# Gr\"obner strata in the Hilbert scheme of points

@article{Lederer2009GrobnerSI,
title={Gr\"obner strata in the Hilbert scheme of points},
author={Mathias Lederer},
journal={arXiv: Algebraic Geometry},
year={2009}
}
• M. Lederer
• Published 2 July 2009
• Mathematics
• arXiv: Algebraic Geometry
The present paper shall provide a framework for working with Gr\"obner bases over arbitrary rings $k$ with a prescribed finite standard set $\Delta$. We show that the functor associating to a $k$-algebra $B$ the set of all reduced Gr\"obner bases with standard set $\Delta$ is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a $k… ## Figures from this paper Natural morphism from the Groebner scheme to the Hilbert scheme Let$k$be a commutative ring and$S=k[x_0, \ldots, x_n]$be a polynomial ring over$k$with a monomial order. For any monomial ideal$J$, there exists an affine$k$-scheme of finite type, called Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals Let$k$be a commutative ring and$S=k[x_0, \ldots, x_n]$be a polynomial ring over$k$with a monomial order. For any monomial ideal$J$, there exists an affine$k$-scheme of finite type, called A Borel open cover of the Hilbert scheme • Mathematics J. Symb. Comput. • 2013 Combinatorial duality of Hilbert schemes of points in the affine plane The Hilbert scheme of$n$points in the affine plane contains the open subscheme parametrizing$n$distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension The close relation between border and Pommaret marked bases • Mathematics • 2020 Given a finite order ideal$\mathcal O$in the polynomial ring$K[x_1,\dots, x_n]$over a field$K$, let$\partial \mathcal O$be the border of$\mathcal O$and$\mathcal P_{\mathcal O}$the Pommaret Components of Gröbner strata in the Hilbert scheme of points We fix the lexicographic order ≺ on the polynomial ring S=k[x1, …, xn] over a ring k. We define HilbS/k≺Δ , the moduli space of reduced Gröbner bases with a given finite standard set Δ, and its open Compatibly Split Subvarieties Of The Hilbert Scheme Of Points In The Plane Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert Functors of liftings of projective schemes • Mathematics J. Symb. Comput. • 2019 Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors • Mathematics • 2012 The Bialynicki-Birula cells on the Hilbert scheme H^n({A}^d) are smooth and reduced in dimension d=2. We prove that there is a schematic structure in higher dimension, the Bialynicki-Birula scheme, ## References SHOWING 1-10 OF 53 REFERENCES Rational components of Hilbert schemes • Mathematics • 2009 The Gr\"obner stratum of a monomial ideal$\id{j}$is an affine variety that parametrizes the family of all ideals having$\id{j}$as initial ideal (with respect to a fixed term ordering). The Incidence relations among the Schubert cells of equivariant punctual Hilbert schemes Abstract. Let${\mathbb H}_{ab}(H)$be the equivariant Hilbert scheme parametrizing the zero dimensional subschemes of the affine plane$k^2\$, fixed under the one dimensional torus
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