Abstract:
A lower bound for the dimensions of the second osculating spaces to any surface scroll is given, relying on the special feature of osculating hyperplane sections to such surfaces. Moreover a class of counterexamples to the even dimensional part of a conjecture of Piene-Tai is provided.

Abstract:
Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic subset of singular elements of $|V|$. We study the components of $\Cal D(X,V)$ in connection with the jumping sets of $(X,V)$, generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of $(X,L,V)$) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.

Abstract:
Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus - deriving from flexes - are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls.

Abstract:
Smooth complex polarized varieties $(X,L)$ with a vector subspace $V \subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the locus ${\Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to $\dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $\dim(X)-4$ while it is $\dim(X)-3$ if and only if $(X,L)$ is a scroll over a smooth curve. When the codimension is $\dim(X)-5$ and the Picard number is one only the Pl\"ucker embedding of the Grassmannian of lines in $\Bbb P^4$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided.

Abstract:
The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that the dimension of X is at least 2, and that L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if L defines a morphism onto a two dimensional cone, in which case B(L) is the inverse image of the vertex of the cone.

Abstract:
Let $X\subset \mathbb P^N$ be a scroll over a $m$-dimensional variety $Y$. We find the locally free sheaves on $X$ governing the osculating behavior of $X$, and, under certain dimension assumptions, we compute the cohomology class and the degree of the inflectional locus of $X$. The case $m=1$ was treated in \cite{LMP}. Here we treat the case $m\ge 2$, which is more complicated for at least two reasons: the expression for the osculating sheaves and the computations of the class of the inflectional locus become more complex, and the dimension requirements needed to ensure validity of the formulas are more severe.

Abstract:
Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let $\L=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of $\L$ are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.

Abstract:
During last decades the use of local varieties was strongly reduced due to introduction of modern cultivars characterized by higher yield, and breed for different traits of agronomic value. However, these cultivars not always have the quality aspects that was found in old traditional and typical crops also depending from the know-how of traditional cultivation. Nowadays the practise of intensive agriculture select only a small number of species and varieties with a consequent reduction of the diversity in agro-ecosystems and risk of loss of important alleles characterizing genetic materials adapted to specific environments. The creation of quality marks of the European Union proved to be a successful system to protect typical products through the Denomination of Origins (PDO- Protected Denomination of Origin and PGI- Protected Geographical Indication). However, the protection of quality needs efficient instruments to discriminate DOP or IGP varieties in the field and to trace them along the agro-food chain. DNA fingerprinting represents an excellent system to discriminate herbaceous and tree species as well as to quantify the amount of genetic variability present in germplasm collections. The paper describes several examples in which AFLPs, SSRs and minisatellite markers were successfully used to identify tomato, artichoke, grape, apple and walnut varieties proving to be effective in discriminating also closely related genetic material. DNA fingerprinting based on SSR is also a powerful tool to trace and authenticate row plant materials in agro-food chains. The paper describes examples of varieties traceability in the food chains durum wheat, olive, apple and tomato pursued through the identification of SSR allelic profiles obtained from DNA isolated from complex highly processed food, such as bread, olive oil, apple pureè and nectar and peeled tomato.

Abstract:
During last decades the use of local varieties was strongly reduced due to introduction of modern cultivars characterized by higher yield, and breed for different traits of agronomic value. However, these cultivars not always have the quality aspects that was found in old traditional and typical crops also depending from the know-how of traditional cultivation. Nowadays the practise of intensive agriculture select only a small number of species and varieties with a consequent reduction of the diversity in agro-ecosystems and risk of loss of important alleles characterizing genetic materials adapted to specific environments. The creation of quality marks of the European Union proved to be a successful system to protect typical products through the Denomination of Origins (PDO- Protected Denomination of Origin and PGI- Protected Geographical Indication). However, the protection of quality needs efficient instruments to discriminate DOP or IGP varieties in the field and to trace them along the agro-food chain. DNA fingerprinting represents an excellent system to discriminate herbaceous and tree species as well as to quantify the amount of genetic variability present in germplasm collections. The paper describes several examples in which AFLPs, SSRs and minisatellite markers were successfully used to identify tomato, artichoke, grape, apple and walnut varieties proving to be effective in discriminating also closely related genetic material. DNA fingerprinting based on SSR is also a powerful tool to trace and authenticate row plant materials in agro-food chains. The paper describes examples of varieties traceability in the food chains durum wheat, olive, apple and tomato pursued through the identification of SSR allelic profiles obtained from DNA isolated from complex highly processed food, such as bread, olive oil, apple pureè and nectar and peeled tomato.