Anna Lorenzini

On a sharp bound for the regularity index of any set of fat points

Abstract We propose an upper bound for the regularity index of fat points of P n with no geometric conditions on the points. Whenever the conjecture is true, the bound is sharp. It is, in fact, reached when there are points with high multiplicities either on a line or on some rational curve. Besides giving an easy proof of the conjecture in P 2 , we prove it in P 3 , by using some preliminary results which hold, more generally, in P n .

Betti numbers of perfect homogeneous ideals

Abstract In this paper we study the graded minimal free resolution of the ideal, I , of any arithmetically Cohen-Macaulay projective variety. First we determine the range of the shifts (twisting numbers) that can possibly occur in the resolution, in terms of the Hilbert function of I . Then we find conditions under which some of the twisting numbers do not occur. Finally, in some ‘good’ cases, all the Betti numbers are (recursively) computed, in terms of the Hilbert function of I or that of Ext n R ( R / I , R ), where R is a polynomial ring over a field and n is the height of I in R .

Fat points of Pn whose support is contained in a linear proper subspace

In this paper we study fat points of whose support is contained in a linear subspace of dimension r, with r<n, and we determine in some cases the Hilbert function, in other cases the regularity index (or an upper bound for it) in , in terms of what is known in .

Segre's Bound and the Case of n+ 2 Fat Points of ℙ n

First we prove that the locus of all sets of fat points with assigned multiplicities whose regularity index is bounded by the so called Segres bound is open and strictly contains the non-empty open set given by the locus of the same number of fat points with the same multiplicities and support in general position. Then we prove that the general Segres bound holds for any set of n + 2 fat points of ℙ n .

Betti numbers of points in projective space

Abstract In this paper we study the graded minimal free resolution of a finite set of points in P n . We find conditions for the minimal number of generators of their ideal and for the Cohen- Macaulay type of their coordinate ring to have the expected minimum value. In some special cases we find that all the Betti numbers of the points are determined. We then explicitly compute them, by means of a compact formula, which allows us to avoid the recursive computation based upon the additivity of the Hilbert function.

CANCELLATION IN RESOLUTIONS AND LEVEL ALGEBRAS

ABSTRACT A difficult open problem (even in codimension 3) asks: what can be the Hilbert function of a standard graded Artinian level algebra? In this paper we solve the related problem (in codimension 3): What are the Hilbert functions H for which the minimal resolution of the lex-segment ideal with Hilbert function H permit (at least theoretically) enough cancellation to give a possible resolution for a level algebra? This gives a necessary (but not sufficient, as we show by example) response to the original open problem. The answer is given in terms of type vectors (which are equivalent to, but different from, Hilbert functions of Artinian algebras). We also give an algorithm (implemented in C.C.A) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree.

Blowing up determinantal space curves and varieties of codim 2 (

Let k be an algebrically closed field with chark = 0 and let C ⊆ P 3 = P 3 k be a curve (i.e. a 1-dimensional, smooth, irreducible scheme). We want to study the scheme X C which is the blow-up of P 3 along C. Let E be the exceptional divisor of the blow-up, and H the strict transform of a generic hyperplane, then PicX C = Z < H, E >, and the kind of questions we would like to address are: