The T -graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the T = (K∗)n action on the Hilbert scheme induced from the T -action on A. It has vertices the T -fixed points, and edges the onedimensional T -orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the pla...

Nowadays, many factors support a growing production of information. In modern large environments, for the user's point of view, it is desirable to have Information Retrieval Systems (IRS) that retrieve documents according to their relevance levels. Relevance levels have been studied in some previous Information Retrieval (IR) works while some others (few) IR research works tackled the questions...

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from mo...

(1.1) Let σ be a rational, polyhedral cone. It induces a (normal) affine toric variety Yσ which may have singularities. We would like to investigate its deformation theory. The vector space T 1 Y of infinitesimal deformations is multigraded, and its homogeneous pieces can be determined by combinatorial formulas developed in [Al 1]. If Yσ only has an isolated Gorenstein singularity, then we can ...

Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem of resolving S/M over S. The difficulty in resolving minimally is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence) into the multigraded Betti numbers of S/M , [St]. In particular, the minimal free...

Let k be a field of positive characteristic. We construct, for each dominant coweight λ of the standard maximal torus in Sln, a closed subvariety D(λ) of the multigraded Hilbert scheme of an affine space over k, such that the k-valued points of D(λ) can be interpreted as lattices in k((z))n endowed with infinitesimal structure. Moreover, for any λ we construct a universal homeomorphism from D(λ...

The Hilbert scheme H n of n points in A contains an irreducible component R n which generically represents n distinct points in A. We show that when n is at most 8, the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 8 ⊂ H 8 is defined by a single explicit equation which serves as a criterion for deciding whether a given idea...