Lucia Marino

Nesting House-designs

A cycle of length 5 with a chordal, i.e.?an edge joining two non-adjacent vertices of the cycle, is called a graph H 5 or also an House-graph. In this paper, the spectrum of House-systems nesting C 3 -systems, C 4 -systems, C 5 -systems and together ( C 3 , C 4 , C 5 ) -systems, of all admissible indices are completely determined, without exceptions.

On 0-dimensional schemes with all permissible conductor degrees

IfX is a set of distinct points in ℙ2 with given graded Betti numbers, we produce a new set of pointsY with the same graded Betti numbers asX which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute the conductor degree for each point.We conclude by generalizing the construction of these schemes, obtaining again the same results.

Minimal Free Resolutions of Zero-dimensional Schemes in ℙ1 × ℙ1

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.