfaces of general type, since for such surfaces less is known than what is proven for other cases. Our result generalizes some results of Chiantini-Sernesi (1997) and of Greuel-Lossen-Shustin (1997 - in the case of nodes) as it is shown by some examples of Severi varieties on blown-up surfaces or surfaces in $\P^3$ which are elements of a component of the Noether-Lefschetz locus. We also consider examples of regular Severi varieties on surfaces in $\P^3$ of general type which contain a line.

Abstract:
In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V(|D|, \delta)), which parametrize universal families of irreducible, \delta-nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and numerical conditions on D and numerical conditions on \delta ensuring that such number coincides with dim(V(|D|, \delta). As related facts, we also determines some sharp results concerning the geometry of some Severi varieties.

Abstract:
In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of smooth projective surfaces, in particular of planes. Our original motivation has been a series of papers of G. Zappa which appeared in the 1940-50's regarding degenerations of scrolls to unions of planes. Here, we present a first set of results on the subject; other aspects are still work in progress and will appear later. We first study the geometry and the combinatorics of a surface like $X$, considered as a reduced, connected surface on its own; then we focus on the case in which X is the central fibre of a degeneration of relative dimension two over the complex unit disk. In this case, we deduce some of the intrinsic and extrinsic invariants of the general fibre from the ones of its central fibre. In the particular case of $X$ a central fibre of a semistable degeneration, i.e. $X$ has only global normal crossing singularities and the total space of the degeneration is smooth, some of the above invariants can be also computed by topological methods (i.e., the Clemens-Schmid exact sequence). Our results are more general, not only because the computations are independent on the fact that $X$ is the central fibre of a degeneration, but also because the degeneration is not semistable in general.

Abstract:
In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques.

Abstract:
In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality.

Abstract:
Polyphenols are the principal compounds associated with health benefic effects of wine consumption and in general are characterized by antioxidant activities. Mass spectrometry is shown to play a very important role in the research of polyphenols in grape and wine and for the quality control of products. The soft ionization of LC/MS makes these techniques suitable to study the structures of polyphenols and anthocyanins in grape extracts and to characterize polyphenolic derivatives formed in wines and correlated to the sensorial characteristics of the product. The coupling of the several MS techniques presented here is shown to be highly effective in structural characterization of the large number of low and high molecular weight polyphenols in grape and wine and also can be highly effective in the study of grape metabolomics. 1. Principal Polyphenols of Grape and Wine Polyphenols are the principal compounds associated to health benefic effects of wine consumption. A French epidemiological study performed in the end of 1970s reported that in France, despite the high consumption of foods rich in saturated fatty acids, the incidence of mortality from cardiovascular diseases was lower than that in other comparable countries. This phenomenon was called “the French paradox” and was related to the beneficial effects of red wine consumption [1]. In general; polyphenols have antioxidant activities. Their activity as peroxyl radical scavengers and in the formation of complexes with metals (Cu, Fe, etc.) has been shown by in vitro studies. Moreover, the ability of polyphenols to cross the intestinal wall of mammals confers their biological properties. Flavan-3-ols are one of the principal classes of grape polyphenols which include (+)-catechin and (？)-epicatechin, and their oligomers called procyanidins, proanthocyanidins, and prodelphinidins. B-type and A-type procyanidins and proanthocyanidins (the latter are condensed tannins) are present in the grape skin and seeds; tannins are mainly present in seeds, and prodelphinidins are polymeric tannins composed of gallocatechin units (structures in Figure 1). During winemaking the condensed (or nonhydrolyzable) tannins are transferred to the wine and contribute strongly to the sensorial characteristic of the product. In the mouth, the formation of complexes between tannins and saliva proteins confers to the wine the sensorial characteristic of astringency: bitterness and astringency of wine is linked to tannins structure, in particular galloylation degree (DG) and polymerization degree (DP) of flavan-3-ols [2, 3].

Abstract:
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.

Abstract:
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 $V$.

Abstract:
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 $X$.