Some Singular Lex-Segments

On the smoothness of lexicographic points on Hilbert schemes

We study the geometry of standard graded Hilbert schemes of polynomial rings and exterior algebras. Our investigation is motivated by a famous theorem of Reeves and Stillman for the Grothendieck Hilbert scheme, which states that the lexicographic point is smooth. By contrast, we show that, in standard graded Hilbert schemes of polynomial rings and exterior algebras, the lexicographic point can be singular, and it can lie in multiple irreducible components. We answer questions of Peeva-Stillman and of Maclagan-Smith.

通过相对标记基的商环上 Hilbert 概形的开覆盖与 Lex 点

Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases

本文引入相对标记基，为商环上的 Hilbert 概形提供计算方法，并给出理论开覆盖与计算 lex 点的两个应用。

Upper Bounds for the Betti Numbers of a given Hilbert Function

From a Macaulay's paper it follows that a lex-segment ideal has the greatest number of generators (the 0-th Betti number $\b_0$) among all the homogeneous ideals with the same Hilbert function. In this paper we prove that this fact extends to every Betti number, in the sense that all the Betti numbers of a minimal free resolution of a lex segment ideal are bigger than or equal to the ones of any homogeneous ideal with the same Hilbert function.

Segments and Hilbert schemes of points

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…