Borel degenerations of arithmetically Cohen-Macaulay curves in \(\mathbb{P}^{3}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1339.14006
DOI: 10.1142/S0218196714500301
author: Gunnar Fløystad
author: Margherita Roggero
published in: International Journal of Algebra and Computation
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.2754
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 6356171
zbMATH Keywords: Borel-fixed ideal
zbMATH Keywords: ACM space curve
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: space curve
MaRDI profile type: Publication
arXiv ID: 1111.2754
cites work: A criterion for detecting m-regularity
cites work: Segments and Hilbert schemes of points
cites work: Q5317677
OpenAlex ID: W3099493549
title: Borel degenerations of arithmetically Cohen-Macaulay curves in \(\mathbb{P}^{3}\)
review text: Consider an algebraically closed field \(k\) of characteristic zero, the projective space \(\mathbb{P}^n = \mathrm{Proj}(S:=k[x_0,\dots,x_n])\) and a numerical polynomial \(p(t) \in \mathbb{Q}[t]\). The Hilbert scheme \(\mathcal{H}\mathrm{ilb}_{p(t)}(\mathbb{P}^n)\) is a projective scheme that parameterizes all closed subschemes in \(\mathbb{P}^n\) whose Hilbert polynomial is \(p(t)\). The understanding of the component structure of a Hilbert scheme is in general very difficult. A major tool to tackle this problem is the study of the class of Borel-fixed ideals in \(k[x_0,\dots,x_n]\). These ideals define the most degenerate objects on the Hilbert scheme, their orbits under the action of \(\mathrm{GL}(n+1)\) are closed and they have nice combinatorial properties. In fact, Borel-fixed ideals have been used in this context since the proof of the connectedness of the Hilbert scheme [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)].NEWLINENEWLINEIn the paper under review, the authors investigate Borel-fixed ideals lying on components of the Hilbert scheme parameterizing arithmetically Cohen-Macaulay schemes of codimension two in \(\mathbb{P}^n\). These components are in 1-to-1 correspondence with Borel-fixed ideals NEWLINE\[NEWLINEJ(a,\boldsymbol{b}) = (x_0^a,x_0^{a-1}x_1^{b_1},\dots,x_0 x_1^{b_{a-1}},x_1^{b_a}) \subset k[x_0,\dots,x_n],NEWLINE\]NEWLINE where \(a \in \mathbb{N}\) and \(\boldsymbol{b}=(b_0,b_1,\dots,b_a) \in \mathbb{N}^{a+1}\) with \(0=b_0<b_1 < \cdots< b_a\). The authors give two necessary conditions for a saturated Borel-fixed ideal \(J\) to define a point on the component corresponding to \(J(a,\boldsymbol{b})\) and they conjecture that this two conditions are also sufficient. The first condition is a consequence of the semi-continuity of ranks of cohomology groups on the fibers of a flat family and says that if a saturated ideal \(J\) defines a point on the component of \(J(a,\boldsymbol{b})\), then \(\di…
reviewed by: Paolo Lella
description: scientific article; zbMATH DE number 6356171