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Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $mathbb{P}^n$. In this context, we look inside properties of several types of segment ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.
It remains an open problem to classify the Hilbert functions of double points in $mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $mathbb{P}^2$, we show how to construct a set of fat points $Z subseteq mathbb{P}^2$ of
In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N$, we presen
For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $ell(R/I) < infty$, we introduce a new invariant $HKd(M, I)$ called the {em Hilbert-Kunz density funct
We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generate
In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic