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- ELENA GUARDO
- 2005

Let I = (F 1 ,. .. , Fr) be a homogeneous ideal of the ring R = k[x 0 ,. .. , xn] generated by a regular sequence of type (d 1 ,. .. , dr). We give an elementary proof for an explicit description of the graded Betti numbers of I s for any s ≥ 1. These numbers depend only upon the type and s. We then use this description to: (1) write H R/I s , the Hilbert… (More)

In this paper we extend the definition of a separator of a point P in P n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes + msPs. We associate to P i a tuple of positive integers of length ν = deg Z − deg Z ′. We call this tuple the degree of the minimal separators of P i of multiplicity m i , and we… (More)

We study the Hilbert functions of fat points in P 1 × P 1. We associate to an arbitrary scheme of fat points Z two tuples of non-negative integers α Z and β Z that depend only upon the multiplicities and relative positions of the points. We then show that all but a finite number of values of the Hilbert function of Z can be computed directly from α Z and β… (More)

We initiate the study of extended bicolorings of Steiner triple systems (STS) which start with a k-bicoloring of an STS(v) and end up with a k-bicoloring of an STS(2v + 1) obtained by a doubling construction , using only the original colors used in coloring the subsystem STS(v). By producing many such extended bicolorings, we obtain several infinite classes… (More)

The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P 2 are determined whenever Z is supported at any 6 or fewer distinct points. We also handle a broad range of cases in which the points can be infinitely near, related to the classification of normal cubic surfaces.… (More)

- ELENA GUARDO
- 2007

If X is a finite set of points in a multiprojective space P n 1 × · · · × P nr with r ≥ 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P 1 × P 1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective… (More)

- ELENA GUARDO
- 2007

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of X and X \ {P} via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a sepa-rator to shifts in the… (More)

- ELENA GUARDO
- 2009

Let Z be a set of fat points in a multiprojective space P n1 × · · · × P nr. We introduce definitions for the separator of a fat point and the degree of a fat point in this context, and we study some of their properties. Our definition has been picked so that when we specialize to the cases: (a) Z is a reduced set of points in P n , (b) Z is a set of fat… (More)

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X, for the bigraded Betti numbers of I Z , the defining ideal of Z. We then relate the total Betti numbers of I Z to the shifts in the graded… (More)

A bicolorable STS(v) is a Steiner triple system whose vertices are colored in such way that every block receives precisely two colors. A k-bicoloring of a STS is a vertex coloring using each of k colors, and the feasible set Ω is a set of integers k for which k-bicolorings exist. In this paper, we study feasible sets of STS(v)s of all orders v < 50.