Minimal free resolutions for subschemes of star configurations
type: Item
instance of: scholarly article
title: Minimal free resolutions for subschemes of star configurations
zbMATH Open document ID: 1333.13034
author: Alfio Ragusa
author: Giuseppe Zappalà
published in: Journal of Pure and Applied Algebra
publication date: 2015-10-27T00:00:00Z
review text: Let \(R=K[x_1,\dots,x_n]\). Let \(C_{c,n}\) denote the set of subsets of \(\{1,2,\dots,n\}\) with cardinality \(c\). For a subset \(S\) of \(C_{c,n}\), one can associate a squarefree monomial ideal \(I_S=\bigcap_{\{a_1,\dots,a_c\}\in S} (x_{a_1},\dots,x_{a_c})\) in \(R\) and vice versa. In particular, \(I_{C_{c,n}}\) defines the \textit{star configuration}. In the paper under review, the authors investigate the subschemes of a star configuration whose support \(S_{n,h}\) is a union of \(n-2h\) consecutive diagonals \(D_k\), where \(D_k=\{\{i,j\}\in C_{2,n} \mid |i-j|=k\}\) and \(S_{n,h}\) is defined recursively as follows: \(S_{n,h}=S_{n,h-1}\setminus (D_h\cup D_{n-h}), 2 \leq h \leq \frac{n-2}{2}\) with \(S_{n,1}=C_{2,n}\setminus P_n\), where \(P_n=\{ \{i,j\}\in C_{2,n}\mid |i-j|=1 \} \cup \{\{1,n\}\}\). The authors compute the minimal graded free resolutions of \(I_{S_{n,h}}\) for various values of \(h\) (Corollary 2.11, Theorem 3.6).
reviewed by: Ramakrishna Nanduri
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14N20
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 6499146
zbMATH Keywords: star configuration
zbMATH Keywords: scheme
zbMATH Keywords: minimal free resolution
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.06.010
OpenAlex ID: W1424155465
cites work: Comparing powers and symbolic powers of ideals
cites work: Tower sets and other configurations with the Cohen-Macaulay property
cites work: Star configurations in \(\mathbb{P}^n\)
cites work: On the first infinitesimal neighborhood of a linear configuration of points in \(\mathbb P^2\)
cites work: Are symbolic powers highly evolved?
cites work: Edge ideals: algebraic and combinatorial properties
cites work: The minimal free graded resolution of a star-configuration in \(\mathbb{P}^n\)
cites work: Liaison des variétés algébriques. I
cites work: Partial intersections and graded Betti numbers
cites work: PROPERTIES OF 3-CODIMENSIONAL GORENSTEIN SCHEMES
DOI: 10.1016/J.JPAA.2015.06.010
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description: scientific article; zbMATH DE number 6499146