Dario Portelli

On Threefolds Covered by Lines

A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, ifX is such a threefold, let Σ denote the Fano scheme of lines onX and μ the number of lines contained inX and passing through a general point ofX. Assume that Σ is generically reduced. Then μ ≤ 6. Moreover,X is birationally a scroll over a surface (μ = 1), orX is a quadric bundle, orX belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −KX = 2HX.

Threefolds in ℙ5 with a 3-dimensional family of plane curveswith a 3-dimensional family of plane curves

A classification theorem is given of smooth threefolds of ℙ5 covered by a family of dimension at least three of plane integral curves of degreed≧2. It is shown that for such a threefoldX there are two possibilities:(1)X is any threefold contained in a hyperquadric;(2)d≦3 andX is either the Bordiga or the Palatini scroll.

Rees algebras and gröbner bases

This paper deals with the following problem. Robbiano showed in [9] that standard bases, Grobner bases, Macaulay bases are all instances of the same general situation. In this paper, we develop this philosophy from the point of view of the Rees algebra R of a ring A w.r.t. a filtration F given on A. The ring R plays a fine job between A and the graded ring G associated to A, F. The use of R and the properties of termorderings and their relate Grobner bases led naturally to the definition of Grobnerfiltrations ingeneral commutative rings.

On the divisor class groups of a two-dimensional local ring and its form ring

Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A : Rad ( A ). We assume that the form ring of A with respect to the ideal I: G = Gr ( A, I ), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl( A ) and Cl( G ).

On the divisor class groups of localizations, completions and veronesean subrings of z-graded krull domains

RiassuntoSiaR un dominioZ-graduato, noetheriano. Questo articolo tratta principalmente il gruppo di classi di divisori di localizzazioni e completamenti rispetto ad una topologia α-adica (α omogeneo) diR. Alcuni risultati tecnici vengono applicati allo studio dei sottoanelli Veronesiani diR.SummaryLetR be aZ-graded, noetherian, integral domain. This paper deals mainly with the divisor class groups of localizations and completions with respect to an α-adic topology (α homogeneous) ofR. Some technical results are used to study the Veronesean subrings ofR.